A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations

KAUST Repository

Bagci, Hakan

2014-11-11

We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off-diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner.

h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems

Science.gov (United States)

Botti, L.; Colombo, A.; Bassi, F.

2017-10-01

In this work we exploit agglomeration based h-multigrid preconditioners to speed-up the iterative solution of discontinuous Galerkin discretizations of the Stokes and Navier-Stokes equations. As a distinctive feature h-coarsened mesh sequences are generated by recursive agglomeration of a fine grid, admitting arbitrarily unstructured grids of complex domains, and agglomeration based discontinuous Galerkin discretizations are employed to deal with agglomerated elements of coarse levels. Both the expense of building coarse grid operators and the performance of the resulting multigrid iteration are investigated. For the sake of efficiency coarse grid operators are inherited through element-by-element L2 projections, avoiding the cost of numerical integration over agglomerated elements. Specific care is devoted to the projection of viscous terms discretized by means of the BR2 dG method. We demonstrate that enforcing the correct amount of stabilization on coarse grids levels is mandatory for achieving uniform convergence with respect to the number of levels. The numerical solution of steady and unsteady, linear and non-linear problems is considered tackling challenging 2D test cases and 3D real life computations on parallel architectures. Significant execution time gains are documented.

Toward robust scalable algebraic multigrid solvers

International Nuclear Information System (INIS)

Waisman, Haim; Schroder, Jacob; Olson, Luke; Hiriyur, Badri; Gaidamour, Jeremie; Siefert, Christopher; Hu, Jonathan Joseph; Tuminaro, Raymond Stephen

2010-01-01

This talk highlights some multigrid challenges that arise from several application areas including structural dynamics, fluid flow, and electromagnetics. A general framework is presented to help introduce and understand algebraic multigrid methods based on energy minimization concepts. Connections between algebraic multigrid prolongators and finite element basis functions are made to explored. It is shown how the general algebraic multigrid framework allows one to adapt multigrid ideas to a number of different situations. Examples are given corresponding to linear elasticity and specifically in the solution of linear systems associated with extended finite elements for fracture problems.

Conjugate gradient coupled with multigrid for an indefinite problem

Science.gov (United States)

Gozani, J.; Nachshon, A.; Turkel, E.

1984-01-01

An iterative algorithm for the Helmholtz equation is presented. This scheme was based on the preconditioned conjugate gradient method for the normal equations. The preconditioning is one cycle of a multigrid method for the discrete Laplacian. The smoothing algorithm is red-black Gauss-Seidel and is constructed so it is a symmetric operator. The total number of iterations needed by the algorithm is independent of h. By varying the number of grids, the number of iterations depends only weakly on k when k(3)h(2) is constant. Comparisons with a SSOR preconditioner are presented.

Multigrid methods for partial differential equations - a short introduction

International Nuclear Information System (INIS)

Linden, J.; Stueben, K.

1993-01-01

These notes summarize the multigrid methods and emphasis is laid on the algorithmic concepts of multigrid for solving linear and non-linear partial differential equations. In this paper there is brief description of the basic structure of multigrid methods. Detailed introduction is also contained with applications to VLSI process simulation. (A.B.)

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

KAUST Repository

Farrell, Patricio

2015-04-30

© 2015John Wiley & Sons, Ltd. Symmetric collocation methods with RBFs allow approximation of the solution of a partial differential equation, even if the right-hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported RBFs and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction. Numerical results verify the effectiveness of the preconditioners.

Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

KAUST Repository

Chavez Chavez, Gustavo Ivan

2017-12-07

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.

A Parallel Algebraic Multigrid Solver on Graphics Processing Units

KAUST Repository

Haase, Gundolf

2010-01-01

The paper presents a multi-GPU implementation of the preconditioned conjugate gradient algorithm with an algebraic multigrid preconditioner (PCG-AMG) for an elliptic model problem on a 3D unstructured grid. An efficient parallel sparse matrix-vector multiplication scheme underlying the PCG-AMG algorithm is presented for the many-core GPU architecture. A performance comparison of the parallel solver shows that a singe Nvidia Tesla C1060 GPU board delivers the performance of a sixteen node Infiniband cluster and a multi-GPU configuration with eight GPUs is about 100 times faster than a typical server CPU core. © 2010 Springer-Verlag.

Multidimensional radiative transfer with multilevel atoms. II. The non-linear multigrid method.

Science.gov (United States)

Fabiani Bendicho, P.; Trujillo Bueno, J.; Auer, L.

1997-08-01

A new iterative method for solving non-LTE multilevel radiative transfer (RT) problems in 1D, 2D or 3D geometries is presented. The scheme obtains the self-consistent solution of the kinetic and RT equations at the cost of only a few (iteration (Brandt, 1977, Math. Comp. 31, 333; Hackbush, 1985, Multi-Grid Methods and Applications, springer-Verlag, Berlin), an efficient multilevel RT scheme based on Gauss-Seidel iterations (cf. Trujillo Bueno & Fabiani Bendicho, 1995ApJ...455..646T), and accurate short-characteristics formal solution techniques. By combining a valid stopping criterion with a nested-grid strategy a converged solution with the desired true error is automatically guaranteed. Contrary to the current operator splitting methods the very high convergence speed of the new RT method does not deteriorate when the grid spatial resolution is increased. With this non-linear multigrid method non-LTE problems discretized on N grid points are solved in O(N) operations. The nested multigrid RT method presented here is, thus, particularly attractive in complicated multilevel transfer problems where small grid-sizes are required. The properties of the method are analyzed both analytically and with illustrative multilevel calculations for Ca II in 1D and 2D schematic model atmospheres.

A survey of parallel multigrid algorithms

Science.gov (United States)

Chan, Tony F.; Tuminaro, Ray S.

1987-01-01

A typical multigrid algorithm applied to well-behaved linear-elliptic partial-differential equations (PDEs) is described. Criteria for designing and evaluating parallel algorithms are presented. Before evaluating the performance of some parallel multigrid algorithms, consideration is given to some theoretical complexity results for solving PDEs in parallel and for executing the multigrid algorithm. The effect of mapping and load imbalance on the partial efficiency of the algorithm is studied.

Incomplete augmented Lagrangian preconditioner for steady incompressible Navier-Stokes equations.

Science.gov (United States)

Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun

2013-01-01

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids.

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

Science.gov (United States)

Molenaar, J.

1996-01-01

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for

Discrete Fourier analysis of multigrid algorithms

NARCIS (Netherlands)

van der Vegt, Jacobus J.W.; Rhebergen, Sander

2011-01-01

The main topic of this report is a detailed discussion of the discrete Fourier multilevel analysis of multigrid algorithms. First, a brief overview of multigrid methods is given for discretizations of both linear and nonlinear partial differential equations. Special attention is given to the

Multigrid methods in structural mechanics

Science.gov (United States)

Raju, I. S.; Bigelow, C. A.; Taasan, S.; Hussaini, M. Y.

1986-01-01

Although the application of multigrid methods to the equations of elasticity has been suggested, few such applications have been reported in the literature. In the present work, multigrid techniques are applied to the finite element analysis of a simply supported Bernoulli-Euler beam, and various aspects of the multigrid algorithm are studied and explained in detail. In this study, six grid levels were used to model half the beam. With linear prolongation and sequential ordering, the multigrid algorithm yielded results which were of machine accuracy with work equivalent to 200 standard Gauss-Seidel iterations on the fine grid. Also with linear prolongation and sequential ordering, the V(1,n) cycle with n greater than 2 yielded better convergence rates than the V(n,1) cycle. The restriction and prolongation operators were derived based on energy principles. Conserving energy during the inter-grid transfers required that the prolongation operator be the transpose of the restriction operator, and led to improved convergence rates. With energy-conserving prolongation and sequential ordering, the multigrid algorithm yielded results of machine accuracy with a work equivalent to 45 Gauss-Seidel iterations on the fine grid. The red-black ordering of relaxations yielded solutions of machine accuracy in a single V(1,1) cycle, which required work equivalent to about 4 iterations on the finest grid level.

Advanced Algebraic Multigrid Solvers for Subsurface Flow Simulation

KAUST Repository

Chen, Meng-Huo

2015-09-13

In this research we are particularly interested in extending the robustness of multigrid solvers to encounter complex systems related to subsurface reservoir applications for flow problems in porous media. In many cases, the step for solving the pressure filed in subsurface flow simulation becomes a bottleneck for the performance of the simulator. For solving large sparse linear system arising from MPFA discretization, we choose multigrid methods as the linear solver. The possible difficulties and issues will be addressed and the corresponding remedies will be studied. As the multigrid methods are used as the linear solver, the simulator can be parallelized (although not trivial) and the high-resolution simulation become feasible, the ultimately goal which we desire to achieve.

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

Science.gov (United States)

Pazner, Will; Persson, Per-Olof

2018-02-01

In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O (p2d) storage and O (p3d) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O (p d + 1) storage, O (p d + 1) work in two spatial dimensions, and O (p d + 2) work in three spatial dimensions. Combined with a matrix-free Newton-Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O (p9) to O (p5) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier-Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p.

An Empirical Analysis of the Performance of Preconditioners for SPD Systems

KAUST Repository

George, Thomas

2012-08-01

Preconditioned iterative solvers have the potential to solve very large sparse linear systems with a fraction of the memory used by direct methods. However, the effectiveness and performance of most preconditioners is not only problem dependent, but also fairly sensitive to the choice of their tunable parameters. As a result, a typical practitioner is faced with an overwhelming number of choices of solvers, preconditioners, and their parameters. The diversity of preconditioners makes it difficult to analyze them in a unified theoretical model. A systematic empirical evaluation of existing preconditioned iterative solvers can help in identifying the relative advantages of various implementations. We present the results of a comprehensive experimental study of the most popular preconditioner and iterative solver combinations for symmetric positive-definite systems. We introduce a methodology for a rigorous comparative evaluation of various preconditioners, including the use of some simple but powerful metrics. The detailed comparison of various preconditioner implementations and a state-of-the-art direct solver gives interesting insights into their relative strengths and weaknesses. We believe that these results would be useful to researchers developing preconditioners and iterative solvers as well as practitioners looking for appropriate sparse solvers for their applications. © 2012 ACM.

Multigrid

CERN Document Server

Trottenberg, Ulrich; Schuller, Anton

2000-01-01

Multigrid presents both an elementary introduction to multigrid methods for solving partial differential equations and a contemporary survey of advanced multigrid techniques and real-life applications.Multigrid methods are invaluable to researchers in scientific disciplines including physics, chemistry, meteorology, fluid and continuum mechanics, geology, biology, and all engineering disciplines. They are also becoming increasingly important in economics and financial mathematics.Readers are presented with an invaluable summary covering 25 years of practical experience acquired by the multigrid research group at the Germany National Research Center for Information Technology. The book presents both practical and theoretical points of view.* Covers the whole field of multigrid methods from its elements up to the most advanced applications* Style is essentially elementary but mathematically rigorous* No other book is so comprehensive and written for both practitioners and students

Extending the applicability of multigrid methods

International Nuclear Information System (INIS)

Brannick, J; Brezina, M; Falgout, R; Manteuffel, T; McCormick, S; Ruge, J; Sheehan, B; Xu, J; Zikatanov, L

2006-01-01

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics

Layout optimization with algebraic multigrid methods

Science.gov (United States)

Regler, Hans; Ruede, Ulrich

1993-01-01

Finding the optimal position for the individual cells (also called functional modules) on the chip surface is an important and difficult step in the design of integrated circuits. This paper deals with the problem of relative placement, that is the minimization of a quadratic functional with a large, sparse, positive definite system matrix. The basic optimization problem must be augmented by constraints to inhibit solutions where cells overlap. Besides classical iterative methods, based on conjugate gradients (CG), we show that algebraic multigrid methods (AMG) provide an interesting alternative. For moderately sized examples with about 10000 cells, AMG is already competitive with CG and is expected to be superior for larger problems. Besides the classical 'multiplicative' AMG algorithm where the levels are visited sequentially, we propose an 'additive' variant of AMG where levels may be treated in parallel and that is suitable as a preconditioner in the CG algorithm.

Algebraic multigrid preconditioning within parallel finite-element solvers for 3-D electromagnetic modelling problems in geophysics

Science.gov (United States)

Koldan, Jelena; Puzyrev, Vladimir; de la Puente, Josep; Houzeaux, Guillaume; Cela, José María

2014-06-01

We present an elaborate preconditioning scheme for Krylov subspace methods which has been developed to improve the performance and reduce the execution time of parallel node-based finite-element (FE) solvers for 3-D electromagnetic (EM) numerical modelling in exploration geophysics. This new preconditioner is based on algebraic multigrid (AMG) that uses different basic relaxation methods, such as Jacobi, symmetric successive over-relaxation (SSOR) and Gauss-Seidel, as smoothers and the wave front algorithm to create groups, which are used for a coarse-level generation. We have implemented and tested this new preconditioner within our parallel nodal FE solver for 3-D forward problems in EM induction geophysics. We have performed series of experiments for several models with different conductivity structures and characteristics to test the performance of our AMG preconditioning technique when combined with biconjugate gradient stabilized method. The results have shown that, the more challenging the problem is in terms of conductivity contrasts, ratio between the sizes of grid elements and/or frequency, the more benefit is obtained by using this preconditioner. Compared to other preconditioning schemes, such as diagonal, SSOR and truncated approximate inverse, the AMG preconditioner greatly improves the convergence of the iterative solver for all tested models. Also, when it comes to cases in which other preconditioners succeed to converge to a desired precision, AMG is able to considerably reduce the total execution time of the forward-problem code-up to an order of magnitude. Furthermore, the tests have confirmed that our AMG scheme ensures grid-independent rate of convergence, as well as improvement in convergence regardless of how big local mesh refinements are. In addition, AMG is designed to be a black-box preconditioner, which makes it easy to use and combine with different iterative methods. Finally, it has proved to be very practical and efficient in the

Fast solution of elliptic partial differential equations using linear combinations of plane waves.

Science.gov (United States)

Pérez-Jordá, José M

2016-02-01

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

A note on the preconditioner Pm=(I+Sm)

Science.gov (United States)

Kohno, Toshiyuki; Niki, Hiroshi

2009-03-01

Kotakemori et al. [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+Smax), Journal of Computational and Applied Mathematics 145 (2002) 373-378] have reported that the convergence rate of the iterative method with a preconditioner Pm=(I+Sm) was superior to one of the modified Gauss-Seidel method under the condition. These authors derived a theorem comparing the Gauss-Seidel method with the proposed method. However, through application of a counter example, Wen Li [Wen Li, A note on the preconditioned GaussSeidel (GS) method for linear systems, Journal of Computational and Applied Mathematics 182 (2005) 81-91] pointed out that there exists a special matrix that does not satisfy this comparison theorem. In this note, we analyze the reason why such a to counter example may be produced, and propose a preconditioner to overcome this problem.

The development of an algebraic multigrid algorithm for symmetric positive definite linear systems

Energy Technology Data Exchange (ETDEWEB)

Vanek, P.; Mandel, J.; Brezina, M. [Univ. of Colorado, Denver, CO (United States)

1996-12-31

An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate blending, and shells.

Approximate Schur complement preconditioning of the lowest order nodal discretizations

Energy Technology Data Exchange (ETDEWEB)

Moulton, J.D.; Ascher, U.M. [Univ. of British Columbia, Vancouver, British Columbia (Canada); Morel, J.E. [Los Alamos National Lab., NM (United States)

1996-12-31

Particular classes of nodal methods and mixed hybrid finite element methods lead to equivalent, robust and accurate discretizations of 2nd order elliptic PDEs. However, widespread popularity of these discretizations has been hindered by the awkward linear systems which result. The present work exploits this awkwardness, which provides a natural partitioning of the linear system, by defining two optimal preconditioners based on approximate Schur complements. Central to the optimal performance of these preconditioners is their sparsity structure which is compatible with Dendy`s black box multigrid code.

A new preconditioner update strategy for the solution of sequences of linear systems in structural mechanics: application to saddle point problems in elasticity

Science.gov (United States)

Mercier, Sylvain; Gratton, Serge; Tardieu, Nicolas; Vasseur, Xavier

2017-12-01

Many applications in structural mechanics require the numerical solution of sequences of linear systems typically issued from a finite element discretization of the governing equations on fine meshes. The method of Lagrange multipliers is often used to take into account mechanical constraints. The resulting matrices then exhibit a saddle point structure and the iterative solution of such preconditioned linear systems is considered as challenging. A popular strategy is then to combine preconditioning and deflation to yield an efficient method. We propose an alternative that is applicable to the general case and not only to matrices with a saddle point structure. In this approach, we consider to update an existing algebraic or application-based preconditioner, using specific available information exploiting the knowledge of an approximate invariant subspace or of matrix-vector products. The resulting preconditioner has the form of a limited memory quasi-Newton matrix and requires a small number of linearly independent vectors. Numerical experiments performed on three large-scale applications in elasticity highlight the relevance of the new approach. We show that the proposed method outperforms the deflation method when considering sequences of linear systems with varying matrices.

Domain decomposition method of stochastic PDEs: a two-level scalable preconditioner

International Nuclear Information System (INIS)

Subber, Waad; Sarkar, Abhijit

2012-01-01

For uncertainty quantification in many practical engineering problems, the stochastic finite element method (SFEM) may be computationally challenging. In SFEM, the size of the algebraic linear system grows rapidly with the spatial mesh resolution and the order of the stochastic dimension. In this paper, we describe a non-overlapping domain decomposition method, namely the iterative substructuring method to tackle the large-scale linear system arising in the SFEM. The SFEM is based on domain decomposition in the geometric space and a polynomial chaos expansion in the probabilistic space. In particular, a two-level scalable preconditioner is proposed for the iterative solver of the interface problem for the stochastic systems. The preconditioner is equipped with a coarse problem which globally connects the subdomains both in the geometric and probabilistic spaces via their corner nodes. This coarse problem propagates the information quickly across the subdomains leading to a scalable preconditioner. For numerical illustrations, a two-dimensional stochastic elliptic partial differential equation (SPDE) with spatially varying non-Gaussian random coefficients is considered. The numerical scalability of the the preconditioner is investigated with respect to the mesh size, subdomain size, fixed problem size per subdomain and order of polynomial chaos expansion. The numerical experiments are performed on a Linux cluster using MPI and PETSc parallel libraries.

Investigations on application of multigrid method to MHD equilibrium analysis

International Nuclear Information System (INIS)

Ikuno, Soichiro

2000-01-01

The potentiality of application for Multi-grid method to MHD equilibrium analysis is investigated. The nonlinear eigenvalue problem often appears when the MHD equilibria are determined by solving the Grad-Shafranov equation numerically. After linearization of the equation, the problem is solved by use of the iterative method. Although the Red-Black SOR method or Gauss-Seidel method is often used for the solution of the linearized equation, it takes much CPU time to solve the problem. The Multi-grid method is compared with the SOR method for the Poisson Problem. The results of computations show that the CPU time required for the Multi-grid method is about 1000 times as small as that for the SOR method. (author)

Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics

Science.gov (United States)

Gilles, Luc; Ellerbroek, Brent L.; Vogel, Curtis R.

2003-02-01

Multi-conjugate adaptive optics (MCAO) systems with 104-105 degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of AO degrees of freedom. In this paper, we develop an iterative sparse matrix implementation of minimum variance wavefront reconstruction for telescope diameters up to 32m with more than 104 actuators. The basic approach is the preconditioned conjugate gradient method, using a multigrid preconditioner incorporating a layer-oriented (block) symmetric Gauss-Seidel iterative smoothing operator. We present open-loop numerical simulation results to illustrate algorithm convergence.

Improved Solver Settings for 3D Exploding Wire Simulations in ALEGRA

Science.gov (United States)

2016-08-01

performance and rate of throughput of ALEGRA-MHD simulations. The algebraic multigrid algorithms in ML approximate the solution of a linear sys- tem on a fine...in the algebraic multigrid preconditioner, Trilinos/ML, as implemented in ALEGRA. Three parameters impacted performance with one dominating. We provide...option (RM << 1) is not quite applicable and not used in this study. ALEGRA uses explicit time integration to solve the solid-dynamic and nonresistive

Design Considerations for a Flexible Multigrid Preconditioning Library

Directory of Open Access Journals (Sweden)

Jérémie Gaidamour

2012-01-01

Full Text Available MueLu is a library within the Trilinos software project [An overview of Trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003] and provides a framework for parallel multigrid preconditioning methods for large sparse linear systems. While providing efficient implementations of modern multigrid methods based on smoothed aggregation and energy minimization concepts, MueLu is designed to be customized and extended. This article gives an overview of design considerations for the MueLu package: user interfaces, internal design, data management, usage of modern software constructs, leveraging Trilinos capabilities, linear algebra operations and advanced application.

The Importance of Structure in Incomplete Factorization Preconditioners

Czech Academy of Sciences Publication Activity Database

Scott, J.; Tůma, Miroslav

2011-01-01

Roč. 51, č. 2 (2011), s. 385-404 ISSN 0006-3835 Grant - others:GA AV ČR(CZ) M100300902 Institutional research plan: CEZ:AV0Z10300504 Keywords : sparse symmetric linear systems * incomplete factorizations * preconditioners * level-based approach Subject RIV: BA - General Mathematics Impact factor: 0.724, year: 2011

The Mixed Finite Element Multigrid Method for Stokes Equations

Science.gov (United States)

Muzhinji, K.; Shateyi, S.; Motsa, S. S.

2015-01-01

The stable finite element discretization of the Stokes problem produces a symmetric indefinite system of linear algebraic equations. A variety of iterative solvers have been proposed for such systems in an attempt to construct efficient, fast, and robust solution techniques. This paper investigates one of such iterative solvers, the geometric multigrid solver, to find the approximate solution of the indefinite systems. The main ingredient of the multigrid method is the choice of an appropriate smoothing strategy. This study considers the application of different smoothers and compares their effects in the overall performance of the multigrid solver. We study the multigrid method with the following smoothers: distributed Gauss Seidel, inexact Uzawa, preconditioned MINRES, and Braess-Sarazin type smoothers. A comparative study of the smoothers shows that the Braess-Sarazin smoothers enhance good performance of the multigrid method. We study the problem in a two-dimensional domain using stable Hood-Taylor Q 2-Q 1 pair of finite rectangular elements. We also give the main theoretical convergence results. We present the numerical results to demonstrate the efficiency and robustness of the multigrid method and confirm the theoretical results. PMID:25945361

A multigrid method for variational inequalities

Energy Technology Data Exchange (ETDEWEB)

Oliveira, S.; Stewart, D.E.; Wu, W.

1996-12-31

Multigrid methods have been used with great success for solving elliptic partial differential equations. Penalty methods have been successful in solving finite-dimensional quadratic programs. In this paper these two techniques are combined to give a fast method for solving obstacle problems. A nonlinear penalized problem is solved using Newton`s method for large values of a penalty parameter. Multigrid methods are used to solve the linear systems in Newton`s method. The overall numerical method developed is based on an exterior penalty function, and numerical results showing the performance of the method have been obtained.

Multi-grid Particle-in-cell Simulations of Plasma Microturbulence

International Nuclear Information System (INIS)

Lewandowski, J.L.V.

2003-01-01

A new scheme to accurately retain kinetic electron effects in particle-in-cell (PIC) simulations for the case of electrostatic drift waves is presented. The splitting scheme, which is based on exact separation between adiabatic and on adiabatic electron responses, is shown to yield more accurate linear growth rates than the standard df scheme. The linear and nonlinear elliptic problems that arise in the splitting scheme are solved using a multi-grid solver. The multi-grid particle-in-cell approach offers an attractive path, both from the physics and numerical points of view, to simulate kinetic electron dynamics in global toroidal plasmas

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Science.gov (United States)

Badia, Santiago; Martín, Alberto F.; Planas, Ramon

2014-10-01

The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 × 2 block matrix, and consider an LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknowns, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

Multigrid methods III

CERN Document Server

Trottenberg, U; Third European Conference on Multigrid Methods

1991-01-01

These proceedings contain a selection of papers presented at the Third European Conference on Multigrid Methods which was held in Bonn on October 1-4, 1990. Following conferences in 1981 and 1985, a platform for the presentation of new Multigrid results was provided for a third time. Multigrid methods no longer have problems being accepted by numerical analysts and users of numerical methods; on the contrary, they have been further developed in such a successful way that they have penetrated a variety of new fields of application. The high number of 154 participants from 18 countries and 76 presented papers show the need to continue the series of the European Multigrid Conferences. The papers of this volume give a survey on the current Multigrid situation; in particular, they correspond to those fields where new developments can be observed. For example, se­ veral papers study the appropriate treatment of time dependent problems. Improvements can also be noticed in the Multigrid approach for semiconductor eq...

Multigrid and multilevel domain decomposition for unstructured grids

Energy Technology Data Exchange (ETDEWEB)

Chan, T.; Smith, B.

1994-12-31

Multigrid has proven itself to be a very versatile method for the iterative solution of linear and nonlinear systems of equations arising from the discretization of PDES. In some applications, however, no natural multilevel structure of grids is available, and these must be generated as part of the solution procedure. In this presentation the authors will consider the problem of generating a multigrid algorithm when only a fine, unstructured grid is given. Their techniques generate a sequence of coarser grids by first forming an approximate maximal independent set of the vertices and then applying a Cavendish type algorithm to form the coarser triangulation. Numerical tests indicate that convergence using this approach can be as fast as standard multigrid on a structured mesh, at least in two dimensions.

Large-Scale Parallel Viscous Flow Computations using an Unstructured Multigrid Algorithm

Science.gov (United States)

Mavriplis, Dimitri J.

1999-01-01

The development and testing of a parallel unstructured agglomeration multigrid algorithm for steady-state aerodynamic flows is discussed. The agglomeration multigrid strategy uses a graph algorithm to construct the coarse multigrid levels from the given fine grid, similar to an algebraic multigrid approach, but operates directly on the non-linear system using the FAS (Full Approximation Scheme) approach. The scalability and convergence rate of the multigrid algorithm are examined on the SGI Origin 2000 and the Cray T3E. An argument is given which indicates that the asymptotic scalability of the multigrid algorithm should be similar to that of its underlying single grid smoothing scheme. For medium size problems involving several million grid points, near perfect scalability is obtained for the single grid algorithm, while only a slight drop-off in parallel efficiency is observed for the multigrid V- and W-cycles, using up to 128 processors on the SGI Origin 2000, and up to 512 processors on the Cray T3E. For a large problem using 25 million grid points, good scalability is observed for the multigrid algorithm using up to 1450 processors on a Cray T3E, even when the coarsest grid level contains fewer points than the total number of processors.

Parallel linear solvers for simulations of reactor thermal hydraulics

International Nuclear Information System (INIS)

Yan, Y.; Antal, S.P.; Edge, B.; Keyes, D.E.; Shaver, D.; Bolotnov, I.A.; Podowski, M.Z.

2011-01-01

The state-of-the-art multiphase fluid dynamics code, NPHASE-CMFD, performs multiphase flow simulations in complex domains using implicit nonlinear treatment of the governing equations and in parallel, which is a very challenging environment for the linear solver. The present work illustrates how the Portable, Extensible Toolkit for Scientific Computation (PETSc) and scalable Algebraic Multigrid (AMG) preconditioner from Hypre can be utilized to construct robust and scalable linear solvers for the Newton correction equation obtained from the discretized system of governing conservation equations in NPHASE-CMFD. The overall long-tem objective of this work is to extend the NPHASE-CMFD code into a fully-scalable solver of multiphase flow and heat transfer problems, applicable to both steady-state and stiff time-dependent phenomena in complete fuel assemblies of nuclear reactors and, eventually, the entire reactor core (such as the Virtual Reactor concept envisioned by CASL). This campaign appropriately begins with the linear algebraic equation solver, which is traditionally a bottleneck to scalability in PDE-based codes. The computational complexity of the solver is usually superlinear in problem size, whereas the rest of the code, the “physics” portion, usually has its complexity linear in the problem size. (author)

The multigrid method for reactor calculations

International Nuclear Information System (INIS)

Douglas, S.R.

1991-07-01

Iterative solutions to linear systems of equations are discussed. The emphasis is on the concepts that affect convergence rates of these solution methods. The multigrid method is described, including the smoothing property, restriction, and prolongation. A simple example is used to illustrate the ideas

Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion

International Nuclear Information System (INIS)

Mousseau, V.A.; Knoll, D.A.; Rider, W.J.

2000-01-01

An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton-Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, the authors employ a Jacobian-free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2 x 2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements

A multigrid solution method for mixed hybrid finite elements

Energy Technology Data Exchange (ETDEWEB)

Schmid, W. [Universitaet Augsburg (Germany)

1996-12-31

We consider the multigrid solution of linear equations arising within the discretization of elliptic second order boundary value problems of the form by mixed hybrid finite elements. Using the equivalence of mixed hybrid finite elements and non-conforming nodal finite elements, we construct a multigrid scheme for the corresponding non-conforming finite elements, and, by this equivalence, for the mixed hybrid finite elements, following guidelines from Arbogast/Chen. For a rectangular triangulation of the computational domain, this non-conforming schemes are the so-called nodal finite elements. We explicitly construct prolongation and restriction operators for this type of non-conforming finite elements. We discuss the use of plain multigrid and the multilevel-preconditioned cg-method and compare their efficiency in numerical tests.

Multigrid Reduction in Time for Nonlinear Parabolic Problems

Energy Technology Data Exchange (ETDEWEB)

Falgout, R. D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Manteuffel, T. A. [Univ. of Colorado, Boulder, CO (United States); O' Neill, B. [Univ. of Colorado, Boulder, CO (United States); Schroder, J. B. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2016-01-04

The need for parallel-in-time is being driven by changes in computer architectures, where future speed-ups will be available through greater concurrency, but not faster clock speeds, which are stagnant.This leads to a bottleneck for sequential time marching schemes, because they lack parallelism in the time dimension. Multigrid Reduction in Time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficient is defined as achieving similar performance when compared to a corresponding linear problem. As our benchmark, we use the p-Laplacian, where p = 4 corresponds to a well-known nonlinear diffusion equation and p = 2 corresponds to our benchmark linear diffusion problem. When considering linear problems and implicit methods, the use of optimal spatial solvers such as spatial multigrid imply that the cost of one time step evaluation is fixed across temporal levels, which have a large variation in time step sizes. This is not the case for nonlinear problems, where the work required increases dramatically on coarser time grids, where relatively large time steps lead to worse conditioned nonlinear solves and increased nonlinear iteration counts per time step evaluation. This is the key difficulty explored by this paper. We show that by using a variety of strategies, most importantly, spatial coarsening and an alternate initial guess to the nonlinear time-step solver, we can reduce the work per time step evaluation over all temporal levels to a range similar with the corresponding linear problem. This allows for parallel scaling behavior comparable to the corresponding linear problem.

Advanced Algebraic Multigrid Solvers for Subsurface Flow Simulation

KAUST Repository

Chen, Meng-Huo; Sun, Shuyu; Salama, Amgad

2015-01-01

and issues will be addressed and the corresponding remedies will be studied. As the multigrid methods are used as the linear solver, the simulator can be parallelized (although not trivial) and the high-resolution simulation become feasible, the ultimately

Block-triangular preconditioners for PDE-constrained optimization

KAUST Repository

Rees, Tyrone

2010-11-26

In this paper we investigate the possibility of using a block-triangular preconditioner for saddle point problems arising in PDE-constrained optimization. In particular, we focus on a conjugate gradient-type method introduced by Bramble and Pasciak that uses self-adjointness of the preconditioned system in a non-standard inner product. We show when the Chebyshev semi-iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble-Pasciak method-the appropriate scaling of the preconditioners-is easily overcome. We present an eigenvalue analysis for the block-triangular preconditioners that gives convergence bounds in the non-standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.

Block-triangular preconditioners for PDE-constrained optimization

KAUST Repository

Rees, Tyrone; Stoll, Martin

2010-01-01

In this paper we investigate the possibility of using a block-triangular preconditioner for saddle point problems arising in PDE-constrained optimization. In particular, we focus on a conjugate gradient-type method introduced by Bramble and Pasciak that uses self-adjointness of the preconditioned system in a non-standard inner product. We show when the Chebyshev semi-iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble-Pasciak method-the appropriate scaling of the preconditioners-is easily overcome. We present an eigenvalue analysis for the block-triangular preconditioners that gives convergence bounds in the non-standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.

Final report on the Copper Mountain conference on multigrid methods

Energy Technology Data Exchange (ETDEWEB)

NONE

1997-10-01

The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners.

A multigrid Newton-Krylov method for flux-limited radiation diffusion

International Nuclear Information System (INIS)

Rider, W.J.; Knoll, D.A.; Olson, G.L.

1998-01-01

The authors focus on the integration of radiation diffusion including flux-limited diffusion coefficients. The nonlinear integration is accomplished with a Newton-Krylov method preconditioned with a multigrid Picard linearization of the governing equations. They investigate the efficiency of the linear and nonlinear iterative techniques

Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

KAUST Repository

Chavez Chavez, Gustavo Ivan; Turkiyyah, George; Zampini, Stefano; Keyes, David E.

2017-01-01

and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a

Development of Multigrid Methods for diffusion, Advection, and the incompressible Navier-Stokes Equations

Energy Technology Data Exchange (ETDEWEB)

Gjesdal, Thor

1997-12-31

This thesis discusses the development and application of efficient numerical methods for the simulation of fluid flows, in particular the flow of incompressible fluids. The emphasis is on practical aspects of algorithm development and on application of the methods either to linear scalar model equations or to the non-linear incompressible Navier-Stokes equations. The first part deals with cell centred multigrid methods and linear correction scheme and presents papers on (1) generalization of the method to arbitrary sized grids for diffusion problems, (2) low order method for advection-diffusion problems, (3) attempt to extend the basic method to advection-diffusion problems, (4) Fourier smoothing analysis of multicolour relaxation schemes, and (5) analysis of high-order discretizations for advection terms. The second part discusses a multigrid based on pressure correction methods, non-linear full approximation scheme, and papers on (1) systematic comparison of the performance of different pressure correction smoothers and some other algorithmic variants, low to moderate Reynolds numbers, and (2) systematic study of implementation strategies for high order advection schemes, high-Re flow. An appendix contains Fortran 90 data structures for multigrid development. 160 refs., 26 figs., 22 tabs.

A parallel sweeping preconditioner for frequency-domain seismic wave propagation

KAUST Repository

Poulson, Jack

2012-09-01

We present a parallel implementation of Engquist and Ying\\'s sweeping preconditioner, which exploits radiation boundary conditions in order to form an approximate block LDLT factorization of the Helmholtz operator with only O(N4/3) work and an application (and memory) cost of only O(N logN). The approximate factorization is then used as a preconditioner for GMRES, and we show that essentially O(1) iterations are required for convergence, even for the full SEG/EAGE over-thrust model at 30 Hz. In particular, we demonstrate the solution of said problem in a mere 15 minutes on 8192 cores of TACC\\'s Lonestar, which may be the largest-scale 3D heterogeneous Helmholtz calculation to date. Generalizations of our parallel strategy are also briefly discussed for time-harmonic linear elasticity and Maxwell\\'s equations.

Application of polynomial preconditioners to conservation laws

NARCIS (Netherlands)

Geurts, Bernardus J.; van Buuren, R.; Lu, H.

2000-01-01

Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of

A multigrid algorithm for the cell-centered finite difference scheme

Science.gov (United States)

Ewing, Richard E.; Shen, Jian

1993-01-01

In this article, we discuss a non-variational V-cycle multigrid algorithm based on the cell-centered finite difference scheme for solving a second-order elliptic problem with discontinuous coefficients. Due to the poor approximation property of piecewise constant spaces and the non-variational nature of our scheme, one step of symmetric linear smoothing in our V-cycle multigrid scheme may fail to be a contraction. Again, because of the simple structure of the piecewise constant spaces, prolongation and restriction are trivial; we save significant computation time with very promising computational results.

A comparison of deflation and the balancing Neumann-Neumann preconditioner

NARCIS (Netherlands)

Nabben, R.; Vuik, C.

2004-01-01

In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing Neumann Neumann preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition

On a multigrid method for the coupled Stokes and porous media flow problem

Science.gov (United States)

Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W.

2017-07-01

The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss-Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, Local Fourier Analysis (LFA) is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs well for small values of the hydraulic conductivity and fluid viscosity, that are relevant for applications.

Scalable and Robust BDDC Preconditioners for Reservoir and Electromagnetics Modeling

KAUST Repository

Zampini, S.; Widlund, O.B.; Keyes, David E.

2015-01-01

The purpose of the study is to show the effectiveness of recent algorithmic advances in Balancing Domain Decomposition by Constraints (BDDC) preconditioners for the solution of elliptic PDEs with highly heterogeneous coefficients, and discretized by means of the finite element method. Applications to large linear systems generated by div- and curl- conforming finite elements discretizations commonly arising in the contexts of modelling reservoirs and electromagnetics will be presented.

Scalable and Robust BDDC Preconditioners for Reservoir and Electromagnetics Modeling

KAUST Repository

Zampini, S.

2015-09-13

The purpose of the study is to show the effectiveness of recent algorithmic advances in Balancing Domain Decomposition by Constraints (BDDC) preconditioners for the solution of elliptic PDEs with highly heterogeneous coefficients, and discretized by means of the finite element method. Applications to large linear systems generated by div- and curl- conforming finite elements discretizations commonly arising in the contexts of modelling reservoirs and electromagnetics will be presented.

Element-topology-independent preconditioners for parallel finite element computations

Science.gov (United States)

Park, K. C.; Alexander, Scott

1992-01-01

A family of preconditioners for the solution of finite element equations are presented, which are element-topology independent and thus can be applicable to element order-free parallel computations. A key feature of the present preconditioners is the repeated use of element connectivity matrices and their left and right inverses. The properties and performance of the present preconditioners are demonstrated via beam and two-dimensional finite element matrices for implicit time integration computations.

Preconditioners based on the Alternating-Direction-Implicit algorithm for the 2D steady-state diffusion equation with orthotropic heterogeneous coefficients

KAUST Repository

Gao, Longfei; Calo, Victor M.

2015-01-01

In this paper, we combine the Alternating Direction Implicit (ADI) algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2D steady-state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions. Specifically, we adopt the compound iteration idea and use ADI iterations as the preconditioner for the outside Krylov subspace method that is used to solve the preconditioned linear system. An efficient algorithm to perform each ADI iteration is crucial to the efficiency of the overall iterative scheme. We exploit the Kronecker product structure in the matrices, inherited from the tensor product basis functions, to achieve high efficiency in each ADI iteration. Meanwhile, in order to reduce the number of Krylov subspace iterations, we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions. Numerical results demonstrated the efficiency and quality of the proposed preconditioner. © 2014 Elsevier B.V. All rights reserved.

Finite element analysis of multi-material models using a balancing domain decomposition method combined with the diagonal scaling preconditioner

International Nuclear Information System (INIS)

Ogino, Masao

2016-01-01

Actual problems in science and industrial applications are modeled by multi-materials and large-scale unstructured mesh, and the finite element analysis has been widely used to solve such problems on the parallel computer. However, for large-scale problems, the iterative methods for linear finite element equations suffer from slow or no convergence. Therefore, numerical methods having both robust convergence and scalable parallel efficiency are in great demand. The domain decomposition method is well known as an iterative substructuring method, and is an efficient approach for parallel finite element methods. Moreover, the balancing preconditioner achieves robust convergence. However, in case of problems consisting of very different materials, the convergence becomes bad. There are some research to solve this issue, however not suitable for cases of complex shape and composite materials. In this study, to improve convergence of the balancing preconditioner for multi-materials, a balancing preconditioner combined with the diagonal scaling preconditioner, called Scaled-BDD method, is proposed. Some numerical results are included which indicate that the proposed method has robust convergence for the number of subdomains and shows high performances compared with the original balancing preconditioner. (author)

Analysis of a parallel multigrid algorithm

Science.gov (United States)

Chan, Tony F.; Tuminaro, Ray S.

1989-01-01

The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made.

Fast wavelet based sparse approximate inverse preconditioner

Energy Technology Data Exchange (ETDEWEB)

Wan, W.L. [Univ. of California, Los Angeles, CA (United States)

1996-12-31

Incomplete LU factorization is a robust preconditioner for both general and PDE problems but unfortunately not easy to parallelize. Recent study of Huckle and Grote and Chow and Saad showed that sparse approximate inverse could be a potential alternative while readily parallelizable. However, for special class of matrix A that comes from elliptic PDE problems, their preconditioners are not optimal in the sense that independent of mesh size. A reason may be that no good sparse approximate inverse exists for the dense inverse matrix. Our observation is that for this kind of matrices, its inverse entries typically have piecewise smooth changes. We can take advantage of this fact and use wavelet compression techniques to construct a better sparse approximate inverse preconditioner. We shall show numerically that our approach is effective for this kind of matrices.

Summary Report: Multigrid for Systems of Elliptic PDEs

Energy Technology Data Exchange (ETDEWEB)

Lee, Barry [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2016-11-17

We are interested in determining if multigrid can be effectively applied to the system. The conclusion that I seem to be drawn to is that it is impossible to develop a blackbox multigrid solver for these general systems. Analysis of the system of PDEs must be conducted first to determine pre-processing procedures on the continuous problem before applying a multigrid method. Determining this pre-processing is currently not incorporated in black-box multigrid strategies. Nevertheless, we characterize some system features that will make the system more amenable to multigrid approaches, techniques that may lead to more amenable systems, and multigrid procedures that are generally more appropriate for these systems.

Accelerated Cyclic Reduction: A Distributed-Memory Fast Solver for Structured Linear Systems

KAUST Repository

Chávez, Gustavo

2017-12-15

We present Accelerated Cyclic Reduction (ACR), a distributed-memory fast solver for rank-compressible block tridiagonal linear systems arising from the discretization of elliptic operators, developed here for three dimensions. Algorithmic synergies between Cyclic Reduction and hierarchical matrix arithmetic operations result in a solver that has O(kNlogN(logN+k2)) arithmetic complexity and O(k Nlog N) memory footprint, where N is the number of degrees of freedom and k is the rank of a block in the hierarchical approximation, and which exhibits substantial concurrency. We provide a baseline for performance and applicability by comparing with the multifrontal method with and without hierarchical semi-separable matrices, with algebraic multigrid and with the classic cyclic reduction method. Over a set of large-scale elliptic systems with features of nonsymmetry and indefiniteness, the robustness of the direct solvers extends beyond that of the multigrid solver, and relative to the multifrontal approach ACR has lower or comparable execution time and size of the factors, with substantially lower numerical ranks. ACR exhibits good strong and weak scaling in a distributed context and, as with any direct solver, is advantageous for problems that require the solution of multiple right-hand sides. Numerical experiments show that the rank k patterns are of O(1) for the Poisson equation and of O(n) for the indefinite Helmholtz equation. The solver is ideal in situations where low-accuracy solutions are sufficient, or otherwise as a preconditioner within an iterative method.

Accelerated Cyclic Reduction: A Distributed-Memory Fast Solver for Structured Linear Systems

KAUST Repository

Chá vez, Gustavo; Turkiyyah, George; Zampini, Stefano; Ltaief, Hatem; Keyes, David E.

2017-01-01

We present Accelerated Cyclic Reduction (ACR), a distributed-memory fast solver for rank-compressible block tridiagonal linear systems arising from the discretization of elliptic operators, developed here for three dimensions. Algorithmic synergies between Cyclic Reduction and hierarchical matrix arithmetic operations result in a solver that has O(kNlogN(logN+k2)) arithmetic complexity and O(k Nlog N) memory footprint, where N is the number of degrees of freedom and k is the rank of a block in the hierarchical approximation, and which exhibits substantial concurrency. We provide a baseline for performance and applicability by comparing with the multifrontal method with and without hierarchical semi-separable matrices, with algebraic multigrid and with the classic cyclic reduction method. Over a set of large-scale elliptic systems with features of nonsymmetry and indefiniteness, the robustness of the direct solvers extends beyond that of the multigrid solver, and relative to the multifrontal approach ACR has lower or comparable execution time and size of the factors, with substantially lower numerical ranks. ACR exhibits good strong and weak scaling in a distributed context and, as with any direct solver, is advantageous for problems that require the solution of multiple right-hand sides. Numerical experiments show that the rank k patterns are of O(1) for the Poisson equation and of O(n) for the indefinite Helmholtz equation. The solver is ideal in situations where low-accuracy solutions are sufficient, or otherwise as a preconditioner within an iterative method.

Semi-coarsening multigrid methods for parallel computing

Energy Technology Data Exchange (ETDEWEB)

Jones, J.E.

1996-12-31

Standard multigrid methods are not well suited for problems with anisotropic coefficients which can occur, for example, on grids that are stretched to resolve a boundary layer. There are several different modifications of the standard multigrid algorithm that yield efficient methods for anisotropic problems. In the paper, we investigate the parallel performance of these multigrid algorithms. Multigrid algorithms which work well for anisotropic problems are based on line relaxation and/or semi-coarsening. In semi-coarsening multigrid algorithms a grid is coarsened in only one of the coordinate directions unlike standard or full-coarsening multigrid algorithms where a grid is coarsened in each of the coordinate directions. When both semi-coarsening and line relaxation are used, the resulting multigrid algorithm is robust and automatic in that it requires no knowledge of the nature of the anisotropy. This is the basic multigrid algorithm whose parallel performance we investigate in the paper. The algorithm is currently being implemented on an IBM SP2 and its performance is being analyzed. In addition to looking at the parallel performance of the basic semi-coarsening algorithm, we present algorithmic modifications with potentially better parallel efficiency. One modification reduces the amount of computational work done in relaxation at the expense of using multiple coarse grids. This modification is also being implemented with the aim of comparing its performance to that of the basic semi-coarsening algorithm.

Acceleration of Multidimensional Discrete Ordinates Methods Via Adjacent-Cell Preconditioners

International Nuclear Information System (INIS)

Azmy, Y.Y.

2000-01-01

The adjacent-cell preconditioner (AP) formalism originally derived in slab geometry is extended to multidimensional Cartesian geometry for generic fixed-weight, weighted diamond difference neutron transport methods. This is accomplished for the thick-cell regime (KAP) and thin-cell regime (NAP). A spectral analysis of the resulting acceleration schemes demonstrates their excellent spectral properties for model problem configurations, characterized by a uniform mesh of infinite extent and homogeneous material composition, each in its own cell-size regime. Thus, the spectral radius of KAP vanishes as the computational cell size approaches infinity, but it exceeds unity for very thin cells, thereby implying instability. In contrast, NAP is stable and robust for all cell sizes, but its spectral radius vanishes more slowly as the cell size increases. For this reason, and to avoid potential complication in the case of cells that are thin in one dimension and thick in another, NAP is adopted in the remainder of this work. The most important feature of AP for practical implementation in production level codes is that it is cell centered, reducing the size of the algebraic system comprising the acceleration stage compared to face-centered schemes. Boundary conditions for finite extent problems and a mixing formula across material and cell-size discontinuity are derived and used to implement NAP in a test code, AHOT, and a production code, TORT. Numerical testing for algebraically linear iterative schemes for the cases embodied in Burre's Suite of Test Problems demonstrates the high efficiency of the new method in reducing the number of iterations required to achieve convergence, especially for optically thick cells where acceleration is most needed. Also, for algebraically nonlinear (adaptive) methods, AP generally performs better than the partial current rebalance method in TORT and the diffusion synthetic acceleration method in TWODANT. Finally, application of the AP

Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

Directory of Open Access Journals (Sweden)

L. Bergamaschi

2013-01-01

the linearized Newton system to solve Au=q(uu, q(u being the Rayleigh quotient. In a previous work (Bergamaschi and Martínez, 2013 the sequence of preconditioned Jacobians is proven to remain close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to 1.5 million unknowns account for the efficiency and the scalability of the proposed low rank update of the RFSAI preconditioner. The overall RFSAI-BFGS preconditioned Newton algorithm has shown comparable efficiencies with a well-established eigenvalue solver on all the test problems.

Object-Oriented Software Tools for the Construction of Preconditioners

Directory of Open Access Journals (Sweden)

Eva Mossberg

1997-01-01

Full Text Available In recent years, there has been considerable progress concerning preconditioned iterative methods for large and sparse systems of equations arising from the discretization of differential equations. Such methods are particularly attractive in the context of high-performance (parallel computers. However, the implementation of a preconditioner is a nontrivial task. The focus of the present contribution is on a set of object-oriented software tools that support the construction of a family of preconditioners based on fast transforms. By combining objects of different classes, it is possible to conveniently construct any preconditioner within this family.

A universal preconditioner for simulating condensed phase materials

Energy Technology Data Exchange (ETDEWEB)

Packwood, David; Ortner, Christoph, E-mail: c.ortner@warwick.ac.uk [Mathematics Institute, University of Warwick, Coventry CV4 7AL (United Kingdom); Kermode, James, E-mail: j.r.kermode@warwick.ac.uk [Warwick Centre for Predictive Modelling, School of Engineering, University of Warwick, Coventry CV4 7AL (United Kingdom); Mones, Letif [Mathematics Institute, University of Warwick, Coventry CV4 7AL (United Kingdom); Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ (United Kingdom); Bernstein, Noam [Center for Materials Physics and Technology, Naval Research Laboratory, Washington, DC 20375 (United States); Woolley, John [Department of Physics, University of Warwick, Coventry CV4 7AL (United Kingdom); Gould, Nicholas [Scientific Computing Department, STFC-Rutherford Appleton Laboratory Chilton, Oxfordshire OX11 0QX (United Kingdom); Csányi, Gábor, E-mail: gc121@cam.ac.uk [Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ (United Kingdom)

2016-04-28

We introduce a universal sparse preconditioner that accelerates geometry optimisation and saddle point search tasks that are common in the atomic scale simulation of materials. Our preconditioner is based on the neighbourhood structure and we demonstrate the gain in computational efficiency in a wide range of materials that include metals, insulators, and molecular solids. The simple structure of the preconditioner means that the gains can be realised in practice not only when using expensive electronic structure models but also for fast empirical potentials. Even for relatively small systems of a few hundred atoms, we observe speedups of a factor of two or more, and the gain grows with system size. An open source Python implementation within the Atomic Simulation Environment is available, offering interfaces to a wide range of atomistic codes.

A calderón multiplicative preconditioner for the combined field integral equation

KAUST Repository

Bagci, Hakan

2009-10-01

A Calderón multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Caldern-preconditioned CFIEs, a localization procedure is employed to ensure that the equation is resonance-free. The iterative solution of the linear system of equations obtained via the CMP-based discretization of the CFIE converges rapidly regardless of the discretization density and the frequency of excitation. © 2009 IEEE.

Two new modified Gauss-Seidel methods for linear system with M-matrices

Science.gov (United States)

Zheng, Bing; Miao, Shu-Xin

2009-12-01

In 2002, H. Kotakemori et al. proposed the modified Gauss-Seidel (MGS) method for solving the linear system with the preconditioner [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner () J. Comput. Appl. Math. 145 (2002) 373-378]. Since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row. In the present paper, to deal with this drawback, we propose two new preconditioners. The convergence and comparison theorems of the modified Gauss-Seidel methods with these two preconditioners for solving the linear system are established. The convergence rates of the new proposed preconditioned methods are compared. In addition, numerical experiments are used to show the effectiveness of the new MGS methods.

Multigrid Algorithms for the Solution of Linear Complementarity Problems Arising from Free Boundary Problems.

Science.gov (United States)

1980-10-01

solving (1.3); PFAS combines the concepts of multigrid algorithms with those of projected SOR. In Section 3, we discuss the implementation of PFAS, and...numerique de la torsion elasto- plastique d’une barre cylindrique. In Approximation et Methodes Iteratives de Resolution d’Inequations Variationelles et

Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretisation

NARCIS (Netherlands)

P.W. Hemker (Piet); W. Hoffmann; M.H. van Raalte (Marc)

2002-01-01

textabstractIn this paper we study a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, andwe give a detailed analysis of the convergence for different block-relaxation strategies.We find that point-wise

Triangular preconditioners for saddle point problems with a penalty term

Energy Technology Data Exchange (ETDEWEB)

Klawonn, A. [Westfaelische Wilhelms-Universitaet, Muenster (Germany)

1996-12-31

Triangular preconditioners for a class of saddle point problems with a penalty term are considered. An important example is the mixed formulation of the pure displacement problem in linear elasticity. It is shown that the spectrum of the preconditioned system is contained in a real, positive interval, and that the interval bounds can be made independent of the discretization and penalty parameters. This fact is used to construct bounds of the convergence rate of the GMRES method used with an energy norm. Numerical results are given for GMRES and BI-CGSTAB.

An Empirical Analysis of the Performance of Preconditioners for SPD Systems

KAUST Repository

George, Thomas; Gupta, Anshul; Sarin, Vivek

2012-01-01

study of the most popular preconditioner and iterative solver combinations for symmetric positive-definite systems. We introduce a methodology for a rigorous comparative evaluation of various preconditioners, including the use of some simple but powerful

Elliptic Preconditioner for Accelerating the Self-Consistent Field Iteration in Kohn--Sham Density Functional Theory

Energy Technology Data Exchange (ETDEWEB)

Lin, Lin [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Yang, Chao [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division

2013-10-28

We discuss techniques for accelerating the self consistent field (SCF) iteration for solving the Kohn-Sham equations. These techniques are all based on constructing approximations to the inverse of the Jacobian associated with a fixed point map satisfied by the total potential. They can be viewed as preconditioners for a fixed point iteration. We point out different requirements for constructing preconditioners for insulating and metallic systems respectively, and discuss how to construct preconditioners to keep the convergence rate of the fixed point iteration independent of the size of the atomistic system. We propose a new preconditioner that can treat insulating and metallic system in a unified way. The new preconditioner, which we call an elliptic preconditioner, is constructed by solving an elliptic partial differential equation. The elliptic preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous systems at low temperature.

Analysis of preconditioning and multigrid for Euler flows with low-subsonic regions

NARCIS (Netherlands)

Koren, B.; Leer, van B.

1995-01-01

For subsonic flows and upwind-discretized, linearized 1-D Euler equations, the smoothing behavior of multigrid-accelerated point Gauss-Seidel relaxation is analyzed. Error decay by convection across domain boundaries is also discussed. A fix to poor convergence rates at low Mach numbers is sought in

Self-correcting Multigrid Solver

International Nuclear Information System (INIS)

Lewandowski, Jerome L.V.

2004-01-01

A new multigrid algorithm based on the method of self-correction for the solution of elliptic problems is described. The method exploits information contained in the residual to dynamically modify the source term (right-hand side) of the elliptic problem. It is shown that the self-correcting solver is more efficient at damping the short wavelength modes of the algebraic error than its standard equivalent. When used in conjunction with a multigrid method, the resulting solver displays an improved convergence rate with no additional computational work

New Multigrid Method Including Elimination Algolithm Based on High-Order Vector Finite Elements in Three Dimensional Magnetostatic Field Analysis

Science.gov (United States)

Hano, Mitsuo; Hotta, Masashi

A new multigrid method based on high-order vector finite elements is proposed in this paper. Low level discretizations in this method are obtained by using low-order vector finite elements for the same mesh. Gauss-Seidel method is used as a smoother, and a linear equation of lowest level is solved by ICCG method. But it is often found that multigrid solutions do not converge into ICCG solutions. An elimination algolithm of constant term using a null space of the coefficient matrix is also described. In three dimensional magnetostatic field analysis, convergence time and number of iteration of this multigrid method are discussed with the convectional ICCG method.

Nonlinear Multigrid solver exploiting AMGe Coarse Spaces with Approximation Properties

DEFF Research Database (Denmark)

Christensen, Max la Cour; Villa, Umberto; Engsig-Karup, Allan Peter

The paper introduces a nonlinear multigrid solver for mixed finite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstructured problems is the guaranteed approximation property of the AMGe coarse...... properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on unstructured meshes has the ability to be as powerful/successful as FAS on geometrically refined meshes. For comparison, Newton’s method and Picard iterations with an inner state-of-the-art linear solver...... are compared to FAS on a nonlinear saddle point problem with applications to porous media flow. It is demonstrated that FAS is faster than Newton’s method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate...

Using Runtime Systems Tools to Implement Efficient Preconditioners for Heterogeneous Architectures

Directory of Open Access Journals (Sweden)

Roussel Adrien

2016-11-01

Full Text Available Solving large sparse linear systems is a time-consuming step in basin modeling or reservoir simulation. The choice of a robust preconditioner strongly impact the performance of the overall simulation. Heterogeneous architectures based on General Purpose computing on Graphic Processing Units (GPGPU or many-core architectures introduce programming challenges which can be managed in a transparent way for developer with the use of runtime systems. Nevertheless, algorithms need to be well suited for these massively parallel architectures. In this paper, we present preconditioning techniques which enable to take advantage of emerging architectures. We also present our task-based implementations through the use of the HARTS (Heterogeneous Abstract RunTime System runtime system, which aims to manage the recent architectures. We focus on two preconditoners. The first is ILU(0 preconditioner implemented on distributing memory systems. The second one is a multi-level domain decomposition method implemented on a shared-memory system. Obtained results are then presented on corresponding architectures, which open the way to discuss on the scalability of such methods according to numerical performances while keeping in mind that the next step is to propose a massively parallel implementations of these techniques.

Subroutine MLTGRD: a multigrid algorithm based on multiplicative correction and implicit non-stationary iteration

International Nuclear Information System (INIS)

Barry, J.M.; Pollard, J.P.

1986-11-01

A FORTRAN subroutine MLTGRD is provided to solve efficiently the large systems of linear equations arising from a five-point finite difference discretisation of some elliptic partial differential equations. MLTGRD is a multigrid algorithm which provides multiplicative correction to iterative solution estimates from successively reduced systems of linear equations. It uses the method of implicit non-stationary iteration for all grid levels

A parallel version of a multigrid algorithm for isotropic transport equations

International Nuclear Information System (INIS)

Manteuffel, T.; McCormick, S.; Yang, G.; Morel, J.; Oliveira, S.

1994-01-01

The focus of this paper is on a parallel algorithm for solving the transport equations in a slab geometry using multigrid. The spatial discretization scheme used is a finite element method called the modified linear discontinuous (MLD) scheme. The MLD scheme represents a lumped version of the standard linear discontinuous (LD) scheme. The parallel algorithm was implemented on the Connection Machine 2 (CM2). Convergence rates and timings for this algorithm on the CM2 and Cray-YMP are shown

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

International Nuclear Information System (INIS)

Adams, Mark F.; Samtaney, Ravi; Brandt, Achi

2010-01-01

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations - so-called 'textbook' multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss-Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations.

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

International Nuclear Information System (INIS)

Adams, Mark F.; Samtaney, Ravi; Brandt, Achi

2013-01-01

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations so-called textbook multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss-Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations.

A first-order multigrid method for bound-constrained convex optimization

Czech Academy of Sciences Publication Activity Database

Kočvara, Michal; Mohammed, S.

2016-01-01

Roč. 31, č. 3 (2016), s. 622-644 ISSN 1055-6788 R&D Projects: GA ČR(CZ) GAP201/12/0671 Grant - others:European Commission - EC(XE) 313781 Institutional support: RVO:67985556 Keywords : bound-constrained optimization * multigrid methods * linear complementarity problems Subject RIV: BA - General Mathematics Impact factor: 1.023, year: 2016 http://library.utia.cas.cz/separaty/2016/MTR/kocvara-0460326.pdf

A Critical Study of Agglomerated Multigrid Methods for Diffusion

Science.gov (United States)

Nishikawa, Hiroaki; Diskin, Boris; Thomas, James L.

2011-01-01

Agglomerated multigrid techniques used in unstructured-grid methods are studied critically for a model problem representative of laminar diffusion in the incompressible limit. The studied target-grid discretizations and discretizations used on agglomerated grids are typical of current node-centered formulations. Agglomerated multigrid convergence rates are presented using a range of two- and three-dimensional randomly perturbed unstructured grids for simple geometries with isotropic and stretched grids. Two agglomeration techniques are used within an overall topology-preserving agglomeration framework. The results show that multigrid with an inconsistent coarse-grid scheme using only the edge terms (also referred to in the literature as a thin-layer formulation) provides considerable speedup over single-grid methods but its convergence deteriorates on finer grids. Multigrid with a Galerkin coarse-grid discretization using piecewise-constant prolongation and a heuristic correction factor is slower and also grid-dependent. In contrast, grid-independent convergence rates are demonstrated for multigrid with consistent coarse-grid discretizations. Convergence rates of multigrid cycles are verified with quantitative analysis methods in which parts of the two-grid cycle are replaced by their idealized counterparts.

Block-accelerated aggregation multigrid for Markov chains with application to PageRank problems

Science.gov (United States)

Shen, Zhao-Li; Huang, Ting-Zhu; Carpentieri, Bruno; Wen, Chun; Gu, Xian-Ming

2018-06-01

Recently, the adaptive algebraic aggregation multigrid method has been proposed for computing stationary distributions of Markov chains. This method updates aggregates on every iterative cycle to keep high accuracies of coarse-level corrections. Accordingly, its fast convergence rate is well guaranteed, but often a large proportion of time is cost by aggregation processes. In this paper, we show that the aggregates on each level in this method can be utilized to transfer the probability equation of that level into a block linear system. Then we propose a Block-Jacobi relaxation that deals with the block system on each level to smooth error. Some theoretical analysis of this technique is presented, meanwhile it is also adapted to solve PageRank problems. The purpose of this technique is to accelerate the adaptive aggregation multigrid method and its variants for solving Markov chains and PageRank problems. It also attempts to shed some light on new solutions for making aggregation processes more cost-effective for aggregation multigrid methods. Numerical experiments are presented to illustrate the effectiveness of this technique.

Numerical Methods for Forward and Inverse Problems in Discontinuous Media

Energy Technology Data Exchange (ETDEWEB)

Chartier, Timothy P.

2011-03-08

The research emphasis under this grant's funding is in the area of algebraic multigrid methods. The research has two main branches: 1) exploring interdisciplinary applications in which algebraic multigrid can make an impact and 2) extending the scope of algebraic multigrid methods with algorithmic improvements that are based in strong analysis.The work in interdisciplinary applications falls primarily in the field of biomedical imaging. Work under this grant demonstrated the effectiveness and robustness of multigrid for solving linear systems that result from highly heterogeneous finite element method models of the human head. The results in this work also give promise to medical advances possible with software that may be developed. Research to extend the scope of algebraic multigrid has been focused in several areas. In collaboration with researchers at the University of Colorado, Lawrence Livermore National Laboratory, and Los Alamos National Laboratory, the PI developed an adaptive multigrid with subcycling via complementary grids. This method has very cheap computing costs per iterate and is showing promise as a preconditioner for conjugate gradient. Recent work with Los Alamos National Laboratory concentrates on developing algorithms that take advantage of the recent advances in adaptive multigrid research. The results of the various efforts in this research could ultimately have direct use and impact to researchers for a wide variety of applications, including, astrophysics, neuroscience, contaminant transport in porous media, bi-domain heart modeling, modeling of tumor growth, and flow in heterogeneous porous media. This work has already led to basic advances in computational mathematics and numerical linear algebra and will continue to do so into the future.

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

KAUST Repository

Adams, Mark F.; Samtaney, Ravi; Brandt, Achi

2010-01-01

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations so-called "textbook" multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss-Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations. (C) 2010 Elsevier Inc. All rights reserved.

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

KAUST Repository

Adams, Mark F.

2010-09-01

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations so-called "textbook" multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss-Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations. (C) 2010 Elsevier Inc. All rights reserved.

Multilevel techniques for Reservoir Simulation

DEFF Research Database (Denmark)

Christensen, Max la Cour

The subject of this thesis is the development, application and study of novel multilevel methods for the acceleration and improvement of reservoir simulation techniques. The motivation for addressing this topic is a need for more accurate predictions of porous media flow and the ability to carry...... Full Approximation Scheme) • Variational (Galerkin) upscaling • Linear solvers and preconditioners First, a nonlinear multigrid scheme in the form of the Full Approximation Scheme (FAS) is implemented and studied for a 3D three-phase compressible rock/fluids immiscible reservoir simulator...... is extended to include a hybrid strategy, where FAS is combined with Newton’s method to construct a multilevel nonlinear preconditioner. This method demonstrates high efficiency and robustness. Second, an improved IMPES formulated reservoir simulator is implemented using a novel variational upscaling approach...

A Parallel Sweeping Preconditioner for Heterogeneous 3D Helmholtz Equations

KAUST Repository

Poulson, Jack

2013-05-02

A parallelization of a sweeping preconditioner for three-dimensional Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ2N4/3) and O(γN logN), where γ(ω) denotes the modestly frequency-dependent number of grid points per perfectly matched layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique. © 2013 Society for Industrial and Applied Mathematics.

High Performance Parallel Multigrid Algorithms for Unstructured Grids

Science.gov (United States)

Frederickson, Paul O.

1996-01-01

We describe a high performance parallel multigrid algorithm for a rather general class of unstructured grid problems in two and three dimensions. The algorithm PUMG, for parallel unstructured multigrid, is related in structure to the parallel multigrid algorithm PSMG introduced by McBryan and Frederickson, for they both obtain a higher convergence rate through the use of multiple coarse grids. Another reason for the high convergence rate of PUMG is its smoother, an approximate inverse developed by Baumgardner and Frederickson.

Progress with multigrid schemes for hypersonic flow problems

International Nuclear Information System (INIS)

Radespiel, R.; Swanson, R.C.

1995-01-01

Several multigrid schemes are considered for the numerical computation of viscous hypersonic flows. For each scheme, the basic solution algorithm employs upwind spatial discretization with explicit multistage time stepping. Two-level versions of the various multigrid algorithms are applied to the two-dimensional advection equation, and Fourier analysis is used to determine their damping properties. The capabilities of the multigrid methods are assessed by solving three different hypersonic flow problems. Some new multigrid schemes based on semicoarsening strategies are shown to be quite effective in relieving the stiffness caused by the high-aspect-ratio cells required to resolve high Reynolds number flows. These schemes exhibit good convergence rates for Reynolds numbers up to 200 X 10 6 and Mach numbers up to 25. 32 refs., 31 figs., 1 tab

A multigrid based 3D space-charge routine in the tracking code GPT

NARCIS (Netherlands)

Pöplau, G.; Rienen, van U.; Loos, de M.J.; Geer, van der S.B.; Berz, M.; Makino, K.

2005-01-01

Fast calculation of3D non-linear space-charge fields is essential for the simulation ofhigh-brightness charged particle beams. We report on our development of a new 3D spacecharge routine in the General Particle Tracer (GPT) code. The model is based on a nonequidistant multigrid Poisson solver that

An extended GS method for dense linear systems

Science.gov (United States)

Niki, Hiroshi; Kohno, Toshiyuki; Abe, Kuniyoshi

2009-09-01

Davey and Rosindale [K. Davey, I. Rosindale, An iterative solution scheme for systems of boundary element equations, Internat. J. Numer. Methods Engrg. 37 (1994) 1399-1411] derived the GSOR method, which uses an upper triangular matrix [Omega] in order to solve dense linear systems. By applying functional analysis, the authors presented an expression for the optimum [Omega]. Moreover, Davey and Bounds [K. Davey, S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Comput. 19 (1998) 953-967] also introduced further interesting results. In this note, we employ a matrix analysis approach to investigate these schemes, and derive theorems that compare these schemes with existing preconditioners for dense linear systems. We show that the convergence rate of the Gauss-Seidel method with preconditioner PG is superior to that of the GSOR method. Moreover, we define some splittings associated with the iterative schemes. Some numerical examples are reported to confirm the theoretical analysis. We show that the EGS method with preconditioner produces an extremely small spectral radius in comparison with the other schemes considered.

Fast multipole preconditioners for sparse matrices arising from elliptic equations

KAUST Repository

Ibeid, Huda

2017-11-09

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the fast multipole method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low-rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.

Fast multipole preconditioners for sparse matrices arising from elliptic equations

KAUST Repository

Ibeid, Huda; Yokota, Rio; Pestana, Jennifer; Keyes, David E.

2017-01-01

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the fast multipole method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. It is capable of algebraic convergence rates down to the truncation error of the discretized PDE comparable to those of multigrid methods, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low-rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.

The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces

KAUST Repository

Chen, Yujia

2015-01-01

© 2015 Society for Industrial and Applied Mathematics. Elliptic partial differential equations are important from both application and analysis points of view. In this paper we apply the closest point method to solve elliptic equations on general curved surfaces. Based on the closest point representation of the underlying surface, we formulate an embedding equation for the surface elliptic problem, then discretize it using standard finite differences and interpolation schemes on banded but uniform Cartesian grids. We prove the convergence of the difference scheme for the Poisson\\'s equation on a smooth closed curve. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method in the setting of the closest point method. Convergence studies in both the accuracy of the difference scheme and the speed of the multigrid algorithm show that our approaches are effective.

Segmental Refinement: A Multigrid Technique for Data Locality

Energy Technology Data Exchange (ETDEWEB)

Adams, Mark [Columbia Univ., New York, NY (United States). Applied Physics and Applied Mathematics Dept.; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

2014-10-27

We investigate a technique - segmental refinement (SR) - proposed by Brandt in the 1970s as a low memory multigrid method. The technique is attractive for modern computer architectures because it provides high data locality, minimizes network communication, is amenable to loop fusion, and is naturally highly parallel and asynchronous. The network communication minimization property was recognized by Brandt and Diskin in 1994; we continue this work by developing a segmental refinement method for a finite volume discretization of the 3D Laplacian on massively parallel computers. An understanding of the asymptotic complexities, required to maintain textbook multigrid efficiency, are explored experimentally with a simple SR method. A two-level memory model is developed to compare the asymptotic communication complexity of a proposed SR method with traditional parallel multigrid. Performance and scalability are evaluated with a Cray XC30 with up to 64K cores. We achieve modest improvement in scalability from traditional parallel multigrid with a simple SR implementation.

Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients [Adaptive Algebraic Multigrid for Sequence of Problems with Slowly Varying Random Coefficients

Energy Technology Data Exchange (ETDEWEB)

Kalchev, D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Ketelsen, C. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Vassilevski, P. S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2013-11-07

Our paper proposes an adaptive strategy for reusing a previously constructed coarse space by algebraic multigrid to construct a two-level solver for a problem with nearby characteristics. Furthermore, a main target application is the solution of the linear problems that appear throughout a sequence of Markov chain Monte Carlo simulations of subsurface flow with uncertain permeability field. We demonstrate the efficacy of the method with extensive set of numerical experiments.

Efficient solvers for coupled models in respiratory mechanics.

Science.gov (United States)

Verdugo, Francesc; Roth, Christian J; Yoshihara, Lena; Wall, Wolfgang A

2017-02-01

We present efficient preconditioners for one of the most physiologically relevant pulmonary models currently available. Our underlying motivation is to enable the efficient simulation of such a lung model on high-performance computing platforms in order to assess mechanical ventilation strategies and contributing to design more protective patient-specific ventilation treatments. The system of linear equations to be solved using the proposed preconditioners is essentially the monolithic system arising in fluid-structure interaction (FSI) extended by additional algebraic constraints. The introduction of these constraints leads to a saddle point problem that cannot be solved with usual FSI preconditioners available in the literature. The key ingredient in this work is to use the idea of the semi-implicit method for pressure-linked equations (SIMPLE) for getting rid of the saddle point structure, resulting in a standard FSI problem that can be treated with available techniques. The numerical examples show that the resulting preconditioners approach the optimal performance of multigrid methods, even though the lung model is a complex multiphysics problem. Moreover, the preconditioners are robust enough to deal with physiologically relevant simulations involving complex real-world patient-specific lung geometries. The same approach is applicable to other challenging biomedical applications where coupling between flow and tissue deformations is modeled with additional algebraic constraints. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

Research on an efficient preconditioner using GMRES method for the MOC

International Nuclear Information System (INIS)

Takeda, Satoshi; Kitada, Takanori; Smith, Michael A.

2011-01-01

The modeling accuracy of reactor analysis techniques has improved considerably with the progressive improvements in computational capabilities. The method of characteristics (MOC) solves the neutron transport equation using tracking lines which simulates the neutron paths. The MOC is an accurate calculation method and is becoming a major solver because of the rapid advancement of the computer. In this methodology, the transport equation is discretized into many spatial meshes and energy wise groups. And the discretization generates a large system which needs a lot of computational costs. To reduce computational costs of MOC calculation, we investigate the Generalized Minimal RESidual (GMRES) method as an accelerator and developed an efficient preconditioner for the MOC calculation. The preconditioner we developed was made by simplifying rigorous preconditioner. And the efficiency was verified by comparing the number of iterations which is calculated by one dimensional MOC code

Segmental Refinement: A Multigrid Technique for Data Locality

KAUST Repository

Adams, Mark F.; Brown, Jed; Knepley, Matt; Samtaney, Ravi

2016-01-01

We investigate a domain decomposed multigrid technique, termed segmental refinement, for solving general nonlinear elliptic boundary value problems. We extend the method first proposed in 1994 by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid is eliminated on fine grids, with modest amounts of extra work and storage, while maintaining the asymptotic exactness of full multigrid. We observe an accuracy dependence on the segmental refinement subdomain size, which was not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores on a Cray XC30.

Segmental Refinement: A Multigrid Technique for Data Locality

KAUST Repository

Adams, Mark F.

2016-08-04

We investigate a domain decomposed multigrid technique, termed segmental refinement, for solving general nonlinear elliptic boundary value problems. We extend the method first proposed in 1994 by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid is eliminated on fine grids, with modest amounts of extra work and storage, while maintaining the asymptotic exactness of full multigrid. We observe an accuracy dependence on the segmental refinement subdomain size, which was not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores on a Cray XC30.

Wide-angle full-vector beam propagation method based on an alternating direction implicit preconditioner

Science.gov (United States)

Chui, Siu Lit; Lu, Ya Yan

2004-03-01

Wide-angle full-vector beam propagation methods (BPMs) for three-dimensional wave-guiding structures can be derived on the basis of rational approximants of a square root operator or its exponential (i.e., the one-way propagator). While the less accurate BPM based on the slowly varying envelope approximation can be efficiently solved by the alternating direction implicit (ADI) method, the wide-angle variants involve linear systems that are more difficult to handle. We present an efficient solver for these linear systems that is based on a Krylov subspace method with an ADI preconditioner. The resulting wide-angle full-vector BPM is used to simulate the propagation of wave fields in a Y branch and a taper.

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

KAUST Repository

Galvis, Juan; Efendiev, Yalchin

2010-01-01

In this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent of the contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper. © 2010 Society for Industrial and Applied Mathematics.

Preconditioner and convergence study for the Quantum Computer Aided Design (QCAD) nonlinear poisson problem posed on the Ottawa Flat 270 design geometry.

Energy Technology Data Exchange (ETDEWEB)

Kalashnikova, Irina

2012-05-01

A numerical study aimed to evaluate different preconditioners within the Trilinos Ifpack and ML packages for the Quantum Computer Aided Design (QCAD) non-linear Poisson problem implemented within the Albany code base and posed on the Ottawa Flat 270 design geometry is performed. This study led to some new development of Albany that allows the user to select an ML preconditioner with Zoltan repartitioning based on nodal coordinates, which is summarized. Convergence of the numerical solutions computed within the QCAD computational suite with successive mesh refinement is examined in two metrics, the mean value of the solution (an L{sup 1} norm) and the field integral of the solution (L{sup 2} norm).

Mapping robust parallel multigrid algorithms to scalable memory architectures

Science.gov (United States)

Overman, Andrea; Vanrosendale, John

1993-01-01

The convergence rate of standard multigrid algorithms degenerates on problems with stretched grids or anisotropic operators. The usual cure for this is the use of line or plane relaxation. However, multigrid algorithms based on line and plane relaxation have limited and awkward parallelism and are quite difficult to map effectively to highly parallel architectures. Newer multigrid algorithms that overcome anisotropy through the use of multiple coarse grids rather than relaxation are better suited to massively parallel architectures because they require only simple point-relaxation smoothers. In this paper, we look at the parallel implementation of a V-cycle multiple semicoarsened grid (MSG) algorithm on distributed-memory architectures such as the Intel iPSC/860 and Paragon computers. The MSG algorithms provide two levels of parallelism: parallelism within the relaxation or interpolation on each grid and across the grids on each multigrid level. Both levels of parallelism must be exploited to map these algorithms effectively to parallel architectures. This paper describes a mapping of an MSG algorithm to distributed-memory architectures that demonstrates how both levels of parallelism can be exploited. The result is a robust and effective multigrid algorithm for distributed-memory machines.

New multigrid solver advances in TOPS

International Nuclear Information System (INIS)

Falgout, R D; Brannick, J; Brezina, M; Manteuffel, T; McCormick, S

2005-01-01

In this paper, we highlight new multigrid solver advances in the Terascale Optimal PDE Simulations (TOPS) project in the Scientific Discovery Through Advanced Computing (SciDAC) program. We discuss two new algebraic multigrid (AMG) developments in TOPS: the adaptive smoothed aggregation method (αSA) and a coarse-grid selection algorithm based on compatible relaxation (CR). The αSA method is showing promising results in initial studies for Quantum Chromodynamics (QCD) applications. The CR method has the potential to greatly improve the applicability of AMG

Multigrid for Staggered Lattice Fermions

Energy Technology Data Exchange (ETDEWEB)

Brower, Richard C. [Boston U.; Clark, M. A. [Unlisted, US; Strelchenko, Alexei [Fermilab; Weinberg, Evan [Boston U.

2018-01-23

Critical slowing down in Krylov methods for the Dirac operator presents a major obstacle to further advances in lattice field theory as it approaches the continuum solution. Here we formulate a multi-grid algorithm for the Kogut-Susskind (or staggered) fermion discretization which has proven difficult relative to Wilson multigrid due to its first-order anti-Hermitian structure. The solution is to introduce a novel spectral transformation by the K\\"ahler-Dirac spin structure prior to the Galerkin projection. We present numerical results for the two-dimensional, two-flavor Schwinger model, however, the general formalism is agnostic to dimension and is directly applicable to four-dimensional lattice QCD.

Matrix-dependent multigrid-homogenization for diffusion problems

Energy Technology Data Exchange (ETDEWEB)

Knapek, S. [Institut fuer Informatik tu Muenchen (Germany)

1996-12-31

We present a method to approximately determine the effective diffusion coefficient on the coarse scale level of problems with strongly varying or discontinuous diffusion coefficients. It is based on techniques used also in multigrid, like Dendy`s matrix-dependent prolongations and the construction of coarse grid operators by means of the Galerkin approximation. In numerical experiments, we compare our multigrid-homogenization method with homogenization, renormalization and averaging approaches.

Highly indefinite multigrid for eigenvalue problems

Energy Technology Data Exchange (ETDEWEB)

Borges, L.; Oliveira, S.

1996-12-31

Eigenvalue problems are extremely important in understanding dynamic processes such as vibrations and control systems. Large scale eigenvalue problems can be very difficult to solve, especially if a large number of eigenvalues and the corresponding eigenvectors need to be computed. For solving this problem a multigrid preconditioned algorithm is presented in {open_quotes}The Davidson Algorithm, preconditioning and misconvergence{close_quotes}. Another approach for solving eigenvalue problems is by developing efficient solutions for highly indefinite problems. In this paper we concentrate on the use of new highly indefinite multigrid algorithms for the eigenvalue problem.

Parallel multigrid smoothing: polynomial versus Gauss-Seidel

International Nuclear Information System (INIS)

Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray

2003-01-01

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines

Parallel multigrid smoothing: polynomial versus Gauss-Seidel

Science.gov (United States)

Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray

2003-07-01

Gauss-Seidel is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for positive definite symmetric systems. Two particular polynomials are considered: Chebyshev and a multilevel specific polynomial. The advantages of polynomial smoothing over traditional smoothers such as Gauss-Seidel are illustrated on several applications: Poisson's equation, thin-body elasticity, and eddy current approximations to Maxwell's equations. While parallelizing the Gauss-Seidel method typically involves a compromise between a scalable convergence rate and maintaining high flop rates, polynomial smoothers achieve parallel scalable multigrid convergence rates without sacrificing flop rates. We show that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.

Solving the Fluid Pressure Poisson Equation Using Multigrid-Evaluation and Improvements.

Science.gov (United States)

Dick, Christian; Rogowsky, Marcus; Westermann, Rudiger

2016-11-01

In many numerical simulations of fluids governed by the incompressible Navier-Stokes equations, the pressure Poisson equation needs to be solved to enforce mass conservation. Multigrid solvers show excellent convergence in simple scenarios, yet they can converge slowly in domains where physically separated regions are combined at coarser scales. Moreover, existing multigrid solvers are tailored to specific discretizations of the pressure Poisson equation, and they cannot easily be adapted to other discretizations. In this paper we analyze the convergence properties of existing multigrid solvers for the pressure Poisson equation in different simulation domains, and we show how to further improve the multigrid convergence rate by using a graph-based extension to determine the coarse grid hierarchy. The proposed multigrid solver is generic in that it can be applied to different kinds of discretizations of the pressure Poisson equation, by using solely the specification of the simulation domain and pre-assembled computational stencils. We analyze the proposed solver in combination with finite difference and finite volume discretizations of the pressure Poisson equation. Our evaluations show that, despite the common assumption, multigrid schemes can exploit their potential even in the most complicated simulation scenarios, yet this behavior is obtained at the price of higher memory consumption.

Modeling of frequency-domain scalar wave equation with the average-derivative optimal scheme based on a multigrid-preconditioned iterative solver

Science.gov (United States)

Cao, Jian; Chen, Jing-Bo; Dai, Meng-Xue

2018-01-01

An efficient finite-difference frequency-domain modeling of seismic wave propagation relies on the discrete schemes and appropriate solving methods. The average-derivative optimal scheme for the scalar wave modeling is advantageous in terms of the storage saving for the system of linear equations and the flexibility for arbitrary directional sampling intervals. However, using a LU-decomposition-based direct solver to solve its resulting system of linear equations is very costly for both memory and computational requirements. To address this issue, we consider establishing a multigrid-preconditioned BI-CGSTAB iterative solver fit for the average-derivative optimal scheme. The choice of preconditioning matrix and its corresponding multigrid components is made with the help of Fourier spectral analysis and local mode analysis, respectively, which is important for the convergence. Furthermore, we find that for the computation with unequal directional sampling interval, the anisotropic smoothing in the multigrid precondition may affect the convergence rate of this iterative solver. Successful numerical applications of this iterative solver for the homogenous and heterogeneous models in 2D and 3D are presented where the significant reduction of computer memory and the improvement of computational efficiency are demonstrated by comparison with the direct solver. In the numerical experiments, we also show that the unequal directional sampling interval will weaken the advantage of this multigrid-preconditioned iterative solver in the computing speed or, even worse, could reduce its accuracy in some cases, which implies the need for a reasonable control of directional sampling interval in the discretization.

Multigrid methods for the computation of propagators in gauge fields

International Nuclear Information System (INIS)

Kalkreuter, T.

1992-11-01

In the present work generalizations of multigrid methods for propagators in gauge fields are investigated. We discuss proper averaging operations for bosons and for staggered fermions. An efficient algorithm for computing C numerically is presented. The averaging kernels C can be used not only in deterministic multigrid computations, but also in multigrid Monte Carlo simulations, and for the definition of block spins and blocked gauge fields in Monte Carlo renormalization group studies of gauge theories. Actual numerical computations of kernels and propagators are performed in compact four-dimensional SU(2) gauge fields. (orig./HSI)

Multicloud: Multigrid convergence with a meshless operator

International Nuclear Information System (INIS)

Katz, Aaron; Jameson, Antony

2009-01-01

The primary objective of this work is to develop and test a new convergence acceleration technique we call multicloud. Multicloud is well-founded in the mathematical basis of multigrid, but relies on a meshless operator on coarse levels. The meshless operator enables extremely simple and automatic coarsening procedures for arbitrary meshes using arbitrary fine level discretization schemes. The performance of multicloud is compared with established multigrid techniques for structured and unstructured meshes for the Euler equations on two-dimensional test cases. Results indicate comparable convergence rates per unit work for multicloud and multigrid. However, because of its mesh and scheme transparency, multicloud may be applied to a wide array of problems with no modification of fine level schemes as is often required with agglomeration techniques. The implication is that multicloud can be implemented in a completely modular fashion, allowing researchers to develop fine level algorithms independent of the convergence accelerator for complex three-dimensional problems.

Unweighted least squares phase unwrapping by means of multigrid techniques

Science.gov (United States)

Pritt, Mark D.

1995-11-01

We present a multigrid algorithm for unweighted least squares phase unwrapping. This algorithm applies Gauss-Seidel relaxation schemes to solve the Poisson equation on smaller, coarser grids and transfers the intermediate results to the finer grids. This approach forms the basis of our multigrid algorithm for weighted least squares phase unwrapping, which is described in a separate paper. The key idea of our multigrid approach is to maintain the partial derivatives of the phase data in separate arrays and to correct these derivatives at the boundaries of the coarser grids. This maintains the boundary conditions necessary for rapid convergence to the correct solution. Although the multigrid algorithm is an iterative algorithm, we demonstrate that it is nearly as fast as the direct Fourier-based method. We also describe how to parallelize the algorithm for execution on a distributed-memory parallel processor computer or a network-cluster of workstations.

Some multigrid algorithms for SIMD machines

Energy Technology Data Exchange (ETDEWEB)

Dendy, J.E. Jr. [Los Alamos National Lab., NM (United States)

1996-12-31

Previously a semicoarsening multigrid algorithm suitable for use on SIMD architectures was investigated. Through the use of new software tools, the performance of this algorithm has been considerably improved. The method has also been extended to three space dimensions. The method performs well for strongly anisotropic problems and for problems with coefficients jumping by orders of magnitude across internal interfaces. The parallel efficiency of this method is analyzed, and its actual performance on the CM-5 is compared with its performance on the CRAY-YMP. A standard coarsening multigrid algorithm is also considered, and we compare its performance on these two platforms as well.

Substructuring preconditioners for an h-p domain decomposition method with interior penalty mortaring

KAUST Repository

Antonietti, P. F.; Ayuso Dios, Blanca; Bertoluzza, S.; Pennacchio, M.

2014-01-01

We propose and study an iterative substructuring method for an h-p Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103–134, (1986) for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory. © 2014, Springer-Verlag Italia.

Substructuring preconditioners for an h-p domain decomposition method with interior penalty mortaring

KAUST Repository

Antonietti, P. F.

2014-05-13

We propose and study an iterative substructuring method for an h-p Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103–134, (1986) for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory. © 2014, Springer-Verlag Italia.

Domain decomposed preconditioners with Krylov subspace methods as subdomain solvers

Energy Technology Data Exchange (ETDEWEB)

Pernice, M. [Univ. of Utah, Salt Lake City, UT (United States)

1994-12-31

Domain decomposed preconditioners for nonsymmetric partial differential equations typically require the solution of problems on the subdomains. Most implementations employ exact solvers to obtain these solutions. Consequently work and storage requirements for the subdomain problems grow rapidly with the size of the subdomain problems. Subdomain solves constitute the single largest computational cost of a domain decomposed preconditioner, and improving the efficiency of this phase of the computation will have a significant impact on the performance of the overall method. The small local memory available on the nodes of most message-passing multicomputers motivates consideration of the use of an iterative method for solving subdomain problems. For large-scale systems of equations that are derived from three-dimensional problems, memory considerations alone may dictate the need for using iterative methods for the subdomain problems. In addition to reduced storage requirements, use of an iterative solver on the subdomains allows flexibility in specifying the accuracy of the subdomain solutions. Substantial savings in solution time is possible if the quality of the domain decomposed preconditioner is not degraded too much by relaxing the accuracy of the subdomain solutions. While some work in this direction has been conducted for symmetric problems, similar studies for nonsymmetric problems appear not to have been pursued. This work represents a first step in this direction, and explores the effectiveness of performing subdomain solves using several transpose-free Krylov subspace methods, GMRES, transpose-free QMR, CGS, and a smoothed version of CGS. Depending on the difficulty of the subdomain problem and the convergence tolerance used, a reduction in solution time is possible in addition to the reduced memory requirements. The domain decomposed preconditioner is a Schur complement method in which the interface operators are approximated using interface probing.

Parallel iterative solvers and preconditioners using approximate hierarchical methods

Energy Technology Data Exchange (ETDEWEB)

Grama, A.; Kumar, V.; Sameh, A. [Univ. of Minnesota, Minneapolis, MN (United States)

1996-12-31

In this paper, we report results of the performance, convergence, and accuracy of a parallel GMRES solver for Boundary Element Methods. The solver uses a hierarchical approximate matrix-vector product based on a hybrid Barnes-Hut / Fast Multipole Method. We study the impact of various accuracy parameters on the convergence and show that with minimal loss in accuracy, our solver yields significant speedups. We demonstrate the excellent parallel efficiency and scalability of our solver. The combined speedups from approximation and parallelism represent an improvement of several orders in solution time. We also develop fast and paralellizable preconditioners for this problem. We report on the performance of an inner-outer scheme and a preconditioner based on truncated Green`s function. Experimental results on a 256 processor Cray T3D are presented.

Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system

KAUST Repository

Cô rtes, A.M.A.; Coutinho, A.L.G.A.; Dalcin, L.; Calo, Victor M.

2015-01-01

The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf-sup stable and pointwise divergence-free. When applied to discretized Stokes equations, these spaces generate a symmetric and indefinite saddle-point linear system. Krylov subspace methods are usually the most efficient procedures to solve such systems. One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES). However, the efficiency and robustness of Krylov subspace methods is closely tied to appropriate preconditioning strategies. For the discrete Stokes system, in particular, block-diagonal strategies provide efficient preconditioners. In this article, we compare the performance of block-diagonal preconditioners for several block choices. We verify how the eigenvalue clustering promoted by the preconditioning strategies affects MINRES convergence. We also compare the number of iterations and wall-clock timings. We conclude that among the building blocks we tested, the strategy with relaxed inner conjugate gradients preconditioned with incomplete Cholesky provided the best results.

Performance evaluation of block-diagonal preconditioners for the divergence-conforming B-spline discretization of the Stokes system

KAUST Repository

Côrtes, A.M.A.

2015-02-20

The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf-sup stable and pointwise divergence-free. When applied to discretized Stokes equations, these spaces generate a symmetric and indefinite saddle-point linear system. Krylov subspace methods are usually the most efficient procedures to solve such systems. One of such methods, for symmetric systems, is the Minimum Residual Method (MINRES). However, the efficiency and robustness of Krylov subspace methods is closely tied to appropriate preconditioning strategies. For the discrete Stokes system, in particular, block-diagonal strategies provide efficient preconditioners. In this article, we compare the performance of block-diagonal preconditioners for several block choices. We verify how the eigenvalue clustering promoted by the preconditioning strategies affects MINRES convergence. We also compare the number of iterations and wall-clock timings. We conclude that among the building blocks we tested, the strategy with relaxed inner conjugate gradients preconditioned with incomplete Cholesky provided the best results.

On several aspects and applications of the multigrid method for solving partial differential equations

Science.gov (United States)

Dinar, N.

1978-01-01

Several aspects of multigrid methods are briefly described. The main subjects include the development of very efficient multigrid algorithms for systems of elliptic equations (Cauchy-Riemann, Stokes, Navier-Stokes), as well as the development of control and prediction tools (based on local mode Fourier analysis), used to analyze, check and improve these algorithms. Preliminary research on multigrid algorithms for time dependent parabolic equations is also described. Improvements in existing multigrid processes and algorithms for elliptic equations were studied.

Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

KAUST Repository

Ayuso Dios, Blanca

2013-10-30

We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society.

Multilevel preconditioners for discontinuous, Galerkin approximations of elliptic problems, with jump coefficients

KAUST Repository

Ayuso Dios, Blanca; Holst, Michael; Zhu, Yunrong; Zikatanov, Ludmil

2013-01-01

We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. © 2013 American Mathematical Society.

Experiences using multigrid for geothermal simulation

Energy Technology Data Exchange (ETDEWEB)

Bullivant, D.P.; O`Sullivan, M.J. [Univ. of Auckland (New Zealand); Yang, Z. [Univ. of New South Wales (Australia)

1995-03-01

Experiences of applying multigrid to the calculation of natural states for geothermal simulations are discussed. The modelling of natural states was chosen for this study because they can take a long time to compute and the computation is often dominated by the development of phase change boundaries that take up a small region in the simulation. For the first part of this work a modified version of TOUGH was used for 2-D vertical problems. A {open_quotes}test-bed{close_quotes} program is now being used to investigate some of the problems encountered with implementing multigrid. This is ongoing work. To date, there have been some encouraging but not startling results.

Multigrid treatment of implicit continuum diffusion

Science.gov (United States)

Francisquez, Manaure; Zhu, Ben; Rogers, Barrett

2017-10-01

Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms can be efficiently and implicitly handled by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for large numbers of processors. We built and present here a multigrid algorithm for these types of problems which outperforms a spectral solution that employs the highly optimized FFTW library. This multigrid algorithm is not only suitable for high performance computing but may also be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with idealized harmonic test functions as well as a turbulent 2D magnetohydrodynamic simulation. It is also shown that an anisotropic operator without cross-terms can improve model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation. This work was supported by DOE-SC-0010508. This research used resources of the National Energy Research Scientific Computing Center (NERSC).

Ground-state projection multigrid for propagators in 4-dimensional SU(2) gauge fields

International Nuclear Information System (INIS)

Kalkreuter, T.

1991-09-01

The ground-state projection multigrid method is studied for computations of slowly decaying bosonic propagators in 4-dimensional SU(2) lattice gauge theory. The defining eigenvalue equation for the restriction operator is solved exactly. Although the critical exponent z is not reduced in nontrivial gauge fields, multigrid still yields considerable speedup compared with conventional relaxation. Multigrid is also able to outperform the conjugate gradient algorithm. (orig.)

NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES

Energy Technology Data Exchange (ETDEWEB)

Christensen, Max La Cour [Technical Univ. of Denmark, Lyngby (Denmark); Villa, Umberto E. [Univ. of Texas, Austin, TX (United States); Engsig-Karup, Allan P. [Technical Univ. of Denmark, Lyngby (Denmark); Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2016-01-22

The paper introduces a nonlinear multigrid solver for mixed nite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The main motivation to use FAS for unstruc- tured problems is the guaranteed approximation property of the AMGe coarse spaces that were developed recently at Lawrence Livermore National Laboratory. These give the ability to derive stable and accurate coarse nonlinear discretization problems. The previous attempts (including ones with the original AMGe method, [5, 11]), were less successful due to lack of such good approximation properties of the coarse spaces. With coarse spaces with approximation properties, our FAS approach on un- structured meshes should be as powerful/successful as FAS on geometrically re ned meshes. For comparison, Newton's method and Picard iterations with an inner state-of-the-art linear solver is compared to FAS on a nonlinear saddle point problem with applications to porous media ow. It is demonstrated that FAS is faster than Newton's method and Picard iterations for the experiments considered here. Due to the guaranteed approximation properties of our AMGe, the coarse spaces are very accurate, providing a solver with the potential for mesh-independent convergence on general unstructured meshes.

BDDC Deluxe for Isogeometric Analysis

KAUST Repository

da Veiga, L. Beirão

2016-01-01

The main goal of this paper is to design, analyze, and test a BDDC (Balancing Domain Decomposition by Constraints, see [12, 23]) preconditioner for Isogeometric Analysis (IGA), based on a novel type of interface averaging, which we will denote by deluxe scaling, with either full or reduced set of primal constraints. IGA is an innovative numerical methodology, introduced in [17] and first analyzed in [1], where the geometry description of the PDE domain is adopted from a Computer Aided Design (CAD) parametrization usually based on Non-Uniform Rational B-Splines (NURBS) and the same NURBS basis functions are also used as the PDEs discrete basis, following an isoparametric paradigm; see the monograph [10]. Recent works on IGA preconditioners have focused on overlapping Schwarz preconditioners [3, 5, 7, 9], multigrid methods [16], and non-overlapping preconditioners [4, 8, 20].

Annual Copper Mountain Conferences on Multigrid and Iterative Methods, Copper Mountain, Colorado

International Nuclear Information System (INIS)

McCormick, Stephen F.

2016-01-01

This project supported the Copper Mountain Conference on Multigrid and Iterative Methods, held from 2007 to 2015, at Copper Mountain, Colorado. The subject of the Copper Mountain Conference Series alternated between Multigrid Methods in odd-numbered years and Iterative Methods in even-numbered years. Begun in 1983, the Series represents an important forum for the exchange of ideas in these two closely related fields. This report describes the Copper Mountain Conference on Multigrid and Iterative Methods, 2007-2015. Information on the conference series is available at http://grandmaster.colorado.edu/~copper/

Annual Copper Mountain Conferences on Multigrid and Iterative Methods, Copper Mountain, Colorado

Energy Technology Data Exchange (ETDEWEB)

McCormick, Stephen F. [Front Range Scientific, Inc., Lake City, CO (United States)

2016-03-25

This project supported the Copper Mountain Conference on Multigrid and Iterative Methods, held from 2007 to 2015, at Copper Mountain, Colorado. The subject of the Copper Mountain Conference Series alternated between Multigrid Methods in odd-numbered years and Iterative Methods in even-numbered years. Begun in 1983, the Series represents an important forum for the exchange of ideas in these two closely related fields. This report describes the Copper Mountain Conference on Multigrid and Iterative Methods, 2007-2015. Information on the conference series is available at http://grandmaster.colorado.edu/~copper/.

Neural multigrid for gauge theories and other disordered systems

International Nuclear Information System (INIS)

Baeker, M.; Kalkreuter, T.; Mack, G.; Speh, M.

1992-09-01

We present evidence that multigrid works for wave equations in disordered systems, e.g. in the presence of gauge fields, no matter how strong the disorder, but one needs to introduce a 'neural computations' point of view into large scale simulations: First, the system must learn how to do the simulations efficiently, then do the simulation (fast). The method can also be used to provide smooth interpolation kernels which are needed in multigrid Monte Carlo updates. (orig.)

Parallel multigrid methods: implementation on message-passing computers and applications to fluid dynamics. A draft

International Nuclear Information System (INIS)

Solchenbach, K.; Thole, C.A.; Trottenberg, U.

1987-01-01

For a wide class of problems in scientific computing, in particular for partial differential equations, the multigrid principle has proved to yield highly efficient numerical methods. However, the principle has to be applied carefully: if the multigrid components are not chosen adequately with respect to the given problem, the efficiency may be much smaller than possible. This has been demonstrated for many practical problems. Unfortunately, the general theories on multigrid convergence do not give much help in constructing really efficient multigrid algorithms. Although some progress has been made in bridging the gap between theory and practice during the last few years, there are still several theoretical approaches which are misleading rather than helpful with respect to the objective of real efficiency. The research in finding highly efficient algorithms for non-model applications therefore is still a sophisticated mixture of theoretical considerations, a transfer of experiences from model to real life problems and systematical experimental work. The emphasis of the practical research activity today lies - among others - in the following fields: - finding efficient multigrid components for really complex problems, - combining the multigrid approach with advanced discretizative techniques: - constructing highly parallel multigrid algorithms. In this paper, we want to deal mainly with the last topic

MGLab3D: An interactive environment for iterative solvers for elliptic PDEs in two and three dimensions

Energy Technology Data Exchange (ETDEWEB)

Bordner, J.; Saied, F. [Univ. of Illinois, Urbana, IL (United States)

1996-12-31

GLab3D is an enhancement of an interactive environment (MGLab) for experimenting with iterative solvers and multigrid algorithms. It is implemented in MATLAB. The new version has built-in 3D elliptic pde`s and several iterative methods and preconditioners that were not available in the original version. A sparse direct solver option has also been included. The multigrid solvers have also been extended to 3D. The discretization and pde domains are restricted to standard finite differences on the unit square/cube. The power of this software studies in the fact that no programming is needed to solve, for example, the convection-diffusion equation in 3D with TFQMR and a customized V-cycle preconditioner, for a variety of problem sizes and mesh Reynolds, numbers. In addition to the graphical user interface, some sample drivers are included to show how experiments can be composed using the underlying suite of problems and solvers.

Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems

Science.gov (United States)

Bramble, James H.; Kwak, Do Y.; Pasciak, Joseph E.

1993-01-01

In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss-Seidel iterations. We also study smoothers based entirely on the original operator. One is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi, and Gauss-Seidel. We show that the uniform estimates for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).

A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations

Science.gov (United States)

Chen, Hao; Lv, Wen; Zhang, Tongtong

2018-05-01

We study preconditioned iterative methods for the linear system arising in the numerical discretization of a two-dimensional space-fractional diffusion equation. Our approach is based on a formulation of the discrete problem that is shown to be the sum of two Kronecker products. By making use of an alternating Kronecker product splitting iteration technique we establish a class of fixed-point iteration methods. Theoretical analysis shows that the new method converges to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameters and the corresponding asymptotic convergence rate are computed exactly when the eigenvalues of the system matrix are all real. The basic iteration is accelerated by a Krylov subspace method like GMRES. The corresponding preconditioner is in a form of a Kronecker product structure and requires at each iteration the solution of a set of discrete one-dimensional fractional diffusion equations. We use structure preserving approximations to the discrete one-dimensional fractional diffusion operators in the action of the preconditioning matrix. Numerical examples are presented to illustrate the effectiveness of this approach.

Multigrid for high dimensional elliptic partial differential equations on non-equidistant grids

NARCIS (Netherlands)

bin Zubair, H.; Oosterlee, C.E.; Wienands, R.

2006-01-01

This work presents techniques, theory and numbers for multigrid in a general d-dimensional setting. The main focus is the multigrid convergence for high-dimensional partial differential equations (PDEs). As a model problem we have chosen the anisotropic diffusion equation, on a unit hypercube. We

Tensor-product preconditioners for higher-order space-time discontinuous Galerkin methods

Science.gov (United States)

Diosady, Laslo T.; Murman, Scott M.

2017-02-01

A space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equations. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high-order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

Tensor-Product Preconditioners for Higher-Order Space-Time Discontinuous Galerkin Methods

Science.gov (United States)

Diosady, Laslo T.; Murman, Scott M.

2016-01-01

space-time discontinuous-Galerkin spectral-element discretization is presented for direct numerical simulation of the compressible Navier-Stokes equat ions. An efficient solution technique based on a matrix-free Newton-Krylov method is developed in order to overcome the stiffness associated with high solution order. The use of tensor-product basis functions is key to maintaining efficiency at high order. Efficient preconditioning methods are presented which can take advantage of the tensor-product formulation. A diagonalized Alternating-Direction-Implicit (ADI) scheme is extended to the space-time discontinuous Galerkin discretization. A new preconditioner for the compressible Euler/Navier-Stokes equations based on the fast-diagonalization method is also presented. Numerical results demonstrate the effectiveness of these preconditioners for the direct numerical simulation of subsonic turbulent flows.

Multigrid solution of the convection-diffusion equation with high-Reynolds number

Energy Technology Data Exchange (ETDEWEB)

Zhang, Jun [George Washington Univ., Washington, DC (United States)

1996-12-31

A fourth-order compact finite difference scheme is employed with the multigrid technique to solve the variable coefficient convection-diffusion equation with high-Reynolds number. Scaled inter-grid transfer operators and potential on vectorization and parallelization are discussed. The high-order multigrid method is unconditionally stable and produces solution of 4th-order accuracy. Numerical experiments are included.

Multigrid Methods for the Computation of Propagators in Gauge Fields

Science.gov (United States)

Kalkreuter, Thomas

Multigrid methods were invented for the solution of discretized partial differential equations in order to overcome the slowness of traditional algorithms by updates on various length scales. In the present work generalizations of multigrid methods for propagators in gauge fields are investigated. Gauge fields are incorporated in algorithms in a covariant way. The kernel C of the restriction operator which averages from one grid to the next coarser grid is defined by projection on the ground-state of a local Hamiltonian. The idea behind this definition is that the appropriate notion of smoothness depends on the dynamics. The ground-state projection choice of C can be used in arbitrary dimension and for arbitrary gauge group. We discuss proper averaging operations for bosons and for staggered fermions. The kernels C can also be used in multigrid Monte Carlo simulations, and for the definition of block spins and blocked gauge fields in Monte Carlo renormalization group studies. Actual numerical computations are performed in four-dimensional SU(2) gauge fields. We prove that our proposals for block spins are “good”, using renormalization group arguments. A central result is that the multigrid method works in arbitrarily disordered gauge fields, in principle. It is proved that computations of propagators in gauge fields without critical slowing down are possible when one uses an ideal interpolation kernel. Unfortunately, the idealized algorithm is not practical, but it was important to answer questions of principle. Practical methods are able to outperform the conjugate gradient algorithm in case of bosons. The case of staggered fermions is harder. Multigrid methods give considerable speed-ups compared to conventional relaxation algorithms, but on lattices up to 184 conjugate gradient is superior.

HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part I. Multilevel Analysis

NARCIS (Netherlands)

van der Vegt, Jacobus J.W.; Rhebergen, Sander

2011-01-01

The hp-Multigrid as Smoother algorithm (hp-MGS) for the solution of higher order accurate space-(time) discontinuous Galerkin discretizations of advection dominated flows is presented. This algorithm combines p-multigrid with h-multigrid at all p-levels, where the h-multigrid acts as smoother in the

Parallel alternating direction preconditioner for isogeometric simulations of explicit dynamics

KAUST Repository

Łoś, Marcin

2015-04-27

In this paper we present a parallel implementation of the alternating direction preconditioner for isogeometric simulations of explicit dynamics. The Alternating Direction Implicit (ADI) algorithm, belongs to the category of matrix-splitting iterative methods, was proposed almost six decades ago for solving parabolic and elliptic partial differential equations, see [1–4]. The new version of this algorithm has been recently developed for isogeometric simulations of two dimensional explicit dynamics [5] and steady-state diffusion equations with orthotropic heterogenous coefficients [6]. In this paper we present a parallel version of the alternating direction implicit algorithm for three dimensional simulations. The algorithm has been incorporated as a part of PETIGA an isogeometric framework [7] build on top of PETSc [8]. We show the scalability of the parallel algorithm on STAMPEDE linux cluster up to 10,000 processors, as well as the convergence rate of the PCG solver with ADI algorithm as preconditioner.

On multigrid solution of the implicit equations of hydrodynamics. Experiments for the compressible Euler equations in general coordinates

Science.gov (United States)

Kifonidis, K.; Müller, E.

2012-08-01

Aims: We describe and study a family of new multigrid iterative solvers for the multidimensional, implicitly discretized equations of hydrodynamics. Schemes of this class are free of the Courant-Friedrichs-Lewy condition. They are intended for simulations in which widely differing wave propagation timescales are present. A preferred solver in this class is identified. Applications to some simple stiff test problems that are governed by the compressible Euler equations, are presented to evaluate the convergence behavior, and the stability properties of this solver. Algorithmic areas are determined where further work is required to make the method sufficiently efficient and robust for future application to difficult astrophysical flow problems. Methods: The basic equations are formulated and discretized on non-orthogonal, structured curvilinear meshes. Roe's approximate Riemann solver and a second-order accurate reconstruction scheme are used for spatial discretization. Implicit Runge-Kutta (ESDIRK) schemes are employed for temporal discretization. The resulting discrete equations are solved with a full-coarsening, non-linear multigrid method. Smoothing is performed with multistage-implicit smoothers. These are applied here to the time-dependent equations by means of dual time stepping. Results: For steady-state problems, our results show that the efficiency of the present approach is comparable to the best implicit solvers for conservative discretizations of the compressible Euler equations that can be found in the literature. The use of red-black as opposed to symmetric Gauss-Seidel iteration in the multistage-smoother is found to have only a minor impact on multigrid convergence. This should enable scalable parallelization without having to seriously compromise the method's algorithmic efficiency. For time-dependent test problems, our results reveal that the multigrid convergence rate degrades with increasing Courant numbers (i.e. time step sizes). Beyond a

Iterative linear solvers in a 2D radiation-hydrodynamics code: Methods and performance

International Nuclear Information System (INIS)

Baldwin, C.; Brown, P.N.; Falgout, R.; Graziani, F.; Jones, J.

1999-01-01

Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion (ICF) phenomena. A crucial aspect of these codes is that they require an implicit solution of the radiation diffusion equations. The authors present in this paper the results of a comparison of five different linear solvers on a range of complex radiation and radiation-hydrodynamics problems. The linear solvers used are diagonally scaled conjugate gradient, GMRES with incomplete LU preconditioning, conjugate gradient with incomplete Cholesky preconditioning, multigrid, and multigrid-preconditioned conjugate gradient. These problems involve shock propagation, opacities varying over 5--6 orders of magnitude, tabular equations of state, and dynamic ALE (Arbitrary Lagrangian Eulerian) meshes. They perform a problem size scalability study by comparing linear solver performance over a wide range of problem sizes from 1,000 to 100,000 zones. The fundamental question they address in this paper is: Is it more efficient to invert the matrix in many inexpensive steps (like diagonally scaled conjugate gradient) or in fewer expensive steps (like multigrid)? In addition, what is the answer to this question as a function of problem size and is the answer problem dependent? They find that the diagonally scaled conjugate gradient method performs poorly with the growth of problem size, increasing in both iteration count and overall CPU time with the size of the problem and also increasing for larger time steps. For all problems considered, the multigrid algorithms scale almost perfectly (i.e., the iteration count is approximately independent of problem size and problem time step). For pure radiation flow problems (i.e., no hydrodynamics), they see speedups in CPU time of factors of ∼15--30 for the largest problems, when comparing the multigrid solvers relative to diagonal scaled conjugate gradient

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Science.gov (United States)

Luo, P.; Rodrigo, C.; Gaspar, F. J.; Oosterlee, C. W.

2018-01-01

The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients.

s-Step Krylov Subspace Methods as Bottom Solvers for Geometric Multigrid

Energy Technology Data Exchange (ETDEWEB)

Williams, Samuel [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Lijewski, Mike [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Almgren, Ann [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Straalen, Brian Van [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Carson, Erin [Univ. of California, Berkeley, CA (United States); Knight, Nicholas [Univ. of California, Berkeley, CA (United States); Demmel, James [Univ. of California, Berkeley, CA (United States)

2014-08-14

Geometric multigrid solvers within adaptive mesh refinement (AMR) applications often reach a point where further coarsening of the grid becomes impractical as individual sub domain sizes approach unity. At this point the most common solution is to use a bottom solver, such as BiCGStab, to reduce the residual by a fixed factor at the coarsest level. Each iteration of BiCGStab requires multiple global reductions (MPI collectives). As the number of BiCGStab iterations required for convergence grows with problem size, and the time for each collective operation increases with machine scale, bottom solves in large-scale applications can constitute a significant fraction of the overall multigrid solve time. In this paper, we implement, evaluate, and optimize a communication-avoiding s-step formulation of BiCGStab (CABiCGStab for short) as a high-performance, distributed-memory bottom solver for geometric multigrid solvers. This is the first time s-step Krylov subspace methods have been leveraged to improve multigrid bottom solver performance. We use a synthetic benchmark for detailed analysis and integrate the best implementation into BoxLib in order to evaluate the benefit of a s-step Krylov subspace method on the multigrid solves found in the applications LMC and Nyx on up to 32,768 cores on the Cray XE6 at NERSC. Overall, we see bottom solver improvements of up to 4.2x on synthetic problems and up to 2.7x in real applications. This results in as much as a 1.5x improvement in solver performance in real applications.

Copper Mountain conference on multigrid methods. Preliminary proceedings -- List of abstracts

Energy Technology Data Exchange (ETDEWEB)

NONE

1995-12-31

This report contains abstracts of the papers presented at the conference. Papers cover multigrid algorithms and applications of multigrid methods. Applications include the following: solution of elliptical problems; electric power grids; fluid mechanics; atmospheric data assimilation; thermocapillary effects on weld pool shape; boundary-value problems; prediction of hurricane tracks; modeling multi-dimensional combustion and detailed chemistry; black-oil reservoir simulation; image processing; and others.

Multigrid Computation of Stratified Flow over Two-Dimensional Obstacles

Science.gov (United States)

Paisley, M. F.

1997-09-01

A robust multigrid method for the incompressible Navier-Stokes equations is presented and applied to the computation of viscous flow over obstacles in a bounded domain under conditions of neutral stability and stable density stratification. Two obstacle shapes have been used, namely a vertical barrier, for which the grid is Cartesian, and a smooth cosine-shaped obstacle, for which a boundary-conforming transformation is incorporated. Results are given for laminar flows at low Reynolds numbers and turbulent flows at a high Reynolds number, when a simple mixing length turbulence model is included. The multigrid algorithm is used to compute steady flows for each obstacle at low and high Reynolds numbers in conditions of weak static stability, defined byK=ND/πU≤ 1, whereU,N, andDare the upstream velocity, bouyancy frequency, and domain height respectively. Results are also presented for the vertical barrier at low and high Reynolds number in conditions of strong static stability,K> 1, when lee wave motions ensure that the flow is unsteady, and the multigrid algorithm is used to compute the flow at each timestep.

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

Energy Technology Data Exchange (ETDEWEB)

Ainsworth, Mark [Brown Univ., Providence, RI (United States). Division of Applied Mathematics; Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division; Glusa, Christian [Brown Univ., Providence, RI (United States). Division of Applied Mathematics

2016-06-30

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.

Analysis and development of stochastic multigrid methods in lattice field theory

International Nuclear Information System (INIS)

Grabenstein, M.

1994-01-01

We study the relation between the dynamical critical behavior and the kinematics of stochastic multigrid algorithms. The scale dependence of acceptance rates for nonlocal Metropolis updates is analyzed with the help of an approximation formula. A quantitative study of the kinematics of multigrid algorithms in several interacting models is performed. We find that for a critical model with Hamiltonian H(Φ) absence of critical slowing down can only be expected if the expansion of (H(Φ+ψ)) in terms of the shift ψ contains no relevant term (mass term). The predictions of this rule was verified in a multigrid Monte Carlo simulation of the Sine Gordon model in two dimensions. Our analysis can serve as a guideline for the development of new algorithms: We propose a new multigrid method for nonabelian lattice gauge theory, the time slice blocking. For SU(2) gauge fields in two dimensions, critical slowing down is almost completely eliminated by this method, in accordance with the theoretical prediction. The generalization of the time slice blocking to SU(2) in four dimensions is investigated analytically and by numerical simulations. Compared to two dimensions, the local disorder in the four dimensional gauge field leads to kinematical problems. (orig.)

Multigrid on unstructured grids using an auxiliary set of structured grids

Energy Technology Data Exchange (ETDEWEB)

Douglas, C.C.; Malhotra, S.; Schultz, M.H. [Yale Univ., New Haven, CT (United States)

1996-12-31

Unstructured grids do not have a convenient and natural multigrid framework for actually computing and maintaining a high floating point rate on standard computers. In fact, just the coarsening process is expensive for many applications. Since unstructured grids play a vital role in many scientific computing applications, many modifications have been proposed to solve this problem. One suggested solution is to map the original unstructured grid onto a structured grid. This can be used as a fine grid in a standard multigrid algorithm to precondition the original problem on the unstructured grid. We show that unless extreme care is taken, this mapping can lead to a system with a high condition number which eliminates the usefulness of the multigrid method. Theorems with lower and upper bounds are provided. Simple examples show that the upper bounds are sharp.

A calderón multiplicative preconditioner for the combined field integral equation

KAUST Repository

Bagci, Hakan; Andriulli, Francesco P.; Cools, Kristof; Olyslager, Femke; Michielssen, Eric

2009-01-01

A Calderón multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Caldern-preconditioned CFIEs, a localization procedure is employed to ensure that the equation

Uzawa smoother in multigrid for the coupleD porous medium and stokes flow system

NARCIS (Netherlands)

P. Luo (Peiyao); C. Rodrigo (Carmen); F.J. Gaspar Lorenz (Franscisco); C.W. Oosterlee (Kees)

2017-01-01

textabstractThe multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the

Fast Multipole-Based Elliptic PDE Solver and Preconditioner

KAUST Repository

Ibeid, Huda

2016-12-07

Exascale systems are predicted to have approximately one billion cores, assuming Gigahertz cores. Limitations on affordable network topologies for distributed memory systems of such massive scale bring new challenges to the currently dominant parallel programing model. Currently, there are many efforts to evaluate the hardware and software bottlenecks of exascale designs. It is therefore of interest to model application performance and to understand what changes need to be made to ensure extrapolated scalability. Fast multipole methods (FMM) were originally developed for accelerating N-body problems for particle-based methods in astrophysics and molecular dynamics. FMM is more than an N-body solver, however. Recent efforts to view the FMM as an elliptic PDE solver have opened the possibility to use it as a preconditioner for even a broader range of applications. In this thesis, we (i) discuss the challenges for FMM on current parallel computers and future exascale architectures, with a focus on inter-node communication, and develop a performance model that considers the communication patterns of the FMM for spatially quasi-uniform distributions, (ii) employ this performance model to guide performance and scaling improvement of FMM for all-atom molecular dynamics simulations of uniformly distributed particles, and (iii) demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Compared with multilevel methods, FMM is capable of comparable algebraic convergence rates down to the truncation error of the discretized PDE, and it has superior multicore and distributed memory scalability properties on commodity

Efficient multigrid computation of steady hypersonic flows

NARCIS (Netherlands)

Koren, B.; Hemker, P.W.; Murthy, T.K.S.

1991-01-01

In steady hypersonic flow computations, Newton iteration as a local relaxation procedure and nonlinear multigrid iteration as an acceleration procedure may both easily fail. In the present chapter, same remedies are presented for overcoming these problems. The equations considered are the steady,

Analysis and performance of adjacent-cell preconditioners for accelerating multidimensional transport calculations

International Nuclear Information System (INIS)

Azmy, Y.Y.

1996-01-01

The formal development of the Adjacent-cell Preconditioner (AP) and its implementation in the TORT code are briefly reviewed. Based on earlier experience with diffusion type acceleration, and excellent results in slab geometry the reciprocal averaging formula is used to mix the preconditioner elements across material and mesh discontinuities. Numerical testing of the method employing the Burre Suite of Test Problems (BSTeP), a collection of 144 cases covering a wide range in parameter space, using AP, Partial Current Rebalance (PCR), and TWODANT's Diffusion Synthetic Acceleration (DSA) is presented. While AP outperforms the other two methods for the majority of the cases included in BSTeP it consumes many more iterations than can be explained by spectral analysis of the homogeneous model problem in cases with sharp material discontinuity. In order to verify this undesirable behavior and explore potential remedies a model problem, the Periodic Horizontal Interface (PHI), is developed that permits discontinuity of nuclear properties and cell height across the interface. Fourier mode decomposition is applied to AP with the reciprocal averaging mixing formula for the PHI configuration and shown to possess a spectral radius that approaches unity as the material discontinuity gets larger. The question of whether an unconditionally stable AP exists for PHI is tackled and preliminary indications are negative. Novel preconditioners that have nontraditional cell-coupling schemes that remain stable in these regimes may have to be sought

On multigrid-CG for efficient topology optimization

DEFF Research Database (Denmark)

Amir, Oded; Aage, Niels; Lazarov, Boyan Stefanov

2014-01-01

reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process...

A Pseudo-Temporal Multi-Grid Relaxation Scheme for Solving the Parabolized Navier-Stokes Equations

Science.gov (United States)

White, J. A.; Morrison, J. H.

1999-01-01

A multi-grid, flux-difference-split, finite-volume code, VULCAN, is presented for solving the elliptic and parabolized form of the equations governing three-dimensional, turbulent, calorically perfect and non-equilibrium chemically reacting flows. The space marching algorithms developed to improve convergence rate and or reduce computational cost are emphasized. The algorithms presented are extensions to the class of implicit pseudo-time iterative, upwind space-marching schemes. A full approximate storage, full multi-grid scheme is also described which is used to accelerate the convergence of a Gauss-Seidel relaxation method. The multi-grid algorithm is shown to significantly improve convergence on high aspect ratio grids.

Higher-order ice-sheet modelling accelerated by multigrid on graphics cards

Science.gov (United States)

Brædstrup, Christian; Egholm, David

2013-04-01

Higher-order ice flow modelling is a very computer intensive process owing primarily to the nonlinear influence of the horizontal stress coupling. When applied for simulating long-term glacial landscape evolution, the ice-sheet models must consider very long time series, while both high temporal and spatial resolution is needed to resolve small effects. The use of higher-order and full stokes models have therefore seen very limited usage in this field. However, recent advances in graphics card (GPU) technology for high performance computing have proven extremely efficient in accelerating many large-scale scientific computations. The general purpose GPU (GPGPU) technology is cheap, has a low power consumption and fits into a normal desktop computer. It could therefore provide a powerful tool for many glaciologists working on ice flow models. Our current research focuses on utilising the GPU as a tool in ice-sheet and glacier modelling. To this extent we have implemented the Integrated Second-Order Shallow Ice Approximation (iSOSIA) equations on the device using the finite difference method. To accelerate the computations, the GPU solver uses a non-linear Red-Black Gauss-Seidel iterator coupled with a Full Approximation Scheme (FAS) multigrid setup to further aid convergence. The GPU finite difference implementation provides the inherent parallelization that scales from hundreds to several thousands of cores on newer cards. We demonstrate the efficiency of the GPU multigrid solver using benchmark experiments.

Efficient Tridiagonal Preconditioner for the Matrix-Free Truncated Newton Method

Czech Academy of Sciences Publication Activity Database

Lukšan, Ladislav; Vlček, Jan

2014-01-01

Roč. 235, 25 May (2014), s. 394-407 ISSN 0096-3003 R&D Projects: GA ČR GA13-06684S Institutional support: RVO:67985807 Keywords : unconstrained optimization * large scale optimization * matrix-free truncated Newton method * preconditioned conjugate gradient method * preconditioners obtained by the directional differentiation * numerical algorithms Subject RIV: BA - General Mathematics Impact factor: 1.551, year: 2014

Morphing Wing Structural Optimization Using Opposite-Based Population-Based Incremental Learning and Multigrid Ground Elements

Directory of Open Access Journals (Sweden)

S. Sleesongsom

2015-01-01

Full Text Available This paper has twin aims. Firstly, a multigrid design approach for optimization of an unconventional morphing wing is proposed. The structural design problem is assigned to optimize wing mass, lift effectiveness, and buckling factor subject to structural safety requirements. Design variables consist of partial topology, nodal positions, and component sizes of a wing internal structure. Such a design process can be accomplished by using multiple resolutions of ground elements, which is called a multigrid approach. Secondly, an opposite-based multiobjective population-based incremental learning (OMPBIL is proposed for comparison with the original multiobjective population-based incremental learning (MPBIL. Multiobjective design problems with single-grid and multigrid design variables are then posed and tackled by OMPBIL and MPBIL. The results show that using OMPBIL in combination with a multigrid design approach is the best design strategy. OMPBIL is superior to MPBIL since the former provides better population diversity. Aeroelastic trim for an elastic morphing wing is also presented.

Electrical Resistivity Tomography using a finite element based BFGS algorithm with algebraic multigrid preconditioning

Science.gov (United States)

Codd, A. L.; Gross, L.

2018-03-01

We present a new inversion method for Electrical Resistivity Tomography which, in contrast to established approaches, minimizes the cost function prior to finite element discretization for the unknown electric conductivity and electric potential. Minimization is performed with the Broyden-Fletcher-Goldfarb-Shanno method (BFGS) in an appropriate function space. BFGS is self-preconditioning and avoids construction of the dense Hessian which is the major obstacle to solving large 3-D problems using parallel computers. In addition to the forward problem predicting the measurement from the injected current, the so-called adjoint problem also needs to be solved. For this problem a virtual current is injected through the measurement electrodes and an adjoint electric potential is obtained. The magnitude of the injected virtual current is equal to the misfit at the measurement electrodes. This new approach has the advantage that the solution process of the optimization problem remains independent to the meshes used for discretization and allows for mesh adaptation during inversion. Computation time is reduced by using superposition of pole loads for the forward and adjoint problems. A smoothed aggregation algebraic multigrid (AMG) preconditioned conjugate gradient is applied to construct the potentials for a given electric conductivity estimate and for constructing a first level BFGS preconditioner. Through the additional reuse of AMG operators and coarse grid solvers inversion time for large 3-D problems can be reduced further. We apply our new inversion method to synthetic survey data created by the resistivity profile representing the characteristics of subsurface fluid injection. We further test it on data obtained from a 2-D surface electrode survey on Heron Island, a small tropical island off the east coast of central Queensland, Australia.

Preisach hysteresis model for non-linear 2D heat diffusion

International Nuclear Information System (INIS)

Jancskar, Ildiko; Ivanyi, Amalia

2006-01-01

This paper analyzes a non-linear heat diffusion process when the thermal diffusivity behaviour is a hysteretic function of the temperature. Modelling this temperature dependence, the discrete Preisach algorithm as general hysteresis model has been integrated into a non-linear multigrid solver. The hysteretic diffusion shows a heating-cooling asymmetry in character. The presented type of hysteresis speeds up the thermal processes in the modelled systems by a very interesting non-linear way

A general class of preconditioners for statistical iterative reconstruction of emission computed tomography

International Nuclear Information System (INIS)

Chinn, G.; Huang, S.C.

1997-01-01

A major drawback of statistical iterative image reconstruction for emission computed tomography is its high computational cost. The ill-posed nature of tomography leads to slow convergence for standard gradient-based iterative approaches such as the steepest descent or the conjugate gradient algorithm. In this paper new theory and methods for a class of preconditioners are developed for accelerating the convergence rate of iterative reconstruction. To demonstrate the potential of this class of preconditioners, a preconditioned conjugate gradient (PCG) iterative algorithm for weighted least squares reconstruction (WLS) was formulated for emission tomography. Using simulated positron emission tomography (PET) data of the Hoffman brain phantom, it was shown that the convergence rate of the PCG can reduce the number of iterations of the standard conjugate gradient algorithm by a factor of 2--8 times depending on the convergence criterion

Improving matrix-vector product performance and multi-level preconditioning for the parallel PCG package

Energy Technology Data Exchange (ETDEWEB)

McLay, R.T.; Carey, G.F.

1996-12-31

In this study we consider parallel solution of sparse linear systems arising from discretized PDE`s. As part of our continuing work on our parallel PCG Solver package, we have made improvements in two areas. The first is improving the performance of the matrix-vector product. Here on regular finite-difference grids, we are able to use the cache memory more efficiently for smaller domains or where there are multiple degrees of freedom. The second problem of interest in the present work is the construction of preconditioners in the context of the parallel PCG solver we are developing. Here the problem is partitioned over a set of processors subdomains and the matrix-vector product for PCG is carried out in parallel for overlapping grid subblocks. For problems of scaled speedup, the actual rate of convergence of the unpreconditioned system deteriorates as the mesh is refined. Multigrid and subdomain strategies provide a logical approach to resolving the problem. We consider the parallel trade-offs between communication and computation and provide a complexity analysis of a representative algorithm. Some preliminary calculations using the parallel package and comparisons with other preconditioners are provided together with parallel performance results.

Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics

Science.gov (United States)

Gilles, Luc; Ellerbroek, Brent L.; Vogel, Curtis R.

2003-09-01

Multiconjugate adaptive optics (MCAO) systems with 104-105 degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wave-front control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of adaptive optics degrees of freedom. We develop scalable open-loop iterative sparse matrix implementations of minimum variance wave-front reconstruction for telescope diameters up to 32 m with more than 104 actuators. The basic approach is the preconditioned conjugate gradient method with an efficient preconditioner, whose block structure is defined by the atmospheric turbulent layers very much like the layer-oriented MCAO algorithms of current interest. Two cost-effective preconditioners are investigated: a multigrid solver and a simpler block symmetric Gauss-Seidel (BSGS) sweep. Both options require off-line sparse Cholesky factorizations of the diagonal blocks of the matrix system. The cost to precompute these factors scales approximately as the three-halves power of the number of estimated phase grid points per atmospheric layer, and their average update rate is typically of the order of 10-2 Hz, i.e., 4-5 orders of magnitude lower than the typical 103 Hz temporal sampling rate. All other computations scale almost linearly with the total number of estimated phase grid points. We present numerical simulation results to illustrate algorithm convergence. Convergence rates of both preconditioners are similar, regardless of measurement noise level, indicating that the layer-oriented BSGS sweep is as effective as the more elaborated multiresolution preconditioner.

Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics.

Science.gov (United States)

Gilles, Luc; Ellerbroek, Brent L; Vogel, Curtis R

2003-09-10

Multiconjugate adaptive optics (MCAO) systems with 10(4)-10(5) degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of adaptive optics degrees of freedom. We develop scalable open-loop iterative sparse matrix implementations of minimum variance wave-front reconstruction for telescope diameters up to 32 m with more than 10(4) actuators. The basic approach is the preconditioned conjugate gradient method with an efficient preconditioner, whose block structure is defined by the atmospheric turbulent layers very much like the layer-oriented MCAO algorithms of current interest. Two cost-effective preconditioners are investigated: a multigrid solver and a simpler block symmetric Gauss-Seidel (BSGS) sweep. Both options require off-line sparse Cholesky factorizations of the diagonal blocks of the matrix system. The cost to precompute these factors scales approximately as the three-halves power of the number of estimated phase grid points per atmospheric layer, and their average update rate is typically of the order of 10(-2) Hz, i.e., 4-5 orders of magnitude lower than the typical 10(3) Hz temporal sampling rate. All other computations scale almost linearly with the total number of estimated phase grid points. We present numerical simulation results to illustrate algorithm convergence. Convergence rates of both preconditioners are similar, regardless of measurement noise level, indicating that the layer-oriented BSGS sweep is as effective as the more elaborated multiresolution preconditioner.

Numerical Evaluation of P-Multigrid Method for the Solution of Discontinuous Galerkin Discretizations of Diffusive Equations

Science.gov (United States)

Atkins, H. L.; Helenbrook, B. T.

2005-01-01

This paper describes numerical experiments with P-multigrid to corroborate analysis, validate the present implementation, and to examine issues that arise in the implementations of the various combinations of relaxation schemes, discretizations and P-multigrid methods. The two approaches to implement P-multigrid presented here are equivalent for most high-order discretization methods such as spectral element, SUPG, and discontinuous Galerkin applied to advection; however it is discovered that the approach that mimics the common geometric multigrid implementation is less robust, and frequently unstable when applied to discontinuous Galerkin discretizations of di usion. Gauss-Seidel relaxation converges 40% faster than block Jacobi, as predicted by analysis; however, the implementation of Gauss-Seidel is considerably more expensive that one would expect because gradients in most neighboring elements must be updated. A compromise quasi Gauss-Seidel relaxation method that evaluates the gradient in each element twice per iteration converges at rates similar to those predicted for true Gauss-Seidel.

A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems

Czech Academy of Sciences Publication Activity Database

Benzi, M.; Tůma, Miroslav

1998-01-01

Roč. 19, č. 3 (1998), s. 968-994 ISSN 1064-8275 R&D Projects: GA ČR GA201/93/0067; GA AV ČR IAA230401 Keywords : large sparse systems * interative methods * preconditioning * approximate inverse * sparse linear systems * sparse matrices * incomplete factorizations * conjugate gradient -type methods Subject RIV: BA - General Mathematics Impact factor: 1.378, year: 1998

SymPix: A Spherical Grid for Efficient Sampling of Rotationally Invariant Operators

Science.gov (United States)

Seljebotn, D. S.; Eriksen, H. K.

2016-02-01

We present SymPix, a special-purpose spherical grid optimized for efficiently sampling rotationally invariant linear operators. This grid is conceptually similar to the Gauss-Legendre (GL) grid, aligning sample points with iso-latitude rings located on Legendre polynomial zeros. Unlike the GL grid, however, the number of grid points per ring varies as a function of latitude, avoiding expensive oversampling near the poles and ensuring nearly equal sky area per grid point. The ratio between the number of grid points in two neighboring rings is required to be a low-order rational number (3, 2, 1, 4/3, 5/4, or 6/5) to maintain a high degree of symmetries. Our main motivation for this grid is to solve linear systems using multi-grid methods, and to construct efficient preconditioners through pixel-space sampling of the linear operator in question. As a benchmark and representative example, we compute a preconditioner for a linear system that involves the operator \\widehat{{\\boldsymbol{D}}}+{\\widehat{{\\boldsymbol{B}}}}T{{\\boldsymbol{N}}}-1\\widehat{{\\boldsymbol{B}}}, where \\widehat{{\\boldsymbol{B}}} and \\widehat{{\\boldsymbol{D}}} may be described as both local and rotationally invariant operators, and {\\boldsymbol{N}} is diagonal in the pixel domain. For a bandwidth limit of {{\\ell }}{max} = 3000, we find that our new SymPix implementation yields average speed-ups of 360 and 23 for {\\widehat{{\\boldsymbol{B}}}}T{{\\boldsymbol{N}}}-1\\widehat{{\\boldsymbol{B}}} and \\widehat{{\\boldsymbol{D}}}, respectively, compared with the previous state-of-the-art implementation.

Towards a multigrid scheme in SU(2) lattice gauge theory

International Nuclear Information System (INIS)

Gutbrod, F.

1992-12-01

The task of constructing a viable updating multigrid scheme for SU(2) lattice gauge theory is discussed in connection with the classical eigenvalue problem. For a nonlocal overrelaxation Monte Carlo update step, the central numerical problem is the search for the minimum of a quadratic approximation to the action under nonlocal constraints. Here approximate eigenfunctions are essential to reduce the numerical work, and these eigenfunctions are to be constructed with multigrid techniques. A simple implementation on asymmetric lattices is described, where the grids are restricted to 3-dimensional hyperplanes. The scheme is shown to be moderately successful in the early stages of the updating history (starting from a cold configuration). The main results of another, less asymmetric scheme are presented briefly. (orig.)

A scalable geometric multigrid solver for nonsymmetric elliptic systems with application to variable-density flows

Science.gov (United States)

Esmaily, M.; Jofre, L.; Mani, A.; Iaccarino, G.

2018-03-01

A geometric multigrid algorithm is introduced for solving nonsymmetric linear systems resulting from the discretization of the variable density Navier-Stokes equations on nonuniform structured rectilinear grids and high-Reynolds number flows. The restriction operation is defined such that the resulting system on the coarser grids is symmetric, thereby allowing for the use of efficient smoother algorithms. To achieve an optimal rate of convergence, the sequence of interpolation and restriction operations are determined through a dynamic procedure. A parallel partitioning strategy is introduced to minimize communication while maintaining the load balance between all processors. To test the proposed algorithm, we consider two cases: 1) homogeneous isotropic turbulence discretized on uniform grids and 2) turbulent duct flow discretized on stretched grids. Testing the algorithm on systems with up to a billion unknowns shows that the cost varies linearly with the number of unknowns. This O (N) behavior confirms the robustness of the proposed multigrid method regarding ill-conditioning of large systems characteristic of multiscale high-Reynolds number turbulent flows. The robustness of our method to density variations is established by considering cases where density varies sharply in space by a factor of up to 104, showing its applicability to two-phase flow problems. Strong and weak scalability studies are carried out, employing up to 30,000 processors, to examine the parallel performance of our implementation. Excellent scalability of our solver is shown for a granularity as low as 104 to 105 unknowns per processor. At its tested peak throughput, it solves approximately 4 billion unknowns per second employing over 16,000 processors with a parallel efficiency higher than 50%.

Interior-Point Method for Non-Linear Non-Convex Optimization

Czech Academy of Sciences Publication Activity Database

Lukšan, Ladislav; Matonoha, Ctirad; Vlček, Jan

2004-01-01

Roč. 11, č. 5-6 (2004), s. 431-453 ISSN 1070-5325 R&D Projects: GA AV ČR IAA1030103 Institutional research plan: CEZ:AV0Z1030915 Keywords : non-linear programming * interior point methods * indefinite systems * indefinite preconditioners * preconditioned conjugate gradient method * merit functions * algorithms * computational experiments Subject RIV: BA - General Mathematics Impact factor: 0.727, year: 2004

Fourier two-level analysis for discontinuous Galerkin discretization with linear elements

NARCIS (Netherlands)

P.W. Hemker (Piet); W. Hoffmann; M.H. van Raalte (Marc)

2002-01-01

textabstractIn this paper we study the convergence of a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence fordifferent block-relaxation strategies. In addition to an

Recent Development of Multigrid Algorithms for Mixed and Noncomforming Methods for Second Order Elliptical Problems

Science.gov (United States)

Chen, Zhangxin; Ewing, Richard E.

1996-01-01

Multigrid algorithms for nonconforming and mixed finite element methods for second order elliptic problems on triangular and rectangular finite elements are considered. The construction of several coarse-to-fine intergrid transfer operators for nonconforming multigrid algorithms is discussed. The equivalence between the nonconforming and mixed finite element methods with and without projection of the coefficient of the differential problems into finite element spaces is described.

On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices

Science.gov (United States)

Freund, Roland

1988-01-01

Conjugate gradient type methods are considered for the solution of large linear systems Ax = b with complex coefficient matrices of the type A = T + i(sigma)I where T is Hermitian and sigma, a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidian error minimization, respectively, are investigated. In particular, numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices are proposed. Error bounds for all three methods are derived. It is shown how the special shift structure of A can be preserved by using polynomial preconditioning. Results on the optimal choice of the polynomial preconditioner are given. Also, some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported.

A Multigrid NLS-4DVar Data Assimilation Scheme with Advanced Research WRF (ARW)

Science.gov (United States)

Zhang, H.; Tian, X.

2017-12-01

The motions of the atmosphere have multiscale properties in space and/or time, and the background error covariance matrix (Β) should thus contain error information at different correlation scales. To obtain an optimal analysis, the multigrid three-dimensional variational data assimilation scheme is used widely when sequentially correcting errors from large to small scales. However, introduction of the multigrid technique into four-dimensional variational data assimilation is not easy, due to its strong dependence on the adjoint model, which has extremely high computational costs in data coding, maintenance, and updating. In this study, the multigrid technique was introduced into the nonlinear least-squares four-dimensional variational assimilation (NLS-4DVar) method, which is an advanced four-dimensional ensemble-variational method that can be applied without invoking the adjoint models. The multigrid NLS-4DVar (MG-NLS-4DVar) scheme uses the number of grid points to control the scale, with doubling of this number when moving from a coarse to a finer grid. Furthermore, the MG-NLS-4DVar scheme not only retains the advantages of NLS-4DVar, but also sufficiently corrects multiscale errors to achieve a highly accurate analysis. The effectiveness and efficiency of the proposed MG-NLS-4DVar scheme were evaluated by several groups of observing system simulation experiments using the Advanced Research Weather Research and Forecasting Model. MG-NLS-4DVar outperformed NLS-4DVar, with a lower computational cost.

Efficient relaxed-Jacobi smoothers for multigrid on parallel computers

Science.gov (United States)

Yang, Xiang; Mittal, Rajat

2017-03-01

In this Technical Note, we present a family of Jacobi-based multigrid smoothers suitable for the solution of discretized elliptic equations. These smoothers are based on the idea of scheduled-relaxation Jacobi proposed recently by Yang & Mittal (2014) [18] and employ two or three successive relaxed Jacobi iterations with relaxation factors derived so as to maximize the smoothing property of these iterations. The performance of these new smoothers measured in terms of convergence acceleration and computational workload, is assessed for multi-domain implementations typical of parallelized solvers, and compared to the lexicographic point Gauss-Seidel smoother. The tests include the geometric multigrid method on structured grids as well as the algebraic grid method on unstructured grids. The tests demonstrate that unlike Gauss-Seidel, the convergence of these Jacobi-based smoothers is unaffected by domain decomposition, and furthermore, they outperform the lexicographic Gauss-Seidel by factors that increase with domain partition count.

Accelerating Lattice QCD Multigrid on GPUs Using Fine-Grained Parallelization

Energy Technology Data Exchange (ETDEWEB)

Clark, M. A. [NVIDIA Corp., Santa Clara; Joó, Bálint [Jefferson Lab; Strelchenko, Alexei [Fermilab; Cheng, Michael [Boston U., Ctr. Comp. Sci.; Gambhir, Arjun [William-Mary Coll.; Brower, Richard [Boston U.

2016-12-22

The past decade has witnessed a dramatic acceleration of lattice quantum chromodynamics calculations in nuclear and particle physics. This has been due to both significant progress in accelerating the iterative linear solvers using multi-grid algorithms, and due to the throughput improvements brought by GPUs. Deploying hierarchical algorithms optimally on GPUs is non-trivial owing to the lack of parallelism on the coarse grids, and as such, these advances have not proved multiplicative. Using the QUDA library, we demonstrate that by exposing all sources of parallelism that the underlying stencil problem possesses, and through appropriate mapping of this parallelism to the GPU architecture, we can achieve high efficiency even for the coarsest of grids. Results are presented for the Wilson-Clover discretization, where we demonstrate up to 10x speedup over present state-of-the-art GPU-accelerated methods on Titan. Finally, we look to the future, and consider the software implications of our findings.

Achieving Textbook Multigrid Efficiency for Hydrostatic Ice Sheet Flow

KAUST Repository

Brown, Jed; Smith, Barry; Ahmadia, Aron

2013-01-01

The hydrostatic equations for ice sheet flow offer improved fidelity compared with the shallow ice approximation and shallow stream approximation popular in today's ice sheet models. Nevertheless, they present a serious bottleneck because they require the solution of a three-dimensional (3D) nonlinear system, as opposed to the two-dimensional system present in the shallow stream approximation. This 3D system is posed on high-aspect domains with strong anisotropy and variation in coefficients, making it expensive to solve with current methods. This paper presents a Newton--Krylov multigrid solver for the hydrostatic equations that demonstrates textbook multigrid efficiency (an order of magnitude reduction in residual per iteration and solution of the fine-level system at a small multiple of the cost of a residual evaluation). Scalability on Blue Gene/P is demonstrated, and the method is compared to various algebraic methods that are in use or have been proposed as viable approaches.

Achieving Textbook Multigrid Efficiency for Hydrostatic Ice Sheet Flow

KAUST Repository

Brown, Jed

2013-03-12

The hydrostatic equations for ice sheet flow offer improved fidelity compared with the shallow ice approximation and shallow stream approximation popular in today\\'s ice sheet models. Nevertheless, they present a serious bottleneck because they require the solution of a three-dimensional (3D) nonlinear system, as opposed to the two-dimensional system present in the shallow stream approximation. This 3D system is posed on high-aspect domains with strong anisotropy and variation in coefficients, making it expensive to solve with current methods. This paper presents a Newton--Krylov multigrid solver for the hydrostatic equations that demonstrates textbook multigrid efficiency (an order of magnitude reduction in residual per iteration and solution of the fine-level system at a small multiple of the cost of a residual evaluation). Scalability on Blue Gene/P is demonstrated, and the method is compared to various algebraic methods that are in use or have been proposed as viable approaches.

Blockspin and multigrid for staggered fermions in non-abelian gauge fields

International Nuclear Information System (INIS)

Kalkreuter, T.; Mack, G.; Speh, M.

1991-07-01

We discuss blockspins for staggered fermions, i.e. averaging and interpolation procedures which are needed in a real space renormalization group approach to gauge theories with staggered fermions and in a multigrid approach to the computation of gauge covariant propagators. The discussion starts from the requirement that the symmetries of the free action should be preserved by the blocking procedure in the limit of a pure gauge. A definition of an averaging kernel as a solution of a gauge covariant eigenvalue equation is proposed, and the properties of a corresponding interpolation kernel are examined in the light of general criteria for good choices of blockspins. Some results of multigrid computation of bosonic propagation in an SU(2) gauge field in 4 dimensions are also presented. (orig.)

Multigrid and defect correction for the steady Navier-Stokes equations

NARCIS (Netherlands)

Koren, B.

1990-01-01

Theoretical and experimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full, steady, 2D, compressible Navier-Stokes equations. Iterative defect correction is introduced for circumventing the difficulty in

A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

Directory of Open Access Journals (Sweden)

Zhao-Li Shen

2015-01-01

Full Text Available Smoothed aggregation multigrid method is considered for computing stationary distributions of Markov chains. A judgement which determines whether to implement the whole aggregation procedure is proposed. Through this strategy, a large amount of time in the aggregation procedure is saved without affecting the convergence behavior. Besides this, we explain the shortage and irrationality of the Neighborhood-Based aggregation which is commonly used in multigrid methods. Then a modified version is presented to remedy and improve it. Numerical experiments on some typical Markov chain problems are reported to illustrate the performance of these methods.

The finite volume element (FVE) and multigrid method for the incompressible Navier-Stokes equations

International Nuclear Information System (INIS)

Gu Lizhen; Bao Weizhu

1992-01-01

The authors apply FVE method to discrete INS equations with the original variable, in which the bilinear square finite element and the square finite volume are chosen. The discrete schemes of INS equations are presented. The FMV multigrid algorithm is applied to solve that discrete system, where DGS iteration is used as smoother, DGS distributive mode for the INS discrete system is also presented. The sample problems for the square cavity flow with Reynolds number Re≤100 are successfully calculated. The numerical solutions show that the results with 1 FMV is satisfactory and when Re is not large, The FVE discrete scheme of the conservative INS equations and that of non-conservative INS equations with linearization both can provide almost same accuracy

Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration

International Nuclear Information System (INIS)

Brown, Peter N.; Shumaker, Dana E.; Woodward, Carol S.

2005-01-01

We present a solution method for fully implicit radiation diffusion problems discretized on meshes having millions of spatial zones. This solution method makes use of high order in time integration techniques, inexact Newton-Krylov nonlinear solvers, and multigrid preconditioners. We explore the advantages and disadvantages of high order time integration methods for the fully implicit formulation on both two- and three-dimensional problems with tabulated opacities and highly nonlinear fusion source terms

Colorado Conference on iterative methods. Volume 1

Energy Technology Data Exchange (ETDEWEB)

NONE

1994-12-31

The conference provided a forum on many aspects of iterative methods. Volume I topics were:Session: domain decomposition, nonlinear problems, integral equations and inverse problems, eigenvalue problems, iterative software kernels. Volume II presents nonsymmetric solvers, parallel computation, theory of iterative methods, software and programming environment, ODE solvers, multigrid and multilevel methods, applications, robust iterative methods, preconditioners, Toeplitz and circulation solvers, and saddle point problems. Individual papers are indexed separately on the EDB.

Multigrid time-accurate integration of Navier-Stokes equations

Science.gov (United States)

Arnone, Andrea; Liou, Meng-Sing; Povinelli, Louis A.

1993-01-01

Efficient acceleration techniques typical of explicit steady-state solvers are extended to time-accurate calculations. Stability restrictions are greatly reduced by means of a fully implicit time discretization. A four-stage Runge-Kutta scheme with local time stepping, residual smoothing, and multigridding is used instead of traditional time-expensive factorizations. Some applications to natural and forced unsteady viscous flows show the capability of the procedure.

Parallel Element Agglomeration Algebraic Multigrid and Upscaling Library

Energy Technology Data Exchange (ETDEWEB)

2017-10-24

ParELAG is a parallel C++ library for numerical upscaling of finite element discretizations and element-based algebraic multigrid solvers. It provides optimal complexity algorithms to build multilevel hierarchies and solvers that can be used for solving a wide class of partial differential equations (elliptic, hyperbolic, saddle point problems) on general unstructured meshes. Additionally, a novel multilevel solver for saddle point problems with divergence constraint is implemented.

Primal-Dual Interior Point Multigrid Method for Topology Optimization

Czech Academy of Sciences Publication Activity Database

Kočvara, Michal; Mohammed, S.

2016-01-01

Roč. 38, č. 5 (2016), B685-B709 ISSN 1064-8275 Grant - others:European Commission - EC(XE) 313781 Institutional support: RVO:67985556 Keywords : topology optimization * multigrid method s * interior point method Subject RIV: BA - General Mathematics Impact factor: 2.195, year: 2016 http://library.utia.cas.cz/separaty/2016/MTR/kocvara-0462418.pdf

Multi-Grid detector for neutron spectroscopy: results obtained on time-of-flight spectrometer CNCS

Science.gov (United States)

Anastasopoulos, M.; Bebb, R.; Berry, K.; Birch, J.; Bryś, T.; Buffet, J.-C.; Clergeau, J.-F.; Deen, P. P.; Ehlers, G.; van Esch, P.; Everett, S. M.; Guerard, B.; Hall-Wilton, R.; Herwig, K.; Hultman, L.; Höglund, C.; Iruretagoiena, I.; Issa, F.; Jensen, J.; Khaplanov, A.; Kirstein, O.; Lopez Higuera, I.; Piscitelli, F.; Robinson, L.; Schmidt, S.; Stefanescu, I.

2017-04-01

The Multi-Grid detector technology has evolved from the proof-of-principle and characterisation stages. Here we report on the performance of the Multi-Grid detector, the MG.CNCS prototype, which has been installed and tested at the Cold Neutron Chopper Spectrometer, CNCS at SNS. This has allowed a side-by-side comparison to the performance of 3He detectors on an operational instrument. The demonstrator has an active area of 0.2 m2. It is specifically tailored to the specifications of CNCS. The detector was installed in June 2016 and has operated since then, collecting neutron scattering data in parallel to the He-3 detectors of CNCS. In this paper, we present a comprehensive analysis of this data, in particular on instrument energy resolution, rate capability, background and relative efficiency. Stability, gamma-ray and fast neutron sensitivity have also been investigated. The effect of scattering in the detector components has been measured and provides input to comparison for Monte Carlo simulations. All data is presented in comparison to that measured by the 3He detectors simultaneously, showing that all features recorded by one detector are also recorded by the other. The energy resolution matches closely. We find that the Multi-Grid is able to match the data collected by 3He, and see an indication of a considerable advantage in the count rate capability. Based on these results, we are confident that the Multi-Grid detector will be capable of producing high quality scientific data on chopper spectrometers utilising the unprecedented neutron flux of the ESS.

Improved L-BFGS diagonal preconditioners for a large-scale 4D-Var inversion system: application to CO2 flux constraints and analysis error calculation

Science.gov (United States)

Bousserez, Nicolas; Henze, Daven; Bowman, Kevin; Liu, Junjie; Jones, Dylan; Keller, Martin; Deng, Feng

2013-04-01

This work presents improved analysis error estimates for 4D-Var systems. From operational NWP models to top-down constraints on trace gas emissions, many of today's data assimilation and inversion systems in atmospheric science rely on variational approaches. This success is due to both the mathematical clarity of these formulations and the availability of computationally efficient minimization algorithms. However, unlike Kalman Filter-based algorithms, these methods do not provide an estimate of the analysis or forecast error covariance matrices, these error statistics being propagated only implicitly by the system. From both a practical (cycling assimilation) and scientific perspective, assessing uncertainties in the solution of the variational problem is critical. For large-scale linear systems, deterministic or randomization approaches can be considered based on the equivalence between the inverse Hessian of the cost function and the covariance matrix of analysis error. For perfectly quadratic systems, like incremental 4D-Var, Lanczos/Conjugate-Gradient algorithms have proven to be most efficient in generating low-rank approximations of the Hessian matrix during the minimization. For weakly non-linear systems though, the Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), a quasi-Newton descent algorithm, is usually considered the best method for the minimization. Suitable for large-scale optimization, this method allows one to generate an approximation to the inverse Hessian using the latest m vector/gradient pairs generated during the minimization, m depending upon the available core memory. At each iteration, an initial low-rank approximation to the inverse Hessian has to be provided, which is called preconditioning. The ability of the preconditioner to retain useful information from previous iterations largely determines the efficiency of the algorithm. Here we assess the performance of different preconditioners to estimate the inverse Hessian of a

Preconditioner considerations for an aerodynamic Newton-Krylov solver

International Nuclear Information System (INIS)

Chisholm, T.; Zingg, D.W.

2003-01-01

A fast Newton-Krylov algorithm is presented for solving the compressible Navier-Stokes equations on structured multi-block grids with application to turbulent aerodynamic flows. The one-equation Spalart-Allmaras model is used to provide the turbulent viscosity. The optimization of the algorithm is discussed. ILU(4) is suggested for a preconditioner, operating on a modified Jacobian matrix. An RCM reordering is used, with a suggested root node in the wake. The advantages of a matrix-free technique for forming matrix-vector products are shown. Three test cases are used to demonstrate convergence rates. Single-element cases are solved in less than 60 seconds on a desktop computer, while the solution of a multi-element case can be found in about 10 minutes. (author)

AZTEC: A parallel iterative package for the solving linear systems

Energy Technology Data Exchange (ETDEWEB)

Hutchinson, S.A.; Shadid, J.N.; Tuminaro, R.S. [Sandia National Labs., Albuquerque, NM (United States)

1996-12-31

We describe a parallel linear system package, AZTEC. The package incorporates a number of parallel iterative methods (e.g. GMRES, biCGSTAB, CGS, TFQMR) and preconditioners (e.g. Jacobi, Gauss-Seidel, polynomial, domain decomposition with LU or ILU within subdomains). Additionally, AZTEC allows for the reuse of previous preconditioning factorizations within Newton schemes for nonlinear methods. Currently, a number of different users are using this package to solve a variety of PDE applications.

A Parallel Multigrid Solver for Viscous Flows on Anisotropic Structured Grids

Science.gov (United States)

Prieto, Manuel; Montero, Ruben S.; Llorente, Ignacio M.; Bushnell, Dennis M. (Technical Monitor)

2001-01-01

This paper presents an efficient parallel multigrid solver for speeding up the computation of a 3-D model that treats the flow of a viscous fluid over a flat plate. The main interest of this simulation lies in exhibiting some basic difficulties that prevent optimal multigrid efficiencies from being achieved. As the computing platform, we have used Coral, a Beowulf-class system based on Intel Pentium processors and equipped with GigaNet cLAN and switched Fast Ethernet networks. Our study not only examines the scalability of the solver but also includes a performance evaluation of Coral where the investigated solver has been used to compare several of its design choices, namely, the interconnection network (GigaNet versus switched Fast-Ethernet) and the node configuration (dual nodes versus single nodes). As a reference, the performance results have been compared with those obtained with the NAS-MG benchmark.

An Optimal Order Nonnested Mixed Multigrid Method for Generalized Stokes Problems

Science.gov (United States)

Deng, Qingping

1996-01-01

A multigrid algorithm is developed and analyzed for generalized Stokes problems discretized by various nonnested mixed finite elements within a unified framework. It is abstractly proved by an element-independent analysis that the multigrid algorithm converges with an optimal order if there exists a 'good' prolongation operator. A technique to construct a 'good' prolongation operator for nonnested multilevel finite element spaces is proposed. Its basic idea is to introduce a sequence of auxiliary nested multilevel finite element spaces and define a prolongation operator as a composite operator of two single grid level operators. This makes not only the construction of a prolongation operator much easier (the final explicit forms of such prolongation operators are fairly simple), but the verification of the approximate properties for prolongation operators is also simplified. Finally, as an application, the framework and technique is applied to seven typical nonnested mixed finite elements.

Fast Multipole-Based Preconditioner for Sparse Iterative Solvers

KAUST Repository

Ibeid, Huda; Yokota, Rio; Keyes, David E.

2014-01-01

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxed global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Compared with multilevel methods, it is capable of comparable algebraic convergence rates down to the truncation error of the discretized PDE, and it has superior multicore and distributed memory scalability properties on commodity architecture supercomputers.

Fast Multipole-Based Preconditioner for Sparse Iterative Solvers

KAUST Repository

Ibeid, Huda

2014-05-04

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxed global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Compared with multilevel methods, it is capable of comparable algebraic convergence rates down to the truncation error of the discretized PDE, and it has superior multicore and distributed memory scalability properties on commodity architecture supercomputers.

Preconditioners for regularized saddle point problems with an application for heterogeneous Darcy flow problems

Czech Academy of Sciences Publication Activity Database

Axelsson, Owe; Blaheta, Radim; Byczanski, Petr; Karátson, J.; Ahmad, B.

2015-01-01

Roč. 280, č. 280 (2015), s. 141-157 ISSN 0377-0427 R&D Projects: GA MŠk ED1.1.00/02.0070 Institutional support: RVO:68145535 Keywords : preconditioners * heterogeneous coefficients * regularized saddle point Inner–outer iterations * Darcy flow Subject RIV: BA - General Mathematics Impact factor: 1.328, year: 2015 http://www.sciencedirect.com/science/article/pii/S0377042714005238

Multigrid Finite Element Method in Calculation of 3D Homogeneous and Composite Solids

Directory of Open Access Journals (Sweden)

A.D. Matveev

2016-12-01

Full Text Available In the present paper, a method of multigrid finite elements to calculate elastic three-dimensional homogeneous and composite solids under static loading has been suggested. The method has been developed based on the finite element method algorithms using homogeneous and composite three-dimensional multigrid finite elements (MFE. The procedures for construction of MFE of both rectangular parallelepiped and complex shapes have been shown. The advantages of MFE are that they take into account, following the rules of the microapproach, heterogeneous and microhomogeneous structures of the bodies, describe the three-dimensional stress-strain state (without any simplifying hypotheses in homogeneous and composite solids, as well as generate small dimensional discrete models and numerical solutions with a high accuracy.

Optimal multigrid algorithms for the massive Gaussian model and path integrals

International Nuclear Information System (INIS)

Brandt, A.; Galun, M.

1996-01-01

Multigrid algorithms are presented which, in addition to eliminating the critical slowing down, can also eliminate the open-quotes volume factorclose quotes. The elimination of the volume factor removes the need to produce many independent fine-grid configurations for averaging out their statistical deviations, by averaging over the many samples produced on coarse grids during the multigrid cycle. Thermodynamic limits of observables can be calculated to relative accuracy var-epsilon r in just O(var-epsilon r -2 ) computer operations, where var-epsilon r is the error relative to the standard deviation of the observable. In this paper, we describe in detail the calculation of the susceptibility in the one-dimensional massive Gaussian model, which is also a simple example of path integrals. Numerical experiments show that the susceptibility can be calculated to relative accuracy var-epsilon r in about 8 var-epsilon r -2 random number generations, independent of the mass size

Multilevel local refinement and multigrid methods for 3-D turbulent flow

Energy Technology Data Exchange (ETDEWEB)

Liao, C.; Liu, C. [UCD, Denver, CO (United States); Sung, C.H.; Huang, T.T. [David Taylor Model Basin, Bethesda, MD (United States)

1996-12-31

A numerical approach based on multigrid, multilevel local refinement, and preconditioning methods for solving incompressible Reynolds-averaged Navier-Stokes equations is presented. 3-D turbulent flow around an underwater vehicle is computed. 3 multigrid levels and 2 local refinement grid levels are used. The global grid is 24 x 8 x 12. The first patch is 40 x 16 x 20 and the second patch is 72 x 32 x 36. 4th order artificial dissipation are used for numerical stability. The conservative artificial compressibility method are used for further improvement of convergence. To improve the accuracy of coarse/fine grid interface of local refinement, flux interpolation method for refined grid boundary is used. The numerical results are in good agreement with experimental data. The local refinement can improve the prediction accuracy significantly. The flux interpolation method for local refinement can keep conservation for a composite grid, therefore further modify the prediction accuracy.

Adjacent-cell Preconditioners for solving optically thick neutron transport problems

International Nuclear Information System (INIS)

Azmy, Y.Y.

1994-01-01

We develop, analyze, and test a new acceleration scheme for neutron transport methods, the Adjacent-cell Preconditioner (AP) that is particularly suited for solving optically thick problems. Our method goes beyond Diffusion Synthetic Acceleration (DSA) methods in that it's spectral radius vanishes with increasing cell thickness. In particular, for the ID case the AP method converges immediately, i.e. in one iteration, to 10 -4 pointwise relative criterion in problems with dominant cell size of 10 mfp or thicker. Also the AP has a simple formalism and is cell-centered hence, multidimensional and high order extensions are easier to develop, and more efficient to implement

Multigrid solution of the Navier-Stokes equations at low speeds with large temperature variations

International Nuclear Information System (INIS)

Sockol, Peter M.

2003-01-01

Multigrid methods for the Navier-Stokes equations at low speeds and large temperature variations are investigated. The compressible equations with time-derivative preconditioning and preconditioned flux-difference splitting of the inviscid terms are used. Three implicit smoothers have been incorporated into a common multigrid procedure. Both full coarsening and semi-coarsening with directional fine-grid defect correction have been studied. The resulting methods have been tested on four 2D laminar problems over a range of Reynolds numbers on both uniform and highly stretched grids. Two of the three methods show efficient and robust performance over the entire range of conditions. In addition, none of the methods has any difficulty with the large temperature variations

On the use of diffusion synthetic acceleration in parallel 3D neutral particle transport calculations

International Nuclear Information System (INIS)

Brown, P.; Chang, B.

1998-01-01

The linear Boltzmann transport equation (BTE) is an integro-differential equation arising in deterministic models of neutral and charged particle transport. In slab (one-dimensional Cartesian) geometry and certain higher-dimensional cases, Diffusion Synthetic Acceleration (DSA) is known to be an effective algorithm for the iterative solution of the discretized BTE. Fourier and asymptotic analyses have been applied to various idealizations (e.g., problems on infinite domains with constant coefficients) to obtain sharp bounds on the convergence rate of DSA in such cases. While DSA has been shown to be a highly effective acceleration (or preconditioning) technique in one-dimensional problems, it has been observed to be less effective in higher dimensions. This is due in part to the expense of solving the related diffusion linear system. We investigate here the effectiveness of a parallel semicoarsening multigrid (SMG) solution approach to DSA preconditioning in several three dimensional problems. In particular, we consider the algorithmic and implementation scalability of a parallel SMG-DSA preconditioner on several types of test problems

Final Report - Subcontract B6183769

Energy Technology Data Exchange (ETDEWEB)

Bank, R. [Univ. of California, San Diego, CA (United States)

2017-06-12

During my visit to LLNL during July 5{15, 2016, we worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear system arising from hp adaptive finite element calculations, and is implemented in the PLTMG software package, version 12 [1]. This preconditioner typically requires 3-20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive definite elliptic equations, convection dominated convection-diffusion equations, and highly inde nite Helmholtz equations, among others). The convergence rate is not independent of the polynomial degree p as p ! 1, but but remains strong for p 9, which is the highest polynomial degree allowed in PLTMG, due to limitations of numerical quadrature formulae. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in [2]. Like traditional geometric multilevel methods, this scheme relies on extensive knowledge of the underlying finite element space in order to construct both the smoother and the coarse grid correction components.

Final Report - Subcontract B623760

Energy Technology Data Exchange (ETDEWEB)

Bank, R. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2017-11-17

During my visit to LLNL during July 17{27, 2017, I worked on linear system solvers. The two level hierarchical solver that initiated our study was developed to solve linear systems arising from hp adaptive finite element calculations, and is implemented in the PLTMG software package, version 12. This preconditioner typically requires 3-20% of the space used by the stiffness matrix for higher order elements. It has multigrid like convergence rates for a wide variety of PDEs (self-adjoint positive de nite elliptic equations, convection dominated convection-diffusion equations, and highly indefinite Helmholtz equations, among others). The convergence rate is not independent of the polynomial degree p as p ! 1, but but remains strong for p 9, which is the highest polynomial degree allowed in PLTMG, due to limitations of the numerical quadrature rules implemented in the software package. A more complete description of the method and some numerical experiments illustrating its effectiveness appear in. Like traditional geometric multilevel methods, this scheme relies on knowledge of the underlying finite element space in order to construct the smoother and the coarse grid correction.

Discontinuous diffusion synthetic acceleration for Sn transport on 2D arbitrary polygonal meshes

International Nuclear Information System (INIS)

Turcksin, Bruno; Ragusa, Jean C.

2014-01-01

In this paper, a Diffusion Synthetic Acceleration (DSA) technique applied to the S n radiation transport equation is developed using Piece-Wise Linear Discontinuous (PWLD) finite elements on arbitrary polygonal grids. The discretization of the DSA equations employs an Interior Penalty technique, as is classically done for the stabilization of the diffusion equation using discontinuous finite element approximations. The penalty method yields a system of linear equations that is Symmetric Positive Definite (SPD). Thus, solution techniques such as Preconditioned Conjugate Gradient (PCG) can be effectively employed. Algebraic MultiGrid (AMG) and Symmetric Gauss–Seidel (SGS) are employed as conjugate gradient preconditioners for the DSA system. AMG is shown to be significantly more efficient than SGS. Fourier analyses are carried out and we show that this discontinuous finite element DSA scheme is always stable and effective at reducing the spectral radius for iterative transport solves, even for grids with high-aspect ratio cells. Numerical results are presented for different grid types: quadrilateral, hexagonal, and polygonal grids as well as grids with local mesh adaptivity

HP-Multigrid as Smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II: Optimization of the Runge-Kutta smoother

NARCIS (Netherlands)

van der Vegt, Jacobus J.W.; Rhebergen, Sander

2012-01-01

Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit

Non-Galerkin Coarse Grids for Algebraic Multigrid

Energy Technology Data Exchange (ETDEWEB)

Falgout, Robert D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Schroder, Jacob B. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2014-06-26

Algebraic multigrid (AMG) is a popular and effective solver for systems of linear equations that arise from discretized partial differential equations. And while AMG has been effectively implemented on large scale parallel machines, challenges remain, especially when moving to exascale. Particularly, stencil sizes (the number of nonzeros in a row) tend to increase further down in the coarse grid hierarchy, and this growth leads to more communication. Therefore, as problem size increases and the number of levels in the hierarchy grows, the overall efficiency of the parallel AMG method decreases, sometimes dramatically. This growth in stencil size is due to the standard Galerkin coarse grid operator, $P^T A P$, where $P$ is the prolongation (i.e., interpolation) operator. For example, the coarse grid stencil size for a simple three-dimensional (3D) seven-point finite differencing approximation to diffusion can increase into the thousands on present day machines, causing an associated increase in communication costs. We therefore consider algebraically truncating coarse grid stencils to obtain a non-Galerkin coarse grid. First, the sparsity pattern of the non-Galerkin coarse grid is determined by employing a heuristic minimal “safe” pattern together with strength-of-connection ideas. Second, the nonzero entries are determined by collapsing the stencils in the Galerkin operator using traditional AMG techniques. The result is a reduction in coarse grid stencil size, overall operator complexity, and parallel AMG solve phase times.

Recent development of the Multi-Grid detector for large area neutron scattering instruments

International Nuclear Information System (INIS)

Guerard, Bruno

2015-01-01

Most of the Neutron Scattering facilities are committed in a continuous program of modernization of their instruments, requiring large area and high performance thermal neutron detectors. Beside scintillators detectors, 3 He detectors, like linear PSDs (Position Sensitive Detectors) and MWPCs (Multi-Wires Proportional Chambers), are the most current techniques nowadays. Time Of Flight instruments are using 3 He PSDs mounted side by side to cover tens of m 2 . As a result of the so-called ' 3 He shortage crisis , the volume of 3He which is needed to build one of these instruments is not accessible anymore. The development of alternative techniques requiring no 3He, has been given high priority to secure the future of neutron scattering instrumentation. This is particularly important in the context where the future ESS (European Spallation Source) will start its operation in 2019-2020. Improved scintillators represent one of the alternative techniques. Another one is the Multi-Grid introduced at the ILL in 2009. A Multi-Grid detector is composed of several independent modules of typically 0.8 m x 3 m sensitive area, mounted side by side in air or in a vacuum TOF chamber. One module is composed of segmented boron-lined proportional counters mounted in a gas vessel; the counters, of square section, are assembled with Aluminium grids electrically insulated and stacked together. This design provides two advantages: First, magnetron sputtering techniques can be used to coat B 4 C films on planar substrates, and second, the neutron position along the anode wires can be measured by reading out individually the grid signals with fast shaping amplifiers followed by comparators. Unlike charge division localisation in linear PSDs, the individual readout of the grids allows operating the Multi-Grid at a low amplification gain, hence this detector is tolerant to mechanical defects and its production accessible to laboratories equipped with standard equipment. Prototypes of

Recent development of the Multi-Grid detector for large area neutron scattering instruments

Energy Technology Data Exchange (ETDEWEB)

Guerard, Bruno [ILL-ESS-LiU collaboration, CRISP project, Institut Laue Langevin - ILL, Grenoble (France)

2015-07-01

Most of the Neutron Scattering facilities are committed in a continuous program of modernization of their instruments, requiring large area and high performance thermal neutron detectors. Beside scintillators detectors, {sup 3}He detectors, like linear PSDs (Position Sensitive Detectors) and MWPCs (Multi-Wires Proportional Chambers), are the most current techniques nowadays. Time Of Flight instruments are using {sup 3}He PSDs mounted side by side to cover tens of m{sup 2}. As a result of the so-called '{sup 3}He shortage crisis{sup ,} the volume of 3He which is needed to build one of these instruments is not accessible anymore. The development of alternative techniques requiring no 3He, has been given high priority to secure the future of neutron scattering instrumentation. This is particularly important in the context where the future ESS (European Spallation Source) will start its operation in 2019-2020. Improved scintillators represent one of the alternative techniques. Another one is the Multi-Grid introduced at the ILL in 2009. A Multi-Grid detector is composed of several independent modules of typically 0.8 m x 3 m sensitive area, mounted side by side in air or in a vacuum TOF chamber. One module is composed of segmented boron-lined proportional counters mounted in a gas vessel; the counters, of square section, are assembled with Aluminium grids electrically insulated and stacked together. This design provides two advantages: First, magnetron sputtering techniques can be used to coat B{sub 4}C films on planar substrates, and second, the neutron position along the anode wires can be measured by reading out individually the grid signals with fast shaping amplifiers followed by comparators. Unlike charge division localisation in linear PSDs, the individual readout of the grids allows operating the Multi-Grid at a low amplification gain, hence this detector is tolerant to mechanical defects and its production accessible to laboratories equipped with standard

An acceleration technique for the Gauss-Seidel method applied to symmetric linear systems

Directory of Open Access Journals (Sweden)

Jesús Cajigas

2014-06-01

Full Text Available A preconditioning technique to improve the convergence of the Gauss-Seidel method applied to symmetric linear systems while preserving symmetry is proposed. The preconditioner is of the form I + K and can be applied an arbitrary number of times. It is shown that under certain conditions the application of the preconditioner a finite number of steps reduces the matrix to a diagonal. A series of numerical experiments using matrices from spatial discretizations of partial differential equations demonstrates that both versions of the preconditioner, point and block version, exhibit lower iteration counts than its non-symmetric version. Resumen. Se propone una técnica de precondicionamiento para mejorar la convergencia del método Gauss-Seidel aplicado a sistemas lineales simétricos pero preservando simetría. El precondicionador es de la forma I + K y puede ser aplicado un número arbitrario de veces. Se demuestra que bajo ciertas condiciones la aplicación del precondicionador un número finito de pasos reduce la matriz del sistema precondicionado a una diagonal. Una serie de experimentos con matrices que provienen de la discretización de ecuaciones en derivadas parciales muestra que ambas versiones del precondicionador, por punto y por bloque, muestran un menor número de iteraciones en comparación con la versión que no preserva simetría.

Multi-Level iterative methods in computational plasma physics

International Nuclear Information System (INIS)

Knoll, D.A.; Barnes, D.C.; Brackbill, J.U.; Chacon, L.; Lapenta, G.

1999-01-01

Plasma physics phenomena occur on a wide range of spatial scales and on a wide range of time scales. When attempting to model plasma physics problems numerically the authors are inevitably faced with the need for both fine spatial resolution (fine grids) and implicit time integration methods. Fine grids can tax the efficiency of iterative methods and large time steps can challenge the robustness of iterative methods. To meet these challenges they are developing a hybrid approach where multigrid methods are used as preconditioners to Krylov subspace based iterative methods such as conjugate gradients or GMRES. For nonlinear problems they apply multigrid preconditioning to a matrix-few Newton-GMRES method. Results are presented for application of these multilevel iterative methods to the field solves in implicit moment method PIC, multidimensional nonlinear Fokker-Planck problems, and their initial efforts in particle MHD

Multigrid direct numerical simulation of the whole process of flow transition in 3-D boundary layers

Science.gov (United States)

Liu, Chaoqun; Liu, Zhining

1993-01-01

A new technology was developed in this study which provides a successful numerical simulation of the whole process of flow transition in 3-D boundary layers, including linear growth, secondary instability, breakdown, and transition at relatively low CPU cost. Most other spatial numerical simulations require high CPU cost and blow up at the stage of flow breakdown. A fourth-order finite difference scheme on stretched and staggered grids, a fully implicit time marching technique, a semi-coarsening multigrid based on the so-called approximate line-box relaxation, and a buffer domain for the outflow boundary conditions were all used for high-order accuracy, good stability, and fast convergence. A new fine-coarse-fine grid mapping technique was developed to keep the code running after the laminar flow breaks down. The computational results are in good agreement with linear stability theory, secondary instability theory, and some experiments. The cost for a typical case with 162 x 34 x 34 grid is around 2 CRAY-YMP CPU hours for 10 T-S periods.

Fast resolution of the neutron diffusion equation through public domain Ode codes

Energy Technology Data Exchange (ETDEWEB)

Garcia, V.M.; Vidal, V.; Garayoa, J. [Universidad Politecnica de Valencia, Departamento de Sistemas Informaticos, Valencia (Spain); Verdu, G. [Universidad Politecnica de Valencia, Departamento de Ingenieria Quimica y Nuclear, Valencia (Spain); Gomez, R. [I.E.S. de Tavernes Blanques, Valencia (Spain)

2003-07-01

The time-dependent neutron diffusion equation is a partial differential equation with source terms. The resolution method usually includes discretizing the spatial domain, obtaining a large system of linear, stiff ordinary differential equations (ODEs), whose resolution is computationally very expensive. Some standard techniques use a fixed time step to solve the ODE system. This can result in errors (if the time step is too large) or in long computing times (if the time step is too little). To speed up the resolution method, two well-known public domain codes have been selected: DASPK and FCVODE that are powerful codes for the resolution of large systems of stiff ODEs. These codes can estimate the error after each time step, and, depending on this estimation can decide which is the new time step and, possibly, which is the integration method to be used in the next step. With these mechanisms, it is possible to keep the overall error below the chosen tolerances, and, when the system behaves smoothly, to take large time steps increasing the execution speed. In this paper we address the use of the public domain codes DASPK and FCVODE for the resolution of the time-dependent neutron diffusion equation. The efficiency of these codes depends largely on the preconditioning of the big systems of linear equations that must be solved. Several pre-conditioners have been programmed and tested; it was found that the multigrid method is the best of the pre-conditioners tested. Also, it has been found that DASPK has performed better than FCVODE, being more robust for our problem.We can conclude that the use of specialized codes for solving large systems of ODEs can reduce drastically the computational work needed for the solution; and combining them with appropriate pre-conditioners, the reduction can be still more important. It has other crucial advantages, since it allows the user to specify the allowed error, which cannot be done in fixed step implementations; this, of course

Scalable parallel elastic-plastic finite element analysis using a quasi-Newton method with a balancing domain decomposition preconditioner

Science.gov (United States)

Yusa, Yasunori; Okada, Hiroshi; Yamada, Tomonori; Yoshimura, Shinobu

2018-04-01

A domain decomposition method for large-scale elastic-plastic problems is proposed. The proposed method is based on a quasi-Newton method in conjunction with a balancing domain decomposition preconditioner. The use of a quasi-Newton method overcomes two problems associated with the conventional domain decomposition method based on the Newton-Raphson method: (1) avoidance of a double-loop iteration algorithm, which generally has large computational complexity, and (2) consideration of the local concentration of nonlinear deformation, which is observed in elastic-plastic problems with stress concentration. Moreover, the application of a balancing domain decomposition preconditioner ensures scalability. Using the conventional and proposed domain decomposition methods, several numerical tests, including weak scaling tests, were performed. The convergence performance of the proposed method is comparable to that of the conventional method. In particular, in elastic-plastic analysis, the proposed method exhibits better convergence performance than the conventional method.

Multigrid and defect correction for the steady Navier-Stokes equations : application to aerodynamics

NARCIS (Netherlands)

Koren, B.

1991-01-01

Theoretical and expcrimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full, steady, 2D, compressible NavierStokes equations. lterative defect correction is introduced for circumventing the difficulty in

Compiler generation and autotuning of communication-avoiding operators for geometric multigrid

Energy Technology Data Exchange (ETDEWEB)

Basu, Protonu [Univ. of Utah, Salt Lake City, UT (United States); Venkat, Anand [Univ. of Utah, Salt Lake City, UT (United States); Hall, Mary [Univ. of Utah, Salt Lake City, UT (United States); Williams, Samuel [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Van Straalen, Brian [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Oliker, Leonid [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

2014-04-17

This paper describes a compiler approach to introducing communication-avoiding optimizations in geometric multigrid (GMG), one of the most popular methods for solving partial differential equations. Communication-avoiding optimizations reduce vertical communication through the memory hierarchy and horizontal communication across processes or threads, usually at the expense of introducing redundant computation. We focus on applying these optimizations to the smooth operator, which successively reduces the error and accounts for the largest fraction of the GMG execution time. Our compiler technology applies both novel and known transformations to derive an implementation comparable to manually-tuned code. To make the approach portable, an underlying autotuning system explores the tradeoff between reduced communication and increased computation, as well as tradeoffs in threading schemes, to automatically identify the best implementation for a particular architecture and at each computation phase. Results show that we are able to quadruple the performance of the smooth operation on the finest grids while attaining performance within 94% of manually-tuned code. Overall we improve the overall multigrid solve time by 2.5× without sacrificing programer productivity.

Cellular Automaton Modeling of Dendritic Growth Using a Multi-grid Method

International Nuclear Information System (INIS)

Natsume, Y; Ohsasa, K

2015-01-01

A two-dimensional cellular automaton model with a multi-grid method was developed to simulate dendritic growth. In the present model, we used a triple-grid system for temperature, solute concentration and solid fraction fields as a new approach of the multi-grid method. In order to evaluate the validity of the present model, we carried out simulations of single dendritic growth, secondary dendrite arm growth, multi-columnar dendritic growth and multi-equiaxed dendritic growth. From the results of the grid dependency from the simulation of single dendritic growth, we confirmed that the larger grid can be used in the simulation and that the computational time can be reduced dramatically. In the simulation of secondary dendrite arm growth, the results from the present model were in good agreement with the experimental data and the simulated results from a phase-field model. Thus, the present model can quantitatively simulate dendritic growth. From the simulated results of multi-columnar and multi-equiaxed dendrites, we confirmed that the present model can perform simulations under practical solidification conditions. (paper)

Construction, classification and parametrization of complex Hadamard matrices

Science.gov (United States)

Szöllősi, Ferenc

To improve the design of nuclear systems, high-fidelity neutron fluxes are required. Leadership-class machines provide platforms on which very large problems can be solved. Computing such fluxes efficiently requires numerical methods with good convergence properties and algorithms that can scale to hundreds of thousands of cores. Many 3-D deterministic transport codes are decomposable in space and angle only, limiting them to tens of thousands of cores. Most codes rely on methods such as Gauss Seidel for fixed source problems and power iteration for eigenvalue problems, which can be slow to converge for challenging problems like those with highly scattering materials or high dominance ratios. Three methods have been added to the 3-D SN transport code Denovo that are designed to improve convergence and enable the full use of cutting-edge computers. The first is a multigroup Krylov solver that converges more quickly than Gauss Seidel and parallelizes the code in energy such that Denovo can use hundreds of thousand of cores effectively. The second is Rayleigh quotient iteration (RQI), an old method applied in a new context. This eigenvalue solver finds the dominant eigenvalue in a mathematically optimal way and should converge in fewer iterations than power iteration. RQI creates energy-block-dense equations that the new Krylov solver treats efficiently. However, RQI can have convergence problems because it creates poorly conditioned systems. This can be overcome with preconditioning. The third method is a multigrid-in-energy preconditioner. The preconditioner takes advantage of the new energy decomposition because the grids are in energy rather than space or angle. The preconditioner greatly reduces iteration count for many problem types and scales well in energy. It also allows RQI to be successful for problems it could not solve otherwise. The methods added to Denovo accomplish the goals of this work. They converge in fewer iterations than traditional methods and

Efficient parallel iterative solvers for the solution of large dense linear systems arising from the boundary element method in electromagnetism

International Nuclear Information System (INIS)

Alleon, G.; Carpentieri, B.; Du, I.S.; Giraud, L.; Langou, J.; Martin, E.

2003-01-01

The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. It discretizes only the surface of the radiating object and gives rise to linear systems that are smaller in size compared to those arising from finite element or finite difference discretizations. However, these systems are prohibitively demanding in terms of memory for direct methods and challenging to solve by iterative methods. In this paper we address the iterative solution via preconditioned Krylov methods of electromagnetic scattering problems expressed in an integral formulation, with main focus on the design of the pre-conditioner. We consider an approximate inverse method based on the Frobenius-norm minimization with a pattern prescribed in advance. The pre-conditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the pre-conditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe the implementation of the approximate inverse in an out-of-core parallel code that uses multipole techniques for the matrix-vector products, and show results on the numerical scalability of our method on systems of size up to one million unknowns. We propose an embedded iterative scheme based on the GMRES method and combined with multipole techniques, aimed at improving the robustness of the approximate inverse for large problems. We prove by numerical experiments that the proposed scheme enables the solution of very large and difficult problems efficiently at reduced computational and memory cost. Finally we perform a preliminary study on a spectral two-level pre-conditioner to enhance the robustness of our method. This numerical technique exploits spectral information of the preconditioned systems to build a low rank-update of the pre-conditioner. (authors)

Efficient parallel iterative solvers for the solution of large dense linear systems arising from the boundary element method in electromagnetism

Energy Technology Data Exchange (ETDEWEB)

Alleon, G. [EADS-CCR, 31 - Blagnac (France); Carpentieri, B.; Du, I.S.; Giraud, L.; Langou, J.; Martin, E. [Cerfacs, 31 - Toulouse (France)

2003-07-01

The boundary element method has become a popular tool for the solution of Maxwell's equations in electromagnetism. It discretizes only the surface of the radiating object and gives rise to linear systems that are smaller in size compared to those arising from finite element or finite difference discretizations. However, these systems are prohibitively demanding in terms of memory for direct methods and challenging to solve by iterative methods. In this paper we address the iterative solution via preconditioned Krylov methods of electromagnetic scattering problems expressed in an integral formulation, with main focus on the design of the pre-conditioner. We consider an approximate inverse method based on the Frobenius-norm minimization with a pattern prescribed in advance. The pre-conditioner is constructed from a sparse approximation of the dense coefficient matrix, and the patterns both for the pre-conditioner and for the coefficient matrix are computed a priori using geometric information from the mesh. We describe the implementation of the approximate inverse in an out-of-core parallel code that uses multipole techniques for the matrix-vector products, and show results on the numerical scalability of our method on systems of size up to one million unknowns. We propose an embedded iterative scheme based on the GMRES method and combined with multipole techniques, aimed at improving the robustness of the approximate inverse for large problems. We prove by numerical experiments that the proposed scheme enables the solution of very large and difficult problems efficiently at reduced computational and memory cost. Finally we perform a preliminary study on a spectral two-level pre-conditioner to enhance the robustness of our method. This numerical technique exploits spectral information of the preconditioned systems to build a low rank-update of the pre-conditioner. (authors)

Multi-grid Beam and Warming scheme for the simulation of unsteady ...

African Journals Online (AJOL)

In this paper, a multi-grid algorithm is applied to a large-scale block matrix that is produced from a Beam and Warming scheme. The Beam and Warming scheme is used in the simulation of unsteady flow in an open channel. The Gauss-Seidel block-wise iteration method is used for a smoothing process with a few iterations.

Multigrid methods for fully implicit oil reservoir simulation

Energy Technology Data Exchange (ETDEWEB)

Molenaar, J.

1995-12-31

In this paper, the authors consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations the material balance or continuity equations, and the equation of motion (Darcy`s law). For the numerical solution of this system of nonlinear partial differential equations, there are two approaches: the fully implicit or simultaneous solution method, and the sequential solution method. In this paper, the authors consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations.

Multigrid Methods for EHL Problems

Science.gov (United States)

Nurgat, Elyas; Berzins, Martin

1996-01-01

In many bearings and contacts, forces are transmitted through thin continuous fluid films which separate two contacting elements. Objects in contact are normally subjected to friction and wear which can be reduced effectively by using lubricants. If the lubricant film is sufficiently thin to prevent the opposing solids from coming into contact and carries the entire load, then we have hydrodynamic lubrication, where the lubricant film is determined by the motion and geometry of the solids. However, for loaded contacts of low geometrical conformity, such as gears, rolling contact bearings and cams, this is not the case due to high pressures and this is referred to as Elasto-Hydrodynamic Lubrication (EHL) In EHL, elastic deformation of the contacting elements and the increase in fluid viscosity with pressure are very significant and cannot be ignored. Since the deformation results in changing the geometry of the lubricating film, which in turn determines the pressure distribution, an EHL mathematical model must simultaneously satisfy the complex elasticity (integral) and the Reynolds lubrication (differential) equations. The nonlinear and coupled nature of the two equations makes numerical calculations computationally intensive. This is especially true for highly loaded problems found in practice. One novel feature of these problems is that the solution may exhibit sharp pressure spikes in the outlet region. To this date both finite element and finite difference methods have been used to solve EHL problems with perhaps greater emphasis on the use of the finite difference approach. In both cases, a major computational difficulty is ensuring convergence of the nonlinear equations solver to a steady state solution. Two successful methods for achieving this are direct iteration and multigrid methods. Direct iteration methods (e.g Gauss Seidel) have long been used in conjunction with finite difference discretizations on regular meshes. Perhaps one of the best examples of

Survey on efficient linear solvers for porous media flow models on recent hardware architectures

International Nuclear Information System (INIS)

Anciaux-Sedrakian, Ani; Gratien, Jean-Marc; Guignon, Thomas; Gottschling, Peter

2014-01-01

In the past few years, High Performance Computing (HPC) technologies led to General Purpose Processing on Graphics Processing Units (GPGPU) and many-core architectures. These emerging technologies offer massive processing units and are interesting for porous media flow simulators may used for CO 2 geological sequestration or Enhanced Oil Recovery (EOR) simulation. However the crucial point is 'are current algorithms and software able to use these new technologies efficiently?' The resolution of large sparse linear systems, almost ill-conditioned, constitutes the most CPU-consuming part of such simulators. This paper proposes a survey on various solver and pre-conditioner algorithms, analyzes their efficiency and performance regarding these distinct architectures. Furthermore it proposes a novel approach based on a hybrid programming model for both GPU and many-core clusters. The proposed optimization techniques are validated through a Krylov subspace solver; BiCGStab and some pre-conditioners like ILU0 on GPU, multi-core and many-core architectures, on various large real study cases in EOR simulation. (authors)

Multigrid preconditioning of the generator two-phase mixture balance equations in the Genepi software

International Nuclear Information System (INIS)

Belliard, M.; Grandotto, M.

2003-01-01

In the framework of the two-phase fluid simulations of the steam generators of pressurized water nuclear reactors, we present in this paper a geometric version of a pseudo-Full MultiGrid (pseudo- FMG) Full Approximation Storage (FAS) preconditioning of balance equations in the GENEPI code. In our application, the 3D steady state flow is reached by a transient computation using a semi-implicit fractional step algorithm for the averaged two-phase mixture balance equations (mass, momentum and energy for the secondary flow). Our application, running on workstation clusters, is based on a CEA code-linker and the PVM package. The difficulties to apply the geometric FAS multigrid method to the momentum and mass balance equations are addressed. The use of a sequential pseudo-FMG FAS twogrid method for both energy and mass/momentum balance equations, using dynamic multigrid cycles, leads to perceptibly improvements in the computation convergences. An original parallel red-black pseudo-FMG FAS three-grid algorithm is presented too. The numerical tests (steam generator mockup simulations) underline the sizable increase in speed of convergence of the computations, essentially for the ones involving a large number of freedom degrees (about 100 thousand cells). The two-phase mixture balance equation residuals are quickly reduced: the reached speed-up stands between 2 and 3 following the number of grids. The effects on the convergence behavior of the numerical parameters are investigated

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

NARCIS (Netherlands)

P. Luo (Peiyao); C. Rodrigo (Carmen); F.J. Gaspar Lorenz (Franscisco); C.W. Oosterlee (Cornelis)

2018-01-01

textabstractThe interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled

Multigrid technique and Optimized Schwarz method on block-structured grids with discontinuous interfaces

DEFF Research Database (Denmark)

Kolmogorov, Dmitry; Sørensen, Niels N.; Shen, Wen Zhong

2013-01-01

An Optimized Schwarz method using Robin boundary conditions for relaxation scheme is presented in the frame of Multigrid method on discontinuous grids. At each iteration the relaxation scheme is performed in two steps: one step with Dirichlet and another step with Robin boundary conditions at inn...

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

Science.gov (United States)

Moatssime, H. Al; Esselaoui, D.; Hakim, A.; Raghay, S.

2001-08-01

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non-linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss-Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright

Multigrid techniques with non-standard coarsening and group relaxation methods

International Nuclear Information System (INIS)

Danaee, A.

1989-06-01

In the usual (standard) multigrid methods, doubling of grid sizes with different smoothing iterations (pointwise, or blockwise) has been considered by different authors. Some have indicated that a large coarsening can also be used, but is not beneficial (cf. H3, p.59). In this paper, it is shown that with a suitable blockwise smoothing scheme, some advantages could be achieved even with a factor of H l-1 /h l = 3. (author). 10 refs, 2 figs, 6 tabs

Subspace orthogonalization for substructuring preconditioners for nonsymmetric systems of linear equations

Energy Technology Data Exchange (ETDEWEB)

Starke, G. [Universitaet Karlsruhe (Germany)

1994-12-31

For nonselfadjoint elliptic boundary value problems which are preconditioned by a substructuring method, i.e., nonoverlapping domain decomposition, the author introduces and studies the concept of subspace orthogonalization. In subspace orthogonalization variants of Krylov methods the computation of inner products and vector updates, and the storage of basis elements is restricted to a (presumably small) subspace, in this case the edge and vertex unknowns with respect to the partitioning into subdomains. The author investigates subspace orthogonalization for two specific iterative algorithms, GMRES and the full orthogonalization method (FOM). This is intended to eliminate certain drawbacks of the Arnoldi-based Krylov subspace methods mentioned above. Above all, the length of the Arnoldi recurrences grows linearly with the iteration index which is therefore restricted to the number of basis elements that can be held in memory. Restarts become necessary and this often results in much slower convergence. The subspace orthogonalization methods, in contrast, require the storage of only the edge and vertex unknowns of each basis element which means that one can iterate much longer before restarts become necessary. Moreover, the computation of inner products is also restricted to the edge and vertex points which avoids the disturbance of the computational flow associated with the solution of subdomain problems. The author views subspace orthogonalization as an alternative to restarting or truncating Krylov subspace methods for nonsymmetric linear systems of equations. Instead of shortening the recurrences, one restricts them to a subset of the unknowns which has to be carefully chosen in order to be able to extend this partial solution to the entire space. The author discusses the convergence properties of these iteration schemes and its advantages compared to restarted or truncated versions of Krylov methods applied to the full preconditioned system.

Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function

KAUST Repository

Pearson, John W.

2012-11-21

Optimal control problems with partial differential equations as constraints play an important role in many applications. The inclusion of bound constraints for the state variable poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared with other approaches. In this paper, we develop robust preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the efficiency of our approach. © 2012 John Wiley & Sons, Ltd.

Adaptive Multigrid Algorithm for the Lattice Wilson-Dirac Operator

International Nuclear Information System (INIS)

Babich, R.; Brower, R. C.; Rebbi, C.; Brannick, J.; Clark, M. A.; Manteuffel, T. A.; McCormick, S. F.; Osborn, J. C.

2010-01-01

We present an adaptive multigrid solver for application to the non-Hermitian Wilson-Dirac system of QCD. The key components leading to the success of our proposed algorithm are the use of an adaptive projection onto coarse grids that preserves the near null space of the system matrix together with a simplified form of the correction based on the so-called γ 5 -Hermitian symmetry of the Dirac operator. We demonstrate that the algorithm nearly eliminates critical slowing down in the chiral limit and that it has weak dependence on the lattice volume.

JTpack90: A parallel, object-based, Fortran 90 linear algebra package

Energy Technology Data Exchange (ETDEWEB)

Turner, J.A.; Kothe, D.B. [Los Alamos National Lab., NM (United States); Ferrell, R.C. [Cambridge Power Computing Associates, Ltd., Brookline, MA (United States)

1997-03-01

The authors have developed an object-based linear algebra package, currently with emphasis on sparse Krylov methods, driven primarily by needs of the Los Alamos National Laboratory parallel unstructured-mesh casting simulation tool Telluride. Support for a number of sparse storage formats, methods, and preconditioners have been implemented, driven primarily by application needs. They describe the object-based Fortran 90 approach, which enhances maintainability, performance, and extensibility, the parallelization approach using a new portable gather/scatter library (PGSLib), current capabilities and future plans, and present preliminary performance results on a variety of platforms.

Iterative and multigrid methods in the finite element solution of incompressible and turbulent fluid flow

Science.gov (United States)

Lavery, N.; Taylor, C.

1999-07-01

Multigrid and iterative methods are used to reduce the solution time of the matrix equations which arise from the finite element (FE) discretisation of the time-independent equations of motion of the incompressible fluid in turbulent motion. Incompressible flow is solved by using the method of reduce interpolation for the pressure to satisfy the Brezzi-Babuska condition. The k-l model is used to complete the turbulence closure problem. The non-symmetric iterative matrix methods examined are the methods of least squares conjugate gradient (LSCG), biconjugate gradient (BCG), conjugate gradient squared (CGS), and the biconjugate gradient squared stabilised (BCGSTAB). The multigrid algorithm applied is based on the FAS algorithm of Brandt, and uses two and three levels of grids with a V-cycling schedule. These methods are all compared to the non-symmetric frontal solver. Copyright

Adaptive parallel multigrid for Euler and incompressible Navier-Stokes equations

Energy Technology Data Exchange (ETDEWEB)

Trottenberg, U.; Oosterlee, K.; Ritzdorf, H. [and others

1996-12-31

The combination of (1) very efficient solution methods (Multigrid), (2) adaptivity, and (3) parallelism (distributed memory) clearly is absolutely necessary for future oriented numerics but still regarded as extremely difficult or even unsolved. We show that very nice results can be obtained for real life problems. Our approach is straightforward (based on {open_quotes}MLAT{close_quotes}). But, of course, reasonable refinement and load-balancing strategies have to be used. Our examples are 2D, but 3D is on the way.

Preconditioner-free Wiener filtering with a dense noise matrix

Science.gov (United States)

Huffenberger, Kevin M.

2018-05-01

This work extends the Elsner & Wandelt (2013) iterative method for efficient, preconditioner-free Wiener filtering to cases in which the noise covariance matrix is dense, but can be decomposed into a sum whose parts are sparse in convenient bases. The new method, which uses multiple messenger fields, reproduces Wiener-filter solutions for test problems, and we apply it to a case beyond the reach of the Elsner & Wandelt (2013) method. We compute the Wiener-filter solution for a simulated Cosmic Microwave Background (CMB) map that contains spatially varying, uncorrelated noise, isotropic 1/f noise, and large-scale horizontal stripes (like those caused by atmospheric noise). We discuss simple extensions that can filter contaminated modes or inverse-noise-filter the data. These techniques help to address complications in the noise properties of maps from current and future generations of ground-based Microwave Background experiments, like Advanced ACTPol, Simons Observatory, and CMB-S4.

Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface

Science.gov (United States)

Coco, Armando; Russo, Giovanni

2018-05-01

In this paper we propose a second-order accurate numerical method to solve elliptic problems with discontinuous coefficients (with general non-homogeneous jumps in the solution and its gradient) in 2D and 3D. The method consists of a finite-difference method on a Cartesian grid in which complex geometries (boundaries and interfaces) are embedded, and is second order accurate in the solution and the gradient itself. In order to avoid the drop in accuracy caused by the discontinuity of the coefficients across the interface, two numerical values are assigned on grid points that are close to the interface: a real value, that represents the numerical solution on that grid point, and a ghost value, that represents the numerical solution extrapolated from the other side of the interface, obtained by enforcing the assigned non-homogeneous jump conditions on the solution and its flux. The method is also extended to the case of matrix coefficient. The linear system arising from the discretization is solved by an efficient multigrid approach. Unlike the 1D case, grid points are not necessarily aligned with the normal derivative and therefore suitable stencils must be chosen to discretize interface conditions in order to achieve second order accuracy in the solution and its gradient. A proper treatment of the interface conditions will allow the multigrid to attain the optimal convergence factor, comparable with the one obtained by Local Fourier Analysis for rectangular domains. The method is robust enough to handle large jump in the coefficients: order of accuracy, monotonicity of the errors and good convergence factor are maintained by the scheme.

Development of a multi-grid FDTD code for three-dimensional simulation of large microwave sintering experiments

Energy Technology Data Exchange (ETDEWEB)

White, M.J.; Iskander, M.F. [Univ. of Utah, Salt Lake City, UT (United States). Electrical Engineering Dept.; Kimrey, H.D. [Oak Ridge National Lab., TN (United States)

1996-12-31

The Finite-Difference Time-Domain (FDTD) code available at the University of Utah has been used to simulate sintering of ceramics in single and multimode cavities, and many useful results have been reported in literature. More detailed and accurate results, specifically around and including the ceramic sample, are often desired to help evaluate the adequacy of the heating procedure. In electrically large multimode cavities, however, computer memory requirements limit the number of the mathematical cells, and the desired resolution is impractical to achieve due to limited computer resources. Therefore, an FDTD algorithm which incorporates multiple-grid regions with variable-grid sizes is required to adequately perform the desired simulations. In this paper the authors describe the development of a three-dimensional multi-grid FDTD code to help focus a large number of cells around the desired region. Test geometries were solved using a uniform-grid and the developed multi-grid code to help validate the results from the developed code. Results from these comparisons, as well as the results of comparisons between the developed FDTD code and other available variable-grid codes are presented. In addition, results from the simulation of realistic microwave sintering experiments showed improved resolution in critical sites inside the three-dimensional sintering cavity. With the validation of the FDTD code, simulations were performed for electrically large, multimode, microwave sintering cavities to fully demonstrate the advantages of the developed multi-grid FDTD code.

Adaptive Selection of Primal Constraints for Isogeometric BDDC Deluxe Preconditioners

KAUST Repository

Beirã o Da Veiga, L.; Pavarino, L. F.; Scacchi, S.; Widlund, O. B.; Zampini, Stefano

2017-01-01

Isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer-aided design (CAD) software and the discretization of variational problems of continuum mechanics. In contrast with the finite element case, the basis functions of isogeometric analysis are often not nodal. As a consequence, there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains. Building on earlier work on the deluxe version of the BDDC (balancing domain decomposition by constraints) family of domain decomposition algorithms, several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global, coarse component of these preconditioners. Numerical experiments provide evidence that this work can be successful, yielding scalable and quasi-optimal adaptive BDDC algorithms for isogeometric discretizations.

Adaptive Selection of Primal Constraints for Isogeometric BDDC Deluxe Preconditioners

KAUST Repository

Beirão Da Veiga, L.

2017-02-23

Isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of computer-aided design (CAD) software and the discretization of variational problems of continuum mechanics. In contrast with the finite element case, the basis functions of isogeometric analysis are often not nodal. As a consequence, there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains. Building on earlier work on the deluxe version of the BDDC (balancing domain decomposition by constraints) family of domain decomposition algorithms, several adaptive algorithms are developed in this paper for scalar elliptic problems in an effort to decrease the dimension of the global, coarse component of these preconditioners. Numerical experiments provide evidence that this work can be successful, yielding scalable and quasi-optimal adaptive BDDC algorithms for isogeometric discretizations.

An h-adaptive finite element solver for the calculations of the electronic structures

International Nuclear Information System (INIS)

Bao Gang; Hu Guanghui; Liu Di

2012-01-01

In this paper, a framework of using h-adaptive finite element method for the Kohn–Sham equation on the tetrahedron mesh is presented. The Kohn–Sham equation is discretized by the finite element method, and the h-adaptive technique is adopted to optimize the accuracy and the efficiency of the algorithm. The locally optimal block preconditioned conjugate gradient method is employed for solving the generalized eigenvalue problem, and an algebraic multigrid preconditioner is used to accelerate the solver. A variety of numerical experiments demonstrate the effectiveness of our algorithm for both the all-electron and the pseudo-potential calculations.

Pseudoinverse preconditioners and iterative methods for large dense linear least-squares problems

Directory of Open Access Journals (Sweden)

Oskar Cahueñas

2013-05-01

Full Text Available We address the issue of approximating the pseudoinverse of the coefficient matrix for dynamically building preconditioning strategies for the numerical solution of large dense linear least-squares problems. The new preconditioning strategies are embedded into simple and well-known iterative schemes that avoid the use of the, usually ill-conditioned, normal equations. We analyze a scheme to approximate the pseudoinverse, based on Schulz iterative method, and also different iterative schemes, based on extensions of Richardson's method, and the conjugate gradient method, that are suitable for preconditioning strategies. We present preliminary numerical results to illustrate the advantages of the proposed schemes.

Adaptive tree multigrids and simplified spherical harmonics approximation in deterministic neutral and charged particle transport

International Nuclear Information System (INIS)

Kotiluoto, P.

2007-05-01

A new deterministic three-dimensional neutral and charged particle transport code, MultiTrans, has been developed. In the novel approach, the adaptive tree multigrid technique is used in conjunction with simplified spherical harmonics approximation of the Boltzmann transport equation. The development of the new radiation transport code started in the framework of the Finnish boron neutron capture therapy (BNCT) project. Since the application of the MultiTrans code to BNCT dose planning problems, the testing and development of the MultiTrans code has continued in conventional radiotherapy and reactor physics applications. In this thesis, an overview of different numerical radiation transport methods is first given. Special features of the simplified spherical harmonics method and the adaptive tree multigrid technique are then reviewed. The usefulness of the new MultiTrans code has been indicated by verifying and validating the code performance for different types of neutral and charged particle transport problems, reported in separate publications. (orig.)

A comparative study of iterative solutions to linear systems arising in quantum mechanics

International Nuclear Information System (INIS)

Jing Yanfei; Huang Tingzhu; Duan Yong; Carpentieri, Bruno

2010-01-01

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.

Development of new multigrid schemes for the method of characteristics in neutron transport theory

International Nuclear Information System (INIS)

Grassi, G.

2006-01-01

This dissertation is based upon our doctoral research that dealt with the conception and development of new non-linear multigrid techniques for the Method of the Characteristics (MOC) within the TDT code. Here we focus upon a two-level scheme consisting of a fine level on which the neutron transport equation is iteratively solved using the MOC algorithm, and a coarse level defined by a more coarsely discretized phase space on which a low-order problem is considered. The solution of this problem is then used in order to correct the angular flux moments resulting from the previous transport iteration. A flux-volume homogenization procedure is employed to evaluate the coarse-level material properties after each transport iteration. This entails the non-linearity of the methods. According to the Generalised Equivalence Theory (GET), additional degrees of freedom are introduced for the low-order problem so that the convergence of the acceleration scheme can be ensured. We present two classes of non-linear methods: transport-like methods and discussion-like methods. Transport-like methods consider a homogenized low-order transport problem on the coarse level. This problem is iteratively solved using the same MOC algorithm as for the transport problem on the fine level. Discontinuity factors are then employed, per region or per surface, in order to reconstruct the currents evaluated by the low-order operator, which ensure the convergence of the acceleration scheme. On the other hand, discussion-like methods consider a low-order problem inspired by diffusion. We studied the non-linear Coarse Mesh Finite Difference (CMFD) method, already present in literature, in the perspective of integrating it into TDT code. Then, we developed a new non-linear method on the model of CMFD. From the latter, we borrowed the idea to establish a simple relation between currents and fluxes in order to obtain a problem involving only coarse fluxes. Finally, those non-linear methods have been

{sup 10}B multi-grid proportional gas counters for large area thermal neutron detectors

Energy Technology Data Exchange (ETDEWEB)

Andersen, K. [ESS, P.O. Box 176, SE-221 00 Lund (Sweden); Bigault, T. [ILL, BP 156, 6, rue Jules Horowitz, 38042 Grenoble Cedex 9 (France); Birch, J. [Linköping University, SE-581, 83 Linköping (Sweden); Buffet, J. C.; Correa, J. [ILL, BP 156, 6, rue Jules Horowitz, 38042 Grenoble Cedex 9 (France); Hall-Wilton, R. [ESS, P.O. Box 176, SE-221 00 Lund (Sweden); Hultman, L. [Linköping University, SE-581, 83 Linköping (Sweden); Höglund, C. [ESS, P.O. Box 176, SE-221 00 Lund (Sweden); Linköping University, SE-581, 83 Linköping (Sweden); Guérard, B., E-mail: guerard@ill.fr [ILL, BP 156, 6, rue Jules Horowitz, 38042 Grenoble Cedex 9 (France); Jensen, J. [Linköping University, SE-581, 83 Linköping (Sweden); Khaplanov, A. [ILL, BP 156, 6, rue Jules Horowitz, 38042 Grenoble Cedex 9 (France); ESS, P.O. Box 176, SE-221 00 Lund (Sweden); Kirstein, O. [Linköping University, SE-581, 83 Linköping (Sweden); Piscitelli, F.; Van Esch, P. [ILL, BP 156, 6, rue Jules Horowitz, 38042 Grenoble Cedex 9 (France); Vettier, C. [ESS, P.O. Box 176, SE-221 00 Lund (Sweden)

2013-08-21

{sup 3}He was a popular material in neutrons detectors until its availability dropped drastically in 2008. The development of techniques based on alternative convertors is now of high priority for neutron research institutes. Thin films of {sup 10}B or {sup 10}B{sub 4}C have been used in gas proportional counters to detect neutrons, but until now, only for small or medium sensitive area. We present here the multi-grid design, introduced at the ILL and developed in collaboration with ESS for LAN (large area neutron) detectors. Typically thirty {sup 10}B{sub 4}C films of 1 μm thickness are used to convert neutrons into ionizing particles which are subsequently detected in a proportional gas counter. The principle and the fabrication of the multi-grid are described and some preliminary results obtained with a prototype of 200 cm×8 cm are reported; a detection efficiency of 48% has been measured at 2.5 Å with a monochromatic neutron beam line, showing the good potential of this new technique.

Development of a 3D non-linear implicit MHD code

International Nuclear Information System (INIS)

Nicolas, T.; Ichiguchi, K.

2016-06-01

This paper details the on-going development of a 3D non-linear implicit MHD code, which aims at making possible large scale simulations of the non-linear phase of the interchange mode. The goal of the paper is to explain the rationale behind the choices made along the development, and the technical difficulties encountered. At the present stage, the development of the code has not been completed yet. Most of the discussion is concerned with the first approach, which utilizes cartesian coordinates in the poloidal plane. This approach shows serious difficulties in writing the preconditioner, closely related to the choice of coordinates. A second approach, based on curvilinear coordinates, also faced significant difficulties, which are detailed. The third and last approach explored involves unstructured tetrahedral grids, and indicates the possibility to solve the problem. The issue to domain meshing is addressed. (author)

On the implementation of an accurate and efficient solver for convection-diffusion equations

Science.gov (United States)

Wu, Chin-Tien

In this dissertation, we examine several different aspects of computing the numerical solution of the convection-diffusion equation. The solution of this equation often exhibits sharp gradients due to Dirichlet outflow boundaries or discontinuities in boundary conditions. Because of the singular-perturbed nature of the equation, numerical solutions often have severe oscillations when grid sizes are not small enough to resolve sharp gradients. To overcome such difficulties, the streamline diffusion discretization method can be used to obtain an accurate approximate solution in regions where the solution is smooth. To increase accuracy of the solution in the regions containing layers, adaptive mesh refinement and mesh movement based on a posteriori error estimations can be employed. An error-adapted mesh refinement strategy based on a posteriori error estimations is also proposed to resolve layers. For solving the sparse linear systems that arise from discretization, goemetric multigrid (MG) and algebraic multigrid (AMG) are compared. In addition, both methods are also used as preconditioners for Krylov subspace methods. We derive some convergence results for MG with line Gauss-Seidel smoothers and bilinear interpolation. Finally, while considering adaptive mesh refinement as an integral part of the solution process, it is natural to set a stopping tolerance for the iterative linear solvers on each mesh stage so that the difference between the approximate solution obtained from iterative methods and the finite element solution is bounded by an a posteriori error bound. Here, we present two stopping criteria. The first is based on a residual-type a posteriori error estimator developed by Verfurth. The second is based on an a posteriori error estimator, using local solutions, developed by Kay and Silvester. Our numerical results show the refined mesh obtained from the iterative solution which satisfies the second criteria is similar to the refined mesh obtained from

Fast multigrid solution of the advection problem with closed characteristics

Energy Technology Data Exchange (ETDEWEB)

Yavneh, I. [Israel Inst. of Technology, Haifa (Israel); Venner, C.H. [Univ. of Twente, Enschede (Netherlands); Brandt, A. [Weizmann Inst. of Science, Rehovot (Israel)

1996-12-31

The numerical solution of the advection-diffusion problem in the inviscid limit with closed characteristics is studied as a prelude to an efficient high Reynolds-number flow solver. It is demonstrated by a heuristic analysis and numerical calculations that using upstream discretization with downstream relaxation-ordering and appropriate residual weighting in a simple multigrid V cycle produces an efficient solution process. We also derive upstream finite-difference approximations to the advection operator, whose truncation terms approximate {open_quotes}physical{close_quotes} (Laplacian) viscosity, thus avoiding spurious solutions to the homogeneous problem when the artificial diffusivity dominates the physical viscosity.

On the fixed-stress split scheme as smoother in multigrid methods for coupling flow and geomechanics

NARCIS (Netherlands)

F.J. Gaspar Lorenz (Franscisco); C. Rodrigo (Carmen)

2017-01-01

textabstractThe fixed-stress split method has been widely used as solution method in the coupling of flow and geomechanics. In this work, we analyze the behavior of an inexact version of this algorithm as smoother within a geometric multigrid method, in order to obtain an efficient monolithic solver

Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction.

Science.gov (United States)

Gilles, Luc; Vogel, Curtis R; Ellerbroek, Brent L

2002-09-01

We introduce a multigrid preconditioned conjugate-gradient (MGCG) iterative scheme for computing open-loop wave-front reconstructors for extreme adaptive optics systems. We present numerical simulations for a 17-m class telescope with n = 48756 sensor measurement grid points within the aperture, which indicate that our MGCG method has a rapid convergence rate for a wide range of subaperture average slope measurement signal-to-noise ratios. The total computational cost is of order n log n. Hence our scheme provides for fast wave-front simulation and control in large-scale adaptive optics systems.

A matrix-free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction

Science.gov (United States)

Lv, X.; Zhao, Y.; Huang, X. Y.; Xia, G. H.; Su, X. H.

2007-07-01

A new three-dimensional (3D) matrix-free implicit unstructured multigrid finite volume (FV) solver for structural dynamics is presented in this paper. The solver is first validated using classical 2D and 3D cantilever problems. It is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates. This method has been integrated into our existing FV compressible solver [X. Lv, Y. Zhao, et al., An efficient parallel/unstructured-multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects, Journal of Computational Physics 215(2) (2006) 661-690] based on the immersed membrane method (IMM) [X. Lv, Y. Zhao, et al., as mentioned above]. Results for the interaction between the fluid and an immersed fixed-free cantilever are also presented to demonstrate the potential of this integrated fluid-structure interaction approach.

On preconditioning techniques for dense linear systems arising from singular boundary integral equations

Energy Technology Data Exchange (ETDEWEB)

Chen, Ke [Univ. of Liverpool (United Kingdom)

1996-12-31

We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.

Multi-grid and ICCG for problems with interfaces

International Nuclear Information System (INIS)

Dendy, J.E.; Hyman, J.M.

1980-01-01

Computation times for the multi-grid (MG) algorithm, the incomplete Cholesky conjugate gradient (ICCG) algorithm [J. Comp. Phys. 26, 43-65 (1978); Math. Comp. 31, 148-162 (1977)], and the modified ICCG (MICCG) algorithm [BIT 18, 142-156 (1978)] to solve elliptic partial differential equations are compared. The MICCG and ICCG algorithms are more robust than the MG for general positive definite systems. A major advantage of the MG algorithm is that the structure of the problem can be exploited to reduce the solution time significantly. Five example problems are discussed. For problems with little structure and for one-shot calculations ICCG is recommended over MG, and MICCG, over ICCG. For problems that are done many times, it is worth investing the effort to study methods like MG. 1 table

S2SA preconditioning for the Sn equations with strictly non negative spatial discretization

International Nuclear Information System (INIS)

Bruss, D. E.; Morel, J. E.; Ragusa, J. C.

2013-01-01

Preconditioners based upon sweeps and diffusion-synthetic acceleration have been constructed and applied to the zeroth and first spatial moments of the 1-D S n transport equation using a strictly non negative nonlinear spatial closure. Linear and nonlinear preconditioners have been analyzed. The effectiveness of various combinations of these preconditioners are compared. In one dimension, nonlinear sweep preconditioning is shown to be superior to linear sweep preconditioning, and DSA preconditioning using nonlinear sweeps in conjunction with a linear diffusion equation is found to be essentially equivalent to nonlinear sweeps in conjunction with a nonlinear diffusion equation. The ability to use a linear diffusion equation has important implications for preconditioning the S n equations with a strictly non negative spatial discretization in multiple dimensions. (authors)

Design and fabrication of multigrid X-ray collimators. [For airborne x-ray spectroscopy

Energy Technology Data Exchange (ETDEWEB)

Acton, L W; Joki, E G; Salmon, R J [Lockheed Missiles and Space Co., Palo Alto, Calif. (USA). Lockheed Palo Alto Research Lab.

1976-08-01

Multigrid X-ray collimators continue to find wide application in space research. This paper treats the principles of their design and fabrication and summarizes the experience obtained in making and flying thirteen such collimators ranging in angular resolution from 10 to 0.7 arc min FWHM. Included is a summary of a survey of scientist-users and industrial producers of collimator grids regarding grid materials, precision, plating, hole quality and results of acceptance testing.

Multigrid solution of diffusion equations on distributed memory multiprocessor systems

International Nuclear Information System (INIS)

Finnemann, H.

1988-01-01

The subject is the solution of partial differential equations for simulation of the reactor core on high-performance computers. The parallelization and implementation of nodal multigrid diffusion algorithms on array and ring configurations of the DIRMU multiprocessor system is outlined. The particular iteration scheme employed in the nodal expansion method appears similarly efficient in serial and parallel environments. The combination of modern multi-level techniques with innovative hardware (vector-multiprocessor systems) provides powerful tools needed for real time simulation of physical systems. The parallel efficiencies range from 70 to 90%. The same performance is estimated for large problems on large multiprocessor systems being designed at present. (orig.) [de

Adaptive Algebraic Multigrid for Finite Element Elliptic Equations with Random Coefficients

Energy Technology Data Exchange (ETDEWEB)

Kalchev, D

2012-04-02

This thesis presents a two-grid algorithm based on Smoothed Aggregation Spectral Element Agglomeration Algebraic Multigrid (SA-{rho}AMGe) combined with adaptation. The aim is to build an efficient solver for the linear systems arising from discretization of second-order elliptic partial differential equations (PDEs) with stochastic coefficients. Examples include PDEs that model subsurface flow with random permeability field. During a Markov Chain Monte Carlo (MCMC) simulation process, that draws PDE coefficient samples from a certain distribution, the PDE coefficients change, hence the resulting linear systems to be solved change. At every such step the system (discretized PDE) needs to be solved and the computed solution used to evaluate some functional(s) of interest that then determine if the coefficient sample is acceptable or not. The MCMC process is hence computationally intensive and requires the solvers used to be efficient and fast. This fact that at every step of MCMC the resulting linear system changes, makes an already existing solver built for the old problem perhaps not as efficient for the problem corresponding to the new sampled coefficient. This motivates the main goal of our study, namely, to adapt an already existing solver to handle the problem (with changed coefficient) with the objective to achieve this goal to be faster and more efficient than building a completely new solver from scratch. Our approach utilizes the local element matrices (for the problem with changed coefficients) to build local problems associated with constructed by the method agglomerated elements (a set of subdomains that cover the given computational domain). We solve a generalized eigenproblem for each set in a subspace spanned by the previous local coarse space (used for the old solver) and a vector, component of the error, that the old solver cannot handle. A portion of the spectrum of these local eigen-problems (corresponding to eigenvalues close to zero) form the

A Multigrid Algorithm for an Elliptic Problem with a Perturbed Boundary Condition

KAUST Repository

Bonito, Andrea; Pasciak, Joseph E.

2013-01-01

We discuss the preconditioning of systems coupling elliptic operators in Ω⊂Rd, d=2,3, with elliptic operators defined on hypersurfaces. These systems arise naturally when physical phenomena are affected by geometric boundary forces, such as the evolution of liquid drops subject to surface tension. The resulting operators are sums of interior and boundary terms weighted by parameters. We investigate the behavior of multigrid algorithms suited to this context and demonstrate numerical results which suggest uniform preconditioning bounds that are level and parameter independent.

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

KAUST Repository

Efendiev, Y.

2012-08-01

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards\\' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results. © 2012 Global-Science Press.

A fully implicit Newton-Krylov-Schwarz method for tokamak magnetohydrodynamics: Jacobian construction and preconditioner formulation

KAUST Repository

Reynolds, Daniel R.

2012-01-01

Single-fluid resistive magnetohydrodynamics (MHD) is a fluid description of fusion plasmas which is often used to investigate macroscopic instabilities in tokamaks. In MHD modeling of tokamaks, it is often desirable to compute MHD phenomena to resistive time scales or a combination of resistive-Alfvén time scales, which can render explicit time stepping schemes computationally expensive. We present recent advancements in the development of preconditioners for fully nonlinearly implicit simulations of single-fluid resistive tokamak MHD. Our work focuses on simulations using a structured mesh mapped into a toroidal geometry with a shaped poloidal cross-section, and a finite-volume spatial discretization of the partial differential equation model. We discretize the temporal dimension using a fully implicit or the backwards differentiation formula method, and solve the resulting nonlinear algebraic system using a standard inexact Newton-Krylov approach, provided by the sundials library. The focus of this paper is on the construction and performance of various preconditioning approaches for accelerating the convergence of the iterative solver algorithms. Effective preconditioners require information about the Jacobian entries; however, analytical formulae for these Jacobian entries may be prohibitive to derive/implement without error. We therefore compute these entries using automatic differentiation with OpenAD. We then investigate a variety of preconditioning formulations inspired by standard solution approaches in modern MHD codes, in order to investigate their utility in a preconditioning context. We first describe the code modifications necessary for the use of the OpenAD tool and sundials solver library. We conclude with numerical results for each of our preconditioning approaches in the context of pellet-injection fueling of tokamak plasmas. Of these, our optimal approach results in a speedup of a factor of 3 compared with non-preconditioned implicit tests, with

Solving Graph Laplacian Systems Through Recursive Bisections and Two-Grid Preconditioning

Energy Technology Data Exchange (ETDEWEB)

Ponce, Colin [Cornell Univ., Ithaca, NY (United States); Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2016-02-18

We present a parallelizable direct method for computing the solution to graph Laplacian-based linear systems derived from graphs that can be hierarchically bipartitioned with small edge cuts. For a graph of size n with constant-size edge cuts, our method decomposes a graph Laplacian in time O(n log n), and then uses that decomposition to perform a linear solve in time O(n log n). We then use the developed technique to design a preconditioner for graph Laplacians that do not have this property. Finally, we augment this preconditioner with a two-grid method that accounts for much of the preconditioner's weaknesses. We present an analysis of this method, as well as a general theorem for the condition number of a general class of two-grid support graph-based preconditioners. Numerical experiments illustrate the performance of the studied methods.

Multigrid methods for S/sub N/ problems

International Nuclear Information System (INIS)

Nowak, P.F.; Larsen, E.W.; Martin, W.R.

1987-01-01

It has long been known that the standard source iteration (SI) method for obtaining iterative solutions of S/sub N/ problems is very slowly converging in optically thick regions with low absorption. The rebalance and diffusion synthetic acceleration (DSA) methods are generalizations of SI that have been developed to accelerate convergence, but neither of these methods has been completely successful. In particular, the rebalance method tends to become unstable in problems where it is needed most (problems with high scattering ratios c = 1), while the DSA method, to be implemented in a stable fashion, requires the solution of a particular system of acceleration equations, and this has been done efficiently in two-dimensional geometries only for the diamond difference S/sub N/ equations. This paper discusses another extension of the SI method, namely, SI combined with the spatial multigrid algorithm (SIMG). This appears to be a viable way to accelerate many S/sub N/ problems in multidimensional geometries, provided the finest mesh consists of cells that are not optically thick

Fast multigrid-based computation of the induced electric field for transcranial magnetic stimulation

Science.gov (United States)

Laakso, Ilkka; Hirata, Akimasa

2012-12-01

In transcranial magnetic stimulation (TMS), the distribution of the induced electric field, and the affected brain areas, depends on the position of the stimulation coil and the individual geometry of the head and brain. The distribution of the induced electric field in realistic anatomies can be modelled using computational methods. However, existing computational methods for accurately determining the induced electric field in realistic anatomical models have suffered from long computation times, typically in the range of tens of minutes or longer. This paper presents a matrix-free implementation of the finite-element method with a geometric multigrid method that can potentially reduce the computation time to several seconds or less even when using an ordinary computer. The performance of the method is studied by computing the induced electric field in two anatomically realistic models. An idealized two-loop coil is used as the stimulating coil. Multiple computational grid resolutions ranging from 2 to 0.25 mm are used. The results show that, for macroscopic modelling of the electric field in an anatomically realistic model, computational grid resolutions of 1 mm or 2 mm appear to provide good numerical accuracy compared to higher resolutions. The multigrid iteration typically converges in less than ten iterations independent of the grid resolution. Even without parallelization, each iteration takes about 1.0 s or 0.1 s for the 1 and 2 mm resolutions, respectively. This suggests that calculating the electric field with sufficient accuracy in real time is feasible.

A Calderón multiplicative preconditioner for coupled surface-volume electric field integral equations

KAUST Repository

Bagci, Hakan

2010-08-01

A well-conditioned coupled set of surface (S) and volume (V) electric field integral equations (S-EFIE and V-EFIE) for analyzing wave interactions with densely discretized composite structures is presented. Whereas the V-EFIE operator is well-posed even when applied to densely discretized volumes, a classically formulated S-EFIE operator is ill-posed when applied to densely discretized surfaces. This renders the discretized coupled S-EFIE and V-EFIE system ill-conditioned, and its iterative solution inefficient or even impossible. The proposed scheme regularizes the coupled set of S-EFIE and V-EFIE using a Calderón multiplicative preconditioner (CMP)-based technique. The resulting scheme enables the efficient analysis of electromagnetic interactions with composite structures containing fine/subwavelength geometric features. Numerical examples demonstrate the efficiency of the proposed scheme. © 2006 IEEE.

On optimal improvements of classical iterative schemes for Z-matrices

Science.gov (United States)

Noutsos, D.; Tzoumas, M.

2006-04-01

Many researchers have considered preconditioners, applied to linear systems, whose matrix coefficient is a Z- or an M-matrix, that make the associated Jacobi and Gauss-Seidel methods converge asymptotically faster than the unpreconditioned ones. Such preconditioners are chosen so that they eliminate the off-diagonal elements of the same column or the elements of the first upper diagonal [Milaszewicz, LAA 93 (1987) 161-170], Gunawardena et al. [LAA 154-156 (1991) 123-143]. In this work we generalize the previous preconditioners to obtain optimal methods. "Good" Jacobi and Gauss-Seidel algorithms are given and preconditioners, that eliminate more than one entry per row, are also proposed and analyzed. Moreover, the behavior of the above preconditioners to the Krylov subspace methods is studied.

A NetCDF version of the two-dimensional energy balance model based on the full multigrid algorithm

Directory of Open Access Journals (Sweden)

Kelin Zhuang

2017-01-01

Full Text Available A NetCDF version of the two-dimensional energy balance model based on the full multigrid method in Fortran is introduced for both pedagogical and research purposes. Based on the land–sea–ice distribution, orbital elements, greenhouse gases concentration, and albedo, the code calculates the global seasonal surface temperature. A step-by-step guide with examples is provided for practice.

A NetCDF version of the two-dimensional energy balance model based on the full multigrid algorithm

Science.gov (United States)

Zhuang, Kelin; North, Gerald R.; Stevens, Mark J.

A NetCDF version of the two-dimensional energy balance model based on the full multigrid method in Fortran is introduced for both pedagogical and research purposes. Based on the land-sea-ice distribution, orbital elements, greenhouse gases concentration, and albedo, the code calculates the global seasonal surface temperature. A step-by-step guide with examples is provided for practice.

A distributed-memory hierarchical solver for general sparse linear systems

Energy Technology Data Exchange (ETDEWEB)

Chen, Chao [Stanford Univ., CA (United States). Inst. for Computational and Mathematical Engineering; Pouransari, Hadi [Stanford Univ., CA (United States). Dept. of Mechanical Engineering; Rajamanickam, Sivasankaran [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research; Boman, Erik G. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research; Darve, Eric [Stanford Univ., CA (United States). Inst. for Computational and Mathematical Engineering and Dept. of Mechanical Engineering

2017-12-20

We present a parallel hierarchical solver for general sparse linear systems on distributed-memory machines. For large-scale problems, this fully algebraic algorithm is faster and more memory-efficient than sparse direct solvers because it exploits the low-rank structure of fill-in blocks. Depending on the accuracy of low-rank approximations, the hierarchical solver can be used either as a direct solver or as a preconditioner. The parallel algorithm is based on data decomposition and requires only local communication for updating boundary data on every processor. Moreover, the computation-to-communication ratio of the parallel algorithm is approximately the volume-to-surface-area ratio of the subdomain owned by every processor. We also provide various numerical results to demonstrate the versatility and scalability of the parallel algorithm.

Evaluating Sparse Linear System Solvers on Scalable Parallel Architectures

National Research Council Canada - National Science Library

Grama, Ananth; Manguoglu, Murat; Koyuturk, Mehmet; Naumov, Maxim; Sameh, Ahmed

2008-01-01

.... The study was motivated primarily by the lack of robustness of Krylov subspace iterative schemes with generic, black-box, pre-conditioners such as approximate (or incomplete) LU-factorizations...

Plasma simulation of electron avalanche in a linear thyratron

International Nuclear Information System (INIS)

Kushner, M.J.

1985-01-01

Thyratrons typically operate at sufficiently small PD (pressure x electrode separation) that holdoff is obtained by operating on the near side of the Paschen curve, and by shielding the slot in the control grid so there is no straight line path for electrons to reach the anode from the cathode. Electron avalanche is initiated by pulsing the control grid to a high voltage. Upon collapse of voltage in the cathode-control grid space, the discharge is sustained by penetration of potential through the control grid slot into the cathode-control grid region. To better understand the electron avalanche process in multi-grid and slotted structures such as thyratrons, a plasma simulation code has been constructed. This effort is in support of a companion program in which a linear thyratron is being electrically and spectroscopically characterized

Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units.

Science.gov (United States)

Qi, Ruxi; Botello-Smith, Wesley M; Luo, Ray

2017-07-11

Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.

Implementation of the Vanka-type multigrid solver for the finite element approximation of the Navier-Stokes equations on GPU

Czech Academy of Sciences Publication Activity Database

Bauer, Petr; Klement, V.; Oberhuber, T.; Žabka, V.

2016-01-01

Roč. 200, March (2016), s. 50-56 ISSN 0010-4655 R&D Projects: GA ČR GB14-36566G Institutional support: RVO:61388998 Keywords : Navier–Stokes equations * mixed finite elements * multigrid * Vanka-type smoothers * Gauss–Seidel * red–black coloring * parallelization * GPU Subject RIV: BK - Fluid Dynamics Impact factor: 3.936, year: 2016

Robust and scalable hierarchical matrix-based fast direct solver and preconditioner for the numerical solution of elliptic partial differential equations

KAUST Repository

Chavez Chavez, Gustavo Ivan

2017-01-01

Numerical experiments corroborate the robustness, accuracy, and complexity claims and provide a baseline of the performance and memory footprint by comparisons with competing approaches such as the multigrid solver hypre, and the STRUMPACK implementation of the multifrontal factorization with hierarchically semi-separable matrices. The companion implementation can utilize many thousands of cores of Shaheen, KAUST's Haswell-based Cray XC-40 supercomputer, and compares favorably with other implementations of hierarchical solvers in terms of time-to-solution and memory consumption.

Least-squares wave-front reconstruction of Shack-Hartmann sensors and shearing interferometers using multigrid techniques

International Nuclear Information System (INIS)

Baker, K.L.

2005-01-01

This article details a multigrid algorithm that is suitable for least-squares wave-front reconstruction of Shack-Hartmann and shearing interferometer wave-front sensors. The algorithm detailed in this article is shown to scale with the number of subapertures in the same fashion as fast Fourier transform techniques, making it suitable for use in applications requiring a large number of subapertures and high Strehl ratio systems such as for high spatial frequency characterization of high-density plasmas, optics metrology, and multiconjugate and extreme adaptive optics systems

Application of Nearly Linear Solvers to Electric Power System Computation

Science.gov (United States)

Grant, Lisa L.

To meet the future needs of the electric power system, improvements need to be made in the areas of power system algorithms, simulation, and modeling, specifically to achieve a time frame that is useful to industry. If power system time-domain simulations could run in real-time, then system operators would have situational awareness to implement online control and avoid cascading failures, significantly improving power system reliability. Several power system applications rely on the solution of a very large linear system. As the demands on power systems continue to grow, there is a greater computational complexity involved in solving these large linear systems within reasonable time. This project expands on the current work in fast linear solvers, developed for solving symmetric and diagonally dominant linear systems, in order to produce power system specific methods that can be solved in nearly-linear run times. The work explores a new theoretical method that is based on ideas in graph theory and combinatorics. The technique builds a chain of progressively smaller approximate systems with preconditioners based on the system's low stretch spanning tree. The method is compared to traditional linear solvers and shown to reduce the time and iterations required for an accurate solution, especially as the system size increases. A simulation validation is performed, comparing the solution capabilities of the chain method to LU factorization, which is the standard linear solver for power flow. The chain method was successfully demonstrated to produce accurate solutions for power flow simulation on a number of IEEE test cases, and a discussion on how to further improve the method's speed and accuracy is included.

Fast linear solver for radiative transport equation with multiple right hand sides in diffuse optical tomography

International Nuclear Information System (INIS)

Jia, Jingfei; Kim, Hyun K.; Hielscher, Andreas H.

2015-01-01

It is well known that radiative transfer equation (RTE) provides more accurate tomographic results than its diffusion approximation (DA). However, RTE-based tomographic reconstruction codes have limited applicability in practice due to their high computational cost. In this article, we propose a new efficient method for solving the RTE forward problem with multiple light sources in an all-at-once manner instead of solving it for each source separately. To this end, we introduce here a novel linear solver called block biconjugate gradient stabilized method (block BiCGStab) that makes full use of the shared information between different right hand sides to accelerate solution convergence. Two parallelized block BiCGStab methods are proposed for additional acceleration under limited threads situation. We evaluate the performance of this algorithm with numerical simulation studies involving the Delta–Eddington approximation to the scattering phase function. The results show that the single threading block RTE solver proposed here reduces computation time by a factor of 1.5–3 as compared to the traditional sequential solution method and the parallel block solver by a factor of 1.5 as compared to the traditional parallel sequential method. This block linear solver is, moreover, independent of discretization schemes and preconditioners used; thus further acceleration and higher accuracy can be expected when combined with other existing discretization schemes or preconditioners. - Highlights: • We solve the multiple-right-hand-side problem in DOT with a block BiCGStab method. • We examine the CPU times of the block solver and the traditional sequential solver. • The block solver is faster than the sequential solver by a factor of 1.5–3.0. • Multi-threading block solvers give additional speedup under limited threads situation.

A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems

Science.gov (United States)

Heinkenschloss, Matthias

2005-01-01

We study a class of time-domain decomposition-based methods for the numerical solution of large-scale linear quadratic optimal control problems. Our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete-time optimal control (DTOC) problem. The optimality conditions for this DTOC problem lead to a linear block tridiagonal system. The diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. This motivates the application of block Gauss-Seidel (GS)-type methods for the solution of the block tridiagonal systems. Numerical experiments show that the spectral radii of the block GS iteration matrices are larger than one for typical applications, but that the eigenvalues of the iteration matrices decay to zero fast. Hence, while the GS method is not expected to convergence for typical applications, it can be effective as a preconditioner for Krylov-subspace methods. This is confirmed by our numerical tests.A byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block GS method applied to the DTOC optimality system.

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

KAUST Repository

Cortes, Adriano Mauricio

2016-10-01

The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf−supinf−sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

Local multigrid mesh refinement in view of nuclear fuel 3D modelling in pressurised water reactors

International Nuclear Information System (INIS)

Barbie, L.

2013-01-01

The aim of this study is to improve the performances, in terms of memory space and computational time, of the current modelling of the Pellet-Cladding mechanical Interaction (PCI), complex phenomenon which may occurs during high power rises in pressurised water reactors. Among the mesh refinement methods - methods dedicated to efficiently treat local singularities - a local multi-grid approach was selected because it enables the use of a black-box solver while dealing few degrees of freedom at each level. The Local Defect Correction (LDC) method, well suited to a finite element discretization, was first analysed and checked in linear elasticity, on configurations resulting from the PCI, since its use in solid mechanics is little widespread. Various strategies concerning the implementation of the multilevel algorithm were also compared. Coupling the LDC method with the Zienkiewicz-Zhu a posteriori error estimator in order to automatically detect the zones to be refined, was then tested. Performances obtained on two-dimensional and three-dimensional cases are very satisfactory, since the algorithm proposed is more efficient than h-adaptive refinement methods. Lastly, the LDC algorithm was extended to nonlinear mechanics. Space/time refinement as well as transmission of the initial conditions during the re-meshing step were looked at. The first results obtained are encouraging and show the interest of using the LDC method for PCI modelling. (author) [fr

Use of a multigrid technique to study effects of limited sampling of heterogeneity on transport prediction

International Nuclear Information System (INIS)

Cole, C.R.; Foote, H.P.

1987-02-01

Reliable ground water transport prediction requires accurate spatial and temporal characterization of a hydrogeologic system. However, cost constraints and the desire to maintain site integrity by minimizing drilling can restrict the amount of spatial sampling that can be obtained to resolve the flow parameter variability associated with heterogeneities. This study quantifies the errors in subsurface transport predictions resulting from incomplete characterization of hydraulic conductivity heterogeneity. A multigrid technique was used to simulate two-dimensional flow velocity fields with high resolution. To obtain these velocity fields, the finite difference code MGRID, which implements a multigrid solution technique, was applied to compute stream functions on a 256-by-256 grid for a variety of hypothetical systems having detailed distributions of hydraulic conductivity. Spatial variability in hydraulic conductivity distributions was characterized by the components in the spectrum of spatial frequencies. A low-pass spatial filtering technique was applied to the base case hydraulic conductivity distribution to produce a data set with lower spatial frequency content. Arrival time curves were then calculated for filtered hydraulic conductivity distribution and compared to base case results to judge the relative importance of the higher spatial frequency components. Results indicate a progression from multimode to single-mode arrival time curves as the number and extent of distinct flow pathways are reduced by low-pass filtering. This relationship between transport predictions and spatial frequencies was used to judge the consequences of sampling the hydraulic conductivity with reduced spatial resolution. 22 refs., 17 figs

Divergence-Conforming Discontinuous Galerkin Methods and $C^0$ Interior Penalty Methods

KAUST Repository

Kanschat, Guido

2014-01-01

© 2014 Society for Industrial and Applied Mathematics. In this paper, we show that recently developed divergence-conforming methods for the Stokes problem have discrete stream functions. These stream functions in turn solve a continuous interior penalty problem for biharmonic equations. The equivalence is established for the most common methods in two dimensions based on interior penalty terms. Then, extensions of the concept to discontinuous Galerkin methods defined through lifting operators, for different weak formulations of the Stokes problem, and to three dimensions are discussed. Application of the equivalence result yields an optimal error estimate for the Stokes velocity without involving the pressure. Conversely, combined with a recent multigrid method for Stokes flow, we obtain a simple and uniform preconditioner for harmonic problems with simply supported and clamped boundary.

Incomplete block factorization preconditioning for indefinite elliptic problems

Energy Technology Data Exchange (ETDEWEB)

Guo, Chun-Hua [Univ. of Calgary, Alberta (Canada)

1996-12-31

The application of the finite difference method to approximate the solution of an indefinite elliptic problem produces a linear system whose coefficient matrix is block tridiagonal and symmetric indefinite. Such a linear system can be solved efficiently by a conjugate residual method, particularly when combined with a good preconditioner. We show that specific incomplete block factorization exists for the indefinite matrix if the mesh size is reasonably small. And this factorization can serve as an efficient preconditioner. Some efforts are made to estimate the eigenvalues of the preconditioned matrix. Numerical results are also given.

3D inversion based on multi-grid approach of magnetotelluric data from Northern Scandinavia

Science.gov (United States)

Cherevatova, M.; Smirnov, M.; Korja, T. J.; Egbert, G. D.

2012-12-01

In this work we investigate the geoelectrical structure of the cratonic margin of Fennoscandian Shield by means of magnetotelluric (MT) measurements carried out in Northern Norway and Sweden during summer 2011-2012. The project Magnetotellurics in the Scandes (MaSca) focuses on the investigation of the crust, upper mantle and lithospheric structure in a transition zone from a stable Precambrian cratonic interior to a passive continental margin beneath the Caledonian Orogen and the Scandes Mountains in western Fennoscandia. Recent MT profiles in the central and southern Scandes indicated a large contrast in resistivity between Caledonides and Precambrian basement. The alum shales as a highly conductive layers between the resistive Precambrian basement and the overlying Caledonian nappes are revealed from this profiles. Additional measurements in the Northern Scandes were required. All together data from 60 synchronous long period (LMT) and about 200 broad band (BMT) sites were acquired. The array stretches from Lofoten and Bodo (Norway) in the west to Kiruna and Skeleftea (Sweden) in the east covering an area of 500x500 square kilometers. LMT sites were occupied for about two months, while most of the BMT sites were measured during one day. We have used new multi-grid approach for 3D electromagnetic (EM) inversion and modelling. Our approach is based on the OcTree discretization where the spatial domain is represented by rectangular cells, each of which might be subdivided (recursively) into eight sub-cells. In this simplified implementation the grid is refined only in the horizontal direction, uniformly in each vertical layer. Using multi-grid we manage to have a high grid resolution near the surface (for instance, to tackle with galvanic distortions) and lower resolution at greater depth as the EM fields decay in the Earth according to the diffusion equation. We also have a benefit in computational costs as number of unknowns decrease. The multi-grid forward

Mathematics and computational methods development in U.S. department of energy-sponsored research (nuclear energy research initiative and nuclear engineering education research). 5. Analysis of Angular V-Cycle Multigrid Formulation for Three-Dimensional Discrete Ordinates Shielding Problems

International Nuclear Information System (INIS)

Kucukboyaci, Vefa; Haghighat, Alireza

2001-01-01

We have developed new angular multigrid formulations, including the Simplified Angular Multigrid (SAM), Nested Iteration (NI), and V-Cycle schemes, that are compatible with the parallel environment and the adaptive differencing strategy of the PENTRAN three-dimensional parallel S N code. Using the Fourier analysis method for an infinite, homogenous medium, we have investigated the effectiveness of the V-Cycle scheme for different problem parameters including scattering ratio, spatial differencing weights, quadrature order, and mesh size. We have further investigated the effectiveness of the new schemes for practical shielding applications such as the Kobayashi benchmark problem and the boiling water reactor core shroud problem. In this paper, we summarize the angular V-Cycle scheme implemented in the PENTRAN code, the Fourier Analysis of the V-Cycle scheme, and results of convergence analysis of the V-Cycle scheme using different problem parameters. The theoretical analysis reveals that the V-Cycle scheme is effective for a large range of scattering ratios and is insensitive to mesh size. Besides the theoretical analysis, we have applied the new angular multigrid schemes to shielding problems. In comparison to the standard PCR formulation, combinations of the new angular multigrid schemes and PCR (e.g., SAM+V-Cycle+PCR) have proved to be very effective for scattering ratios in a range of 0.6 to 0.9. (authors)

Approximate inverse preconditioning of iterative methods for nonsymmetric linear systems

Energy Technology Data Exchange (ETDEWEB)

Benzi, M. [Universita di Bologna (Italy); Tuma, M. [Inst. of Computer Sciences, Prague (Czech Republic)

1996-12-31

A method for computing an incomplete factorization of the inverse of a nonsymmetric matrix A is presented. The resulting factorized sparse approximate inverse is used as a preconditioner in the iterative solution of Ax = b by Krylov subspace methods.

The preconditioned Gauss-Seidel method faster than the SOR method

Science.gov (United States)

Niki, Hiroshi; Kohno, Toshiyuki; Morimoto, Munenori

2008-09-01

In recent years, a number of preconditioners have been applied to linear systems [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154-156 (1991) 123-143; T. Kohno, H. Kotakemori, H. Niki, M. Usui, Improving modified Gauss-Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997) 113-123; H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner (I+Smax), J. Comput. Appl. Math. 145 (2002) 373-378; H. Kotakemori, H. Niki, N. Okamoto, Accelerated iteration method for Z-matrices, J. Comput. Appl. Math. 75 (1996) 87-97; M. Usui, H. Niki, T.Kohno, Adaptive Gauss-Seidel method for linear systems, Internat. J. Comput. Math. 51(1994)119-125 [10

Convergence analysis of variational and non-variational multigrid algorithms for the Laplace-Beltrami operator

KAUST Repository

Bonito, Andrea

2012-09-01

We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the V -cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio. © 2011 American Mathematical Society.

An implicit multigrid algorithm for computing hypersonic, chemically reacting viscous flows

International Nuclear Information System (INIS)

Edwards, J.R.

1996-01-01

An implicit algorithm for computing viscous flows in chemical nonequilibrium is presented. Emphasis is placed on the numerical efficiency of the time integration scheme, both in terms of periteration workload and overall convergence rate. In this context, several techniques are introduced, including a stable, O(m 2 ) approximate factorization of the chemical source Jacobian and implementations of V-cycle and filtered multigrid acceleration methods. A five species-seventeen reaction air model is used to calculate hypersonic viscous flow over a cylinder at conditions corresponding to flight at 5 km/s, 60 km altitude and at 11.36 km/s, 76.42 km altitude. Inviscid calculations using an eleven-species reaction mechanism including ionization are presented for a case involving 11.37 km/s flow at an altitude of 84.6 km. Comparisons among various options for the implicit treatment of the chemical source terms and among different multilevel approaches for convergence acceleration are presented for all simulations

A second-order virtual node algorithm for nearly incompressible linear elasticity in irregular domains

Science.gov (United States)

Zhu, Yongning; Wang, Yuting; Hellrung, Jeffrey; Cantarero, Alejandro; Sifakis, Eftychios; Teran, Joseph M.

2012-08-01

We present a cut cell method in R2 for enforcing Dirichlet and Neumann boundary conditions with nearly incompressible linear elastic materials in irregular domains. Virtual nodes on cut uniform grid cells are used to provide geometric flexibility in the domain boundary shape without sacrificing accuracy. We use a mixed formulation utilizing a MAC-type staggered grid with piecewise bilinear displacements centered at cell faces and piecewise constant pressures at cell centers. These discretization choices provide the necessary stability in the incompressible limit and the necessary accuracy in cut cells. Numerical experiments suggest second order accuracy in L∞. We target high-resolution problems and present a class of geometric multigrid methods for solving the discrete equations for displacements and pressures that achieves nearly optimal convergence rates independent of grid resolution.

New iterative solvers for the NAG Libraries

Energy Technology Data Exchange (ETDEWEB)

Salvini, S.; Shaw, G. [Numerical Algorithms Group Ltd., Oxford (United Kingdom)

1996-12-31

The purpose of this paper is to introduce the work which has been carried out at NAG Ltd to update the iterative solvers for sparse systems of linear equations, both symmetric and unsymmetric, in the NAG Fortran 77 Library. Our current plans to extend this work and include it in our other numerical libraries in our range are also briefly mentioned. We have added to the Library the new Chapter F11, entirely dedicated to sparse linear algebra. At Mark 17, the F11 Chapter includes sparse iterative solvers, preconditioners, utilities and black-box routines for sparse symmetric (both positive-definite and indefinite) linear systems. Mark 18 will add solvers, preconditioners, utilities and black-boxes for sparse unsymmetric systems: the development of these has already been completed.

MAPCUMBA: A fast iterative multi-grid map-making algorithm for CMB experiments

Science.gov (United States)

Doré, O.; Teyssier, R.; Bouchet, F. R.; Vibert, D.; Prunet, S.

2001-07-01

The data analysis of current Cosmic Microwave Background (CMB) experiments like BOOMERanG or MAXIMA poses severe challenges which already stretch the limits of current (super-) computer capabilities, if brute force methods are used. In this paper we present a practical solution for the optimal map making problem which can be used directly for next generation CMB experiments like ARCHEOPS and TopHat, and can probably be extended relatively easily to the full PLANCK case. This solution is based on an iterative multi-grid Jacobi algorithm which is both fast and memory sparing. Indeed, if there are Ntod data points along the one dimensional timeline to analyse, the number of operations is of O (Ntod \\ln Ntod) and the memory requirement is O (Ntod). Timing and accuracy issues have been analysed on simulated ARCHEOPS and TopHat data, and we discuss as well the issue of the joint evaluation of the signal and noise statistical properties.

Sparse symmetric preconditioners for dense linear systems in electromagnetism

NARCIS (Netherlands)

Carpentieri, Bruno; Duff, Iain S.; Giraud, Luc; Monga Made, M. Magolu

2004-01-01

We consider symmetric preconditioning strategies for the iterative solution of dense complex symmetric non-Hermitian systems arising in computational electromagnetics. In particular, we report on the numerical behaviour of the classical incomplete Cholesky factorization as well as some of its recent

Asynchronous Task-Based Parallelization of Algebraic Multigrid

KAUST Repository

AlOnazi, Amani A.

2017-06-23

As processor clock rates become more dynamic and workloads become more adaptive, the vulnerability to global synchronization that already complicates programming for performance in today\\'s petascale environment will be exacerbated. Algebraic multigrid (AMG), the solver of choice in many large-scale PDE-based simulations, scales well in the weak sense, with fixed problem size per node, on tightly coupled systems when loads are well balanced and core performance is reliable. However, its strong scaling to many cores within a node is challenging. Reducing synchronization and increasing concurrency are vital adaptations of AMG to hybrid architectures. Recent communication-reducing improvements to classical additive AMG by Vassilevski and Yang improve concurrency and increase communication-computation overlap, while retaining convergence properties close to those of standard multiplicative AMG, but remain bulk synchronous.We extend the Vassilevski and Yang additive AMG to asynchronous task-based parallelism using a hybrid MPI+OmpSs (from the Barcelona Supercomputer Center) within a node, along with MPI for internode communications. We implement a tiling approach to decompose the grid hierarchy into parallel units within task containers. We compare against the MPI-only BoomerAMG and the Auxiliary-space Maxwell Solver (AMS) in the hypre library for the 3D Laplacian operator and the electromagnetic diffusion, respectively. In time to solution for a full solve an MPI-OmpSs hybrid improves over an all-MPI approach in strong scaling at full core count (32 threads per single Haswell node of the Cray XC40) and maintains this per node advantage as both weak scale to thousands of cores, with MPI between nodes.

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-based Algebraic Multigrid (AMGe) Approach

DEFF Research Database (Denmark)

Christensen, Max la Cour; Villa, Umberto; Engsig-Karup, Allan Peter

2017-01-01

associated with non-planar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart-Thomas and piecewise......We study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element agglomeration based Algebraic Multigrid (AMGe) technique with improved approximation proper ties [37] is used, for the first...... discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured...

Toward an optimal solver for time-spectral fluid-dynamic and aeroelastic solutions on unstructured meshes

Science.gov (United States)

Mundis, Nathan L.; Mavriplis, Dimitri J.

2017-09-01

The time-spectral method applied to the Euler and coupled aeroelastic equations theoretically offers significant computational savings for purely periodic problems when compared to standard time-implicit methods. However, attaining superior efficiency with time-spectral methods over traditional time-implicit methods hinges on the ability rapidly to solve the large non-linear system resulting from time-spectral discretizations which become larger and stiffer as more time instances are employed or the period of the flow becomes especially short (i.e. the maximum resolvable wave-number increases). In order to increase the efficiency of these solvers, and to improve robustness, particularly for large numbers of time instances, the Generalized Minimal Residual Method (GMRES) is used to solve the implicit linear system over all coupled time instances. The use of GMRES as the linear solver makes time-spectral methods more robust, allows them to be applied to a far greater subset of time-accurate problems, including those with a broad range of harmonic content, and vastly improves the efficiency of time-spectral methods. In previous work, a wave-number independent preconditioner that mitigates the increased stiffness of the time-spectral method when applied to problems with large resolvable wave numbers has been developed. This preconditioner, however, directly inverts a large matrix whose size increases in proportion to the number of time instances. As a result, the computational time of this method scales as the cube of the number of time instances. In the present work, this preconditioner has been reworked to take advantage of an approximate-factorization approach that effectively decouples the spatial and temporal systems. Once decoupled, the time-spectral matrix can be inverted in frequency space, where it has entries only on the main diagonal and therefore can be inverted quite efficiently. This new GMRES/preconditioner combination is shown to be over an order of

Multigrid solution of incompressible turbulent flows by using two-equation turbulence models

Energy Technology Data Exchange (ETDEWEB)

Zheng, X.; Liu, C. [Front Range Scientific Computations, Inc., Denver, CO (United States); Sung, C.H. [David Taylor Model Basin, Bethesda, MD (United States)

1996-12-31

Most of practical flows are turbulent. From the interest of engineering applications, simulation of realistic flows is usually done through solution of Reynolds-averaged Navier-Stokes equations and turbulence model equations. It has been widely accepted that turbulence modeling plays a very important role in numerical simulation of practical flow problem, particularly when the accuracy is of great concern. Among the most used turbulence models today, two-equation models appear to be favored for the reason that they are more general than algebraic models and affordable with current available computer resources. However, investigators using two-equation models seem to have been more concerned with the solution of N-S equations. Less attention is paid to the solution method for the turbulence model equations. In most cases, the turbulence model equations are loosely coupled with N-S equations, multigrid acceleration is only applied to the solution of N-S equations due to perhaps the fact the turbulence model equations are source-term dominant and very stiff in sublayer region.

Conservative multigrid methods for Cahn-Hilliard fluids

International Nuclear Information System (INIS)

Kim, Junseok; Kang, Kyungkeun; Lowengrub, John

2004-01-01

We develop a conservative, second-order accurate fully implicit discretization of the Navier-Stokes (NS) and Cahn-Hilliard (CH) system that has an associated discrete energy functional. This system provides a diffuse-interface description of binary fluid flows with compressible or incompressible flow components [R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998) 2617]. In this work, we focus on the case of flows containing two immiscible, incompressible and density-matched components. The scheme, however, has a straightforward extension to multi-component systems. To efficiently solve the discrete system at the implicit time-level, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an applied external shear, the evolution of the flow is nontrivial and the flow morphology repeats itself in time as multiple pinchoff and reconnection events occur. Eventually, the periodic motion ceases and the system relaxes to a global equilibrium. The equilibria we observe appears has a similar structure in all cases although the dynamics of the evolution is quite different. We view the work presented in this paper as preparatory for a detailed investigation of liquid-liquid interfaces with surface tension where the interfaces separate two immiscible fluids [On the pinchoff of liquid-liquid jets with surface tension, in preparation]. To this end, we also include a simulation of the pinchoff of a liquid thread under the Rayleigh instability at finite Reynolds number

First-order system least squares for the pure traction problem in planar linear elasticity

Energy Technology Data Exchange (ETDEWEB)

Cai, Z.; Manteuffel, T.; McCormick, S.; Parter, S.

1996-12-31

This talk will develop two first-order system least squares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are two-stage algorithms that first solve for the gradients of displacement, then for the displacement itself. One approach, which uses L{sup 2} norms to define the FOSLS functional, is shown under certain H{sup 2} regularity assumptions to admit optimal H{sup 1}-like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the Poisson ratio for all variables. The second approach, which is based on H{sup -1} norms, is shown under general assumptions to admit optimal uniform performance for displacement flux in an L{sup 2} norm and for displacement in an H{sup 1} norm. These methods do not degrade as other methods generally do when the material properties approach the incompressible limit.

A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations

Science.gov (United States)

Li, Meng; Gu, Xian-Ming; Huang, Chengming; Fei, Mingfa; Zhang, Guoyu

2018-04-01

In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schrödinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical induction, the discrete scheme is proved to be unconditionally convergent in the sense of L2-norm and H α / 2-norm, which means that there are no any constraints on the grid ratios. Then, the prior bound of the discrete solution in L2-norm and L∞-norm are also obtained. Moreover, we propose an iterative algorithm, by which the coefficient matrix is independent of the time level, and thus it leads to Toeplitz-like linear systems that can be efficiently solved by Krylov subspace solvers with circulant preconditioners. This method can reduce the memory requirement of the proposed linearized finite element scheme from O (M2) to O (M) and the computational complexity from O (M3) to O (Mlog ⁡ M) in each iterative step, where M is the number of grid nodes. Finally, numerical results are carried out to verify the correction of the theoretical analysis, simulate the collision of two solitary waves, and show the utility of the fast numerical solution techniques.

Iterative convergence acceleration of neutral particle transport methods via adjacent-cell preconditioners

International Nuclear Information System (INIS)

Azmy, Y.Y.

1999-01-01

The author proposes preconditioning as a viable acceleration scheme for the inner iterations of transport calculations in slab geometry. In particular he develops Adjacent-Cell Preconditioners (AP) that have the same coupling stencil as cell-centered diffusion schemes. For lowest order methods, e.g., Diamond Difference, Step, and 0-order Nodal Integral Method (ONIM), cast in a Weighted Diamond Difference (WDD) form, he derives AP for thick (KAP) and thin (NAP) cells that for model problems are unconditionally stable and efficient. For the First-Order Nodal Integral Method (INIM) he derives a NAP that possesses similarly excellent spectral properties for model problems. The two most attractive features of the new technique are:(1) its cell-centered coupling stencil, which makes it more adequate for extension to multidimensional, higher order situations than the standard edge-centered or point-centered Diffusion Synthetic Acceleration (DSA) methods; and (2) its decreasing spectral radius with increasing cell thickness to the extent that immediate pointwise convergence, i.e., in one iteration, can be achieved for problems with sufficiently thick cells. He implemented these methods, augmented with appropriate boundary conditions and mixing formulas for material heterogeneities, in the test code APID that he uses to successfully verify the analytical spectral properties for homogeneous problems. Furthermore, he conducts numerical tests to demonstrate the robustness of the KAP and NAP in the presence of sharp mesh or material discontinuities. He shows that the AP for WDD is highly resilient to such discontinuities, but for INIM a few cases occur in which the scheme does not converge; however, when it converges, AP greatly reduces the number of iterations required to achieve convergence

Comparative Performance Analysis of Coarse Solvers for Algebraic Multigrid on Multicore and Manycore Architectures

Energy Technology Data Exchange (ETDEWEB)

Druinsky, A; Ghysels, P; Li, XS; Marques, O; Williams, S; Barker, A; Kalchev, D; Vassilevski, P

2016-04-02

In this paper, we study the performance of a two-level algebraic-multigrid algorithm, with a focus on the impact of the coarse-grid solver on performance. We consider two algorithms for solving the coarse-space systems: the preconditioned conjugate gradient method and a new robust HSS-embedded low-rank sparse-factorization algorithm. Our test data comes from the SPE Comparative Solution Project for oil-reservoir simulations. We contrast the performance of our code on one 12-core socket of a Cray XC30 machine with performance on a 60-core Intel Xeon Phi coprocessor. To obtain top performance, we optimized the code to take full advantage of fine-grained parallelism and made it thread-friendly for high thread count. We also developed a bounds-and-bottlenecks performance model of the solver which we used to guide us through the optimization effort, and also carried out performance tuning in the solver’s large parameter space. Finally, as a result, significant speedups were obtained on both machines.

Analysis of secondary particle behavior in multiaperture, multigrid accelerator for the ITER neutral beam injector.

Science.gov (United States)

Mizuno, T; Taniguchi, M; Kashiwagi, M; Umeda, N; Tobari, H; Watanabe, K; Dairaku, M; Sakamoto, K; Inoue, T

2010-02-01

Heat load on acceleration grids by secondary particles such as electrons, neutrals, and positive ions, is a key issue for long pulse acceleration of negative ion beams. Complicated behaviors of the secondary particles in multiaperture, multigrid (MAMuG) accelerator have been analyzed using electrostatic accelerator Monte Carlo code. The analytical result is compared to experimental one obtained in a long pulse operation of a MeV accelerator, of which second acceleration grid (A2G) was removed for simplification of structure. The analytical results show that relatively high heat load on the third acceleration grid (A3G) since stripped electrons were deposited mainly on A3G. This heat load on the A3G can be suppressed by installing the A2G. Thus, capability of MAMuG accelerator is demonstrated for suppression of heat load due to secondary particles by the intermediate grids.

DL_MG: A Parallel Multigrid Poisson and Poisson-Boltzmann Solver for Electronic Structure Calculations in Vacuum and Solution.

Science.gov (United States)

Womack, James C; Anton, Lucian; Dziedzic, Jacek; Hasnip, Phil J; Probert, Matt I J; Skylaris, Chris-Kriton

2018-03-13

The solution of the Poisson equation is a crucial step in electronic structure calculations, yielding the electrostatic potential-a key component of the quantum mechanical Hamiltonian. In recent decades, theoretical advances and increases in computer performance have made it possible to simulate the electronic structure of extended systems in complex environments. This requires the solution of more complicated variants of the Poisson equation, featuring nonhomogeneous dielectric permittivities, ionic concentrations with nonlinear dependencies, and diverse boundary conditions. The analytic solutions generally used to solve the Poisson equation in vacuum (or with homogeneous permittivity) are not applicable in these circumstances, and numerical methods must be used. In this work, we present DL_MG, a flexible, scalable, and accurate solver library, developed specifically to tackle the challenges of solving the Poisson equation in modern large-scale electronic structure calculations on parallel computers. Our solver is based on the multigrid approach and uses an iterative high-order defect correction method to improve the accuracy of solutions. Using two chemically relevant model systems, we tested the accuracy and computational performance of DL_MG when solving the generalized Poisson and Poisson-Boltzmann equations, demonstrating excellent agreement with analytic solutions and efficient scaling to ∼10 9 unknowns and 100s of CPU cores. We also applied DL_MG in actual large-scale electronic structure calculations, using the ONETEP linear-scaling electronic structure package to study a 2615 atom protein-ligand complex with routinely available computational resources. In these calculations, the overall execution time with DL_MG was not significantly greater than the time required for calculations using a conventional FFT-based solver.

Exploring the interplay of resilience and energy consumption for a task-based partial differential equations preconditioner

KAUST Repository

Rizzi, F.

2017-05-25

We discuss algorithm-based resilience to silent data corruptions (SDCs) in a task-based domain-decomposition preconditioner for partial differential equations (PDEs). The algorithm exploits a reformulation of the PDE as a sampling problem, followed by a solution update through data manipulation that is resilient to SDCs. The implementation is based on a server-client model where all state information is held by the servers, while clients are designed solely as computational units. Scalability tests run up to ∼ 51K cores show a parallel efficiency greater than 90%. We use a 2D elliptic PDE and a fault model based on random single and double bit-flip to demonstrate the resilience of the application to synthetically injected SDC. We discuss two fault scenarios: one based on the corruption of all data of a target task, and the other involving the corruption of a single data point. We show that for our application, given the test problem considered, a four-fold increase in the number of faults only yields a 2% change in the overhead to overcome their presence, from 7% to 9%. We then discuss potential savings in energy consumption via dynamic voltage/frequency scaling, and its interplay with fault-rates, and application overhead.

Exploring the interplay of resilience and energy consumption for a task-based partial differential equations preconditioner

KAUST Repository

Rizzi, F.; Morris, K.; Sargsyan, K.; Mycek, P.; Safta, C.; Le Maî tre, O.; Knio, Omar; Debusschere, B.J.

2017-01-01

We discuss algorithm-based resilience to silent data corruptions (SDCs) in a task-based domain-decomposition preconditioner for partial differential equations (PDEs). The algorithm exploits a reformulation of the PDE as a sampling problem, followed by a solution update through data manipulation that is resilient to SDCs. The implementation is based on a server-client model where all state information is held by the servers, while clients are designed solely as computational units. Scalability tests run up to ∼ 51K cores show a parallel efficiency greater than 90%. We use a 2D elliptic PDE and a fault model based on random single and double bit-flip to demonstrate the resilience of the application to synthetically injected SDC. We discuss two fault scenarios: one based on the corruption of all data of a target task, and the other involving the corruption of a single data point. We show that for our application, given the test problem considered, a four-fold increase in the number of faults only yields a 2% change in the overhead to overcome their presence, from 7% to 9%. We then discuss potential savings in energy consumption via dynamic voltage/frequency scaling, and its interplay with fault-rates, and application overhead.

On the Robustness and Prospects of Adaptive BDDC Methods for Finite Element Discretizations of Elliptic PDEs with High-Contrast Coefficients

KAUST Repository

Zampini, Stefano; Keyes, David E.

2016-01-01

Balancing Domain Decomposition by Constraints (BDDC) methods have proven to be powerful preconditioners for large and sparse linear systems arising from the finite element discretization of elliptic PDEs. Condition number bounds can be theoretically

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

Science.gov (United States)

Costiner, Sorin; Taasan, Shlomo

1994-01-01

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.

Fast isogeometric solvers for explicit dynamics

KAUST Repository

Gao, Longfei

2014-06-01

In finite element analysis, solving time-dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix. For mass matrices that can be expressed as tensor products of lower dimensional matrices, we present a direct method that has linear computational complexity, i.e., O(N), where N is the total number of degrees of freedom in the system. We refer to these matrices as separable matrices. For non-separable mass matrices, we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative. We demonstrate that these preconditioners, which are easy to construct and cheap to apply (O(N)), can deliver significant convergence acceleration. The performances of these preconditioners are independent of the polynomial order (p independence) and mesh resolution (h independence) for maximum continuity B-splines, as verified by various numerical tests. © 2014 Elsevier B.V.

Natural Preconditioning and Iterative Methods for Saddle Point Systems

KAUST Repository

Pestana, Jennifer

2015-01-01

© 2015 Society for Industrial and Applied Mathematics. The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness - in terms of rapidity of convergence - is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends.

Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems

Czech Academy of Sciences Publication Activity Database

Lukšan, Ladislav; Vlček, Jan

1998-01-01

Roč. 5, č. 3 (1998), s. 219-247 ISSN 1070-5325 R&D Projects: GA ČR GA201/96/0918 Keywords : nonlinear programming * sparse problems * equality constraints * truncated Newton method * augmented Lagrangian function * indefinite systems * indefinite preconditioners * conjugate gradient method * residual smoothing Subject RIV: BA - General Mathematics Impact factor: 0.741, year: 1998

Implicit Unstructured Computational Aerodynamics on Many-Integrated Core Architecture

KAUST Repository

Al Farhan, Mohammed A.; Keyes, David E.

2014-01-01

the linear algebraic kernel is limited by the bottleneck of memory bandwidth, the flux kernel arising in control volume discretization of the conservation law residuals and the preconditioner for the Jacobian exploits the Phi hardware well.

Comparison of different precondtioners for nonsymmtric finite volume element methods

Energy Technology Data Exchange (ETDEWEB)

Mishev, I.D.

1996-12-31

We consider a few different preconditioners for the linear systems arising from the discretization of 3-D convection-diffusion problems with the finite volume element method. Their theoretical and computational convergence rates are compared and discussed.

High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative Preconditioning of the EFIE

KAUST Repository

Valdes, Felipe; Andriulli, Francesco P.; Cools, Kristof; Michielssen, Eric

2011-01-01

A new high-order Calderón multiplicative preconditioner (HO-CMP) for the electric field integral equation (EFIE) is presented. In contrast to previous CMPs, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of high-order quasi curl-conforming basis functions. Like its predecessors, the HO-CMP can be seamlessly integrated into existing EFIE codes. Numerical results demonstrate that the linear systems of equations obtained using the proposed HO-CMP converge rapidly, regardless of the mesh density and of the order of the current expansion. © 2006 IEEE.

High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative Preconditioning of the EFIE

KAUST Repository

Valdes, Felipe

2011-04-01

A new high-order Calderón multiplicative preconditioner (HO-CMP) for the electric field integral equation (EFIE) is presented. In contrast to previous CMPs, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of high-order quasi curl-conforming basis functions. Like its predecessors, the HO-CMP can be seamlessly integrated into existing EFIE codes. Numerical results demonstrate that the linear systems of equations obtained using the proposed HO-CMP converge rapidly, regardless of the mesh density and of the order of the current expansion. © 2006 IEEE.

Damage mapping in structural health monitoring using a multi-grid architecture

Energy Technology Data Exchange (ETDEWEB)

Mathews, V. John [Dept. of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT 84112 (United States)

2015-03-31

This paper presents a multi-grid architecture for tomography-based damage mapping of composite aerospace structures. The system employs an array of piezo-electric transducers bonded on the structure. Each transducer may be used as an actuator as well as a sensor. The structure is excited sequentially using the actuators and the guided waves arriving at the sensors in response to the excitations are recorded for further analysis. The sensor signals are compared to their baseline counterparts and a damage index is computed for each actuator-sensor pair. These damage indices are then used as inputs to the tomographic reconstruction system. Preliminary damage maps are reconstructed on multiple coordinate grids defined on the structure. These grids are shifted versions of each other where the shift is a fraction of the spatial sampling interval associated with each grid. These preliminary damage maps are then combined to provide a reconstruction that is more robust to measurement noise in the sensor signals and the ill-conditioned problem formulation for single-grid algorithms. Experimental results on a composite structure with complexity that is representative of aerospace structures included in the paper demonstrate that for sufficiently high sensor densities, the algorithm of this paper is capable of providing damage detection and characterization with accuracy comparable to traditional C-scan and A-scan-based ultrasound non-destructive inspection systems quickly and without human supervision.

Adaptive local refinement and multi-level methods for simulating multiphasic flows

International Nuclear Information System (INIS)

Minjeaud, Sebastian

2010-01-01

This thesis describes some numerical and mathematical aspects of incompressible multiphase flows simulations with a diffuse interface Cahn-Hilliard / Navier-Stokes model (interfaces have a small but a positive thickness). The space discretization is performed thanks to a Galerkin formulation and the finite elements method. The presence of different scales in the system (interfaces have a very small thickness compared to the characteristic lengths of the domain) suggests the use of a local adaptive refinement method. The algorithm that is introduced allows to implicitly handle the non-conformities of the generated meshes to produce conformal finite elements approximation spaces. It consists in refining basis functions instead of cells. The refinement of a basis function is made possible by the conceptual existence of a nested sequence of uniformly refined grids from which 'parent-child' relationships are deduced, linking the basis functions of two consecutive refinement levels. Moreover, it is shown how this method can be exploited to build multigrid pre-conditioners. From a composite finite elements approximation space, it is indeed possible to rebuild, by 'coarsening', a sequence of auxiliary nested spaces which allows to enter in the abstract multigrid framework. Concerning the time discretization, it begins with the study of the Cahn-Hilliard system. A semi-implicit scheme is proposed to remedy to convergence failures of the Newton method used to solve this (non linear) system. It guarantees the decrease of the discrete free energy ensuring the stability of the scheme. The existence and convergence of discrete solutions towards the weak solution of the system are shown. The study continues with providing an unconditionally stable time discretization of the complete Cahn-Hilliard / Navier-Stokes model. An important point is that this discretization does not strongly couple the Cahn-Hilliard and Navier-Stokes systems allowing to independently solve the two systems

Preconditioners Based on the ISM Factorization

Czech Academy of Sciences Publication Activity Database

Bru, R.; Cerdán, J.; Marín, J.; Mas, J.; Tůma, Miroslav

2015-01-01

Roč. 23, č. 3 (2015), s. 17-27 ISSN 1224-1784 R&D Projects: GA ČR GA13-06684S Institutional support: RVO:67985807 Keywords : preconditioned iterative methods * incomplete decompositions * approximate inverses * linear least squares Subject RIV: BA - General Mathematics Impact factor: 0.383, year: 2015 http://emis.muni.cz/journals/ASUO/mathematics_/anale2015vol3/Bru_R.__Cerdan_J.__Marin_J.__Mas_J.__Tuma_M..pdf

An application of multigrid methods for a discrete elastic model for epitaxial systems

International Nuclear Information System (INIS)

Caflisch, R.E.; Lee, Y.-J.; Shu, S.; Xiao, Y.-X.; Xu, J.

2006-01-01

We apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems, recently introduced by Schindler et al. [Phys. Rev. B 67, 075316 (2003)]. The discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth, but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge. In this paper, we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model. Using this method, simulations are performed for several representative physical problems, including an infinite periodic step train, a layered nanocrystal, and a system of quantum dots. The results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties, regardless of the problem size, the geometry and the physical parameters. The effects of substrate depth and of invariance due to traction-free boundary conditions are assessed. For a system of quantum dots, the simulated strain energy density supports the observations that trench formation near the dots provides strain relief

Massively Parallel Geostatistical Inversion of Coupled Processes in Heterogeneous Porous Media

Science.gov (United States)

Ngo, A.; Schwede, R. L.; Li, W.; Bastian, P.; Ippisch, O.; Cirpka, O. A.

2012-04-01

The quasi-linear geostatistical approach is an inversion scheme that can be used to estimate the spatial distribution of a heterogeneous hydraulic conductivity field. The estimated parameter field is considered to be a random variable that varies continuously in space, meets the measurements of dependent quantities (such as the hydraulic head, the concentration of a transported solute or its arrival time) and shows the required spatial correlation (described by certain variogram models). This is a method of conditioning a parameter field to observations. Upon discretization, this results in as many parameters as elements of the computational grid. For a full three dimensional representation of the heterogeneous subsurface it is hardly sufficient to work with resolutions (up to one million parameters) of the model domain that can be achieved on a serial computer. The forward problems to be solved within the inversion procedure consists of the elliptic steady-state groundwater flow equation and the formally elliptic but nearly hyperbolic steady-state advection-dominated solute transport equation in a heterogeneous porous medium. Both equations are discretized by Finite Element Methods (FEM) using fully scalable domain decomposition techniques. Whereas standard conforming FEM is sufficient for the flow equation, for the advection dominated transport equation, which rises well known numerical difficulties at sharp fronts or boundary layers, we use the streamline diffusion approach. The arising linear systems are solved using efficient iterative solvers with an AMG (algebraic multigrid) pre-conditioner. During each iteration step of the inversion scheme one needs to solve a multitude of forward and adjoint problems in order to calculate the sensitivities of each measurement and the related cross-covariance matrix of the unknown parameters and the observations. In order to reduce interprocess communications and to improve the scalability of the code on larger clusters

On the Preconditioning of a Newton-Krylov Solver for a High-Order reconstructed Discontinuous Galerkin Discretization of All-Speed Compressible Flow with Phase Change for Application in Laser-Based Additive Manufacturing

Energy Technology Data Exchange (ETDEWEB)

Weston, Brian T. [Univ. of California, Davis, CA (United States)

2017-05-17

This dissertation focuses on the development of a fully-implicit, high-order compressible ow solver with phase change. The work is motivated by laser-induced phase change applications, particularly by the need to develop large-scale multi-physics simulations of the selective laser melting (SLM) process in metal additive manufacturing (3D printing). Simulations of the SLM process require precise tracking of multi-material solid-liquid-gas interfaces, due to laser-induced melting/ solidi cation and evaporation/condensation of metal powder in an ambient gas. These rapid density variations and phase change processes tightly couple the governing equations, requiring a fully compressible framework to robustly capture the rapid density variations of the ambient gas and the melting/evaporation of the metal powder. For non-isothermal phase change, the velocity is gradually suppressed through the mushy region by a variable viscosity and Darcy source term model. The governing equations are discretized up to 4th-order accuracy with our reconstructed Discontinuous Galerkin spatial discretization scheme and up to 5th-order accuracy with L-stable fully implicit time discretization schemes (BDF2 and ESDIRK3-5). The resulting set of non-linear equations is solved using a robust Newton-Krylov method, with the Jacobian-free version of the GMRES solver for linear iterations. Due to the sti nes associated with the acoustic waves and thermal and viscous/material strength e ects, preconditioning the GMRES solver is essential. A robust and scalable approximate block factorization preconditioner was developed, which utilizes the velocity-pressure (vP) and velocity-temperature (vT) Schur complement systems. This multigrid block reduction preconditioning technique converges for high CFL/Fourier numbers and exhibits excellent parallel and algorithmic scalability on classic benchmark problems in uid dynamics (lid-driven cavity ow and natural convection heat transfer) as well as for laser

Analysis of multigrid methods on massively parallel computers: Architectural implications

Science.gov (United States)

Matheson, Lesley R.; Tarjan, Robert E.

1993-01-01

We study the potential performance of multigrid algorithms running on massively parallel computers with the intent of discovering whether presently envisioned machines will provide an efficient platform for such algorithms. We consider the domain parallel version of the standard V cycle algorithm on model problems, discretized using finite difference techniques in two and three dimensions on block structured grids of size 10(exp 6) and 10(exp 9), respectively. Our models of parallel computation were developed to reflect the computing characteristics of the current generation of massively parallel multicomputers. These models are based on an interconnection network of 256 to 16,384 message passing, 'workstation size' processors executing in an SPMD mode. The first model accomplishes interprocessor communications through a multistage permutation network. The communication cost is a logarithmic function which is similar to the costs in a variety of different topologies. The second model allows single stage communication costs only. Both models were designed with information provided by machine developers and utilize implementation derived parameters. With the medium grain parallelism of the current generation and the high fixed cost of an interprocessor communication, our analysis suggests an efficient implementation requires the machine to support the efficient transmission of long messages, (up to 1000 words) or the high initiation cost of a communication must be significantly reduced through an alternative optimization technique. Furthermore, with variable length message capability, our analysis suggests the low diameter multistage networks provide little or no advantage over a simple single stage communications network.

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

KAUST Repository

Efendiev, Y.; Galvis, J.; Kang, S. Ki; Lazarov, R.D.

2012-01-01

needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer

RECTC/RECTCF, 2. Order Elliptical Partial Differential Equation, Arbitrary Boundary Conditions

International Nuclear Information System (INIS)

Hackbusch, W.

1983-01-01

1 - Description of problem or function: A general linear elliptical second order partial differential equation on a rectangle with arbitrary boundary conditions is solved. 2 - Method of solution: Multi-grid iteration

Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators

Directory of Open Access Journals (Sweden)

Rami Nammour

2011-01-01

a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

Primal Domain Decomposition Method with Direct and Iterative Solver for Circuit-Field-Torque Coupled Parallel Finite Element Method to Electric Machine Modelling

Directory of Open Access Journals (Sweden)

Daniel Marcsa

2015-01-01

Full Text Available The analysis and design of electromechanical devices involve the solution of large sparse linear systems, and require therefore high performance algorithms. In this paper, the primal Domain Decomposition Method (DDM with parallel forward-backward and with parallel Preconditioned Conjugate Gradient (PCG solvers are introduced in two-dimensional parallel time-stepping finite element formulation to analyze rotating machine considering the electromagnetic field, external circuit and rotor movement. The proposed parallel direct and the iterative solver with two preconditioners are analyzed concerning its computational efficiency and number of iterations of the solver with different preconditioners. Simulation results of a rotating machine is also presented.

Parallel preconditioning techniques for sparse CG solvers

Energy Technology Data Exchange (ETDEWEB)

Basermann, A.; Reichel, B.; Schelthoff, C. [Central Institute for Applied Mathematics, Juelich (Germany)

1996-12-31

Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial differential equations. The large size and the condition of many technical or physical applications in this area result in the need for efficient parallelization and preconditioning techniques of the CG method. In particular for very ill-conditioned matrices, sophisticated preconditioner are necessary to obtain both acceptable convergence and accuracy of CG. Here, we investigate variants of polynomial and incomplete Cholesky preconditioners that markedly reduce the iterations of the simply diagonally scaled CG and are shown to be well suited for massively parallel machines.

Supersonic Turbulent Fuel-Air Mixing and Evaporation

National Research Council Canada - National Science Library

Moukalled, Fadi

2003-01-01

.... The convergence rate will be accelerated using a full non-linear multi-grid method. The discretization will use a second order scheme for diffusion and a pseudo-third order bounded scheme for convection...

Preconditioning for Allen–Cahn variational inequalities with non-local constraints

KAUST Repository

Blank, Luise; Sarbu, Lavinia; Stoll, Martin

2012-01-01

-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate

An FMM based on dual tree traversal for many-core architectures

KAUST Repository

Yokota, Rio

2013-01-01

interest to use FMM as a preconditioner for sparse linear solvers. A direct comparison with other state-of-the-art fast N-body codes demonstrates that orders of magnitude increase in performance can be achieved by careful selection of the optimal algorithm

Domain Decomposition for Computing Extremely Low Frequency Induced Current in the Human Body

OpenAIRE

Perrussel , Ronan; Voyer , Damien; Nicolas , Laurent; Scorretti , Riccardo; Burais , Noël

2011-01-01

International audience; Computation of electromagnetic fields in high resolution computational phantoms requires solving large linear systems. We present an application of Schwarz preconditioners with Krylov subspace methods for computing extremely low frequency induced fields in a phantom issued from the Visible Human.

Optimization of Regional Geodynamic Models for Mantle Dynamics

Science.gov (United States)

Knepley, M.; Isaac, T.; Jadamec, M. A.

2016-12-01

The SubductionGenerator program is used to construct high resolution, 3D regional thermal structures for mantle convection simulations using a variety of data sources, including sea floor ages and geographically referenced 3D slab locations based on seismic observations. The initial bulk temperature field is constructed using a half-space cooling model or plate cooling model, and related smoothing functions based on a diffusion length-scale analysis. In this work, we seek to improve the 3D thermal model and test different model geometries and dynamically driven flow fields using constraints from observed seismic velocities and plate motions. Through a formal adjoint analysis, we construct the primal-dual version of the multi-objective PDE-constrained optimization problem for the plate motions and seismic misfit. We have efficient, scalable preconditioners for both the forward and adjoint problems based upon a block preconditioning strategy, and a simple gradient update is used to improve the control residual. The full optimal control problem is formulated on a nested hierarchy of grids, allowing a nonlinear multigrid method to accelerate the solution.

Curvilinear immersed-boundary method for simulating unsteady flows in shallow natural streams with arbitrarily complex obstacles

Science.gov (United States)

Kang, Seokkoo; Borazjani, Iman; Sotiropoulos, Fotis

2008-11-01

Unsteady 3D simulations of flows in natural streams is a challenging task due to the complexity of the bathymetry, the shallowness of the flow, and the presence of multiple nature- and man-made obstacles. This work is motivated by the need to develop a powerful numerical method for simulating such flows using coherent-structure-resolving turbulence models. We employ the curvilinear immersed boundary method of Ge and Sotiropoulos (Journal of Computational Physics, 2007) and address the critical issue of numerical efficiency in large aspect ratio computational domains and grids such as those encountered in long and shallow open channels. We show that the matrix-free Newton-Krylov method for solving the momentum equations coupled with an algebraic multigrid method with incomplete LU preconditioner for solving the Poisson equation yield a robust and efficient procedure for obtaining time-accurate solutions in such problems. We demonstrate the potential of the numerical approach by carrying out a direct numerical simulation of flow in a long and shallow meandering stream with multiple hydraulic structures.

Higher Order, Hybrid BEM/FEM Methods Applied to Antenna Modeling

Science.gov (United States)

Fink, P. W.; Wilton, D. R.; Dobbins, J. A.

2002-01-01

In this presentation, the authors address topics relevant to higher order modeling using hybrid BEM/FEM formulations. The first of these is the limitation on convergence rates imposed by geometric modeling errors in the analysis of scattering by a dielectric sphere. The second topic is the application of an Incomplete LU Threshold (ILUT) preconditioner to solve the linear system resulting from the BEM/FEM formulation. The final tOpic is the application of the higher order BEM/FEM formulation to antenna modeling problems. The authors have previously presented work on the benefits of higher order modeling. To achieve these benefits, special attention is required in the integration of singular and near-singular terms arising in the surface integral equation. Several methods for handling these terms have been presented. It is also well known that achieving he high rates of convergence afforded by higher order bases may als'o require the employment of higher order geometry models. A number of publications have described the use of quadratic elements to model curved surfaces. The authors have shown in an EFIE formulation, applied to scattering by a PEC .sphere, that quadratic order elements may be insufficient to prevent the domination of modeling errors. In fact, on a PEC sphere with radius r = 0.58 Lambda(sub 0), a quartic order geometry representation was required to obtain a convergence benefi.t from quadratic bases when compared to the convergence rate achieved with linear bases. Initial trials indicate that, for a dielectric sphere of the same radius, - requirements on the geometry model are not as severe as for the PEC sphere. The authors will present convergence results for higher order bases as a function of the geometry model order in the hybrid BEM/FEM formulation applied to dielectric spheres. It is well known that the system matrix resulting from the hybrid BEM/FEM formulation is ill -conditioned. For many real applications, a good preconditioner is required

An extended algebraic variational multiscale-multigrid-multifractal method (XAVM4) for large-eddy simulation of turbulent two-phase flow

Science.gov (United States)

Rasthofer, U.; Wall, W. A.; Gravemeier, V.

2018-04-01

A novel and comprehensive computational method, referred to as the eXtended Algebraic Variational Multiscale-Multigrid-Multifractal Method (XAVM4), is proposed for large-eddy simulation of the particularly challenging problem of turbulent two-phase flow. The XAVM4 involves multifractal subgrid-scale modeling as well as a Nitsche-type extended finite element method as an approach for two-phase flow. The application of an advanced structural subgrid-scale modeling approach in conjunction with a sharp representation of the discontinuities at the interface between two bulk fluids promise high-fidelity large-eddy simulation of turbulent two-phase flow. The high potential of the XAVM4 is demonstrated for large-eddy simulation of turbulent two-phase bubbly channel flow, that is, turbulent channel flow carrying a single large bubble of the size of the channel half-width in this particular application.

StagBL : A Scalable, Portable, High-Performance Discretization and Solver Layer for Geodynamic Simulation

Science.gov (United States)

Sanan, P.; Tackley, P. J.; Gerya, T.; Kaus, B. J. P.; May, D.

2017-12-01

StagBL is an open-source parallel solver and discretization library for geodynamic simulation,encapsulating and optimizing operations essential to staggered-grid finite volume Stokes flow solvers.It provides a parallel staggered-grid abstraction with a high-level interface in C and Fortran.On top of this abstraction, tools are available to define boundary conditions and interact with particle systems.Tools and examples to efficiently solve Stokes systems defined on the grid are provided in small (direct solver), medium (simple preconditioners), and large (block factorization and multigrid) model regimes.By working directly with leading application codes (StagYY, I3ELVIS, and LaMEM) and providing an API and examples to integrate with others, StagBL aims to become a community tool supplying scalable, portable, reproducible performance toward novel science in regional- and planet-scale geodynamics and planetary science.By implementing kernels used by many research groups beneath a uniform abstraction layer, the library will enable optimization for modern hardware, thus reducing community barriers to large- or extreme-scale parallel simulation on modern architectures. In particular, the library will include CPU-, Manycore-, and GPU-optimized variants of matrix-free operators and multigrid components.The common layer provides a framework upon which to introduce innovative new tools.StagBL will leverage p4est to provide distributed adaptive meshes, and incorporate a multigrid convergence analysis tool.These options, in addition to a wealth of solver options provided by an interface to PETSc, will make the most modern solution techniques available from a common interface. StagBL in turn provides a PETSc interface, DMStag, to its central staggered grid abstraction.We present public version 0.5 of StagBL, including preliminary integration with application codes and demonstrations with its own demonstration application, StagBLDemo. Central to StagBL is the notion of an

Application Of Multi-grid Method On China Seas' Temperature Forecast

Science.gov (United States)

Li, W.; Xie, Y.; He, Z.; Liu, K.; Han, G.; Ma, J.; Li, D.

2006-12-01

Correlation scales have been used in traditional scheme of 3-dimensional variational (3D-Var) data assimilation to estimate the background error covariance for the numerical forecast and reanalysis of atmosphere and ocean for decades. However there are still some drawbacks of this scheme. First, the correlation scales are difficult to be determined accurately. Second, the positive definition of the first-guess error covariance matrix cannot be guaranteed unless the correlation scales are sufficiently small. Xie et al. (2005) indicated that a traditional 3D-Var only corrects some certain wavelength errors and its accuracy depends on the accuracy of the first-guess covariance. And in general, short wavelength error can not be well corrected until long one is corrected and then inaccurate first-guess covariance may mistakenly take long wave error as short wave ones and result in erroneous analysis. For the purpose of quickly minimizing the errors of long and short waves successively, a new 3D-Var data assimilation scheme, called multi-grid data assimilation scheme, is proposed in this paper. By assimilating the shipboard SST and temperature profiles data into a numerical model of China Seas, we applied this scheme in two-month data assimilation and forecast experiment which ended in a favorable result. Comparing with the traditional scheme of 3D-Var, the new scheme has higher forecast accuracy and a lower forecast Root-Mean-Square (RMS) error. Furthermore, this scheme was applied to assimilate the SST of shipboard, AVHRR Pathfinder Version 5.0 SST and temperature profiles at the same time, and a ten-month forecast experiment on sea temperature of China Seas was carried out, in which a successful forecast result was obtained. Particularly, the new scheme is demonstrated a great numerical efficiency in these analyses.

Non-linear triangle-based polynomial expansion nodal method for hexagonal core analysis

International Nuclear Information System (INIS)

Cho, Jin Young; Cho, Byung Oh; Joo, Han Gyu; Zee, Sung Qunn; Park, Sang Yong

2000-09-01

This report is for the implementation of triangle-based polynomial expansion nodal (TPEN) method to MASTER code in conjunction with the coarse mesh finite difference(CMFD) framework for hexagonal core design and analysis. The TPEN method is a variation of the higher order polynomial expansion nodal (HOPEN) method that solves the multi-group neutron diffusion equation in the hexagonal-z geometry. In contrast with the HOPEN method, only two-dimensional intranodal expansion is considered in the TPEN method for a triangular domain. The axial dependence of the intranodal flux is incorporated separately here and it is determined by the nodal expansion method (NEM) for a hexagonal node. For the consistency of node geometry of the MASTER code which is based on hexagon, TPEN solver is coded to solve one hexagonal node which is composed of 6 triangular nodes directly with Gauss elimination scheme. To solve the CMFD linear system efficiently, stabilized bi-conjugate gradient(BiCG) algorithm and Wielandt eigenvalue shift method are adopted. And for the construction of the efficient preconditioner of BiCG algorithm, the incomplete LU(ILU) factorization scheme which has been widely used in two-dimensional problems is used. To apply the ILU factorization scheme to three-dimensional problem, a symmetric Gauss-Seidel Factorization scheme is used. In order to examine the accuracy of the TPEN solution, several eigenvalue benchmark problems and two transient problems, i.e., a realistic VVER1000 and VVER440 rod ejection benchmark problems, were solved and compared with respective references. The results of eigenvalue benchmark problems indicate that non-linear TPEN method is very accurate showing less than 15 pcm of eigenvalue errors and 1% of maximum power errors, and fast enough to solve the three-dimensional VVER-440 problem within 5 seconds on 733MHz PENTIUM-III. In the case of the transient problems, the non-linear TPEN method also shows good results within a few minute of

Parallel supercomputing: Advanced methods, algorithms, and software for large-scale linear and nonlinear problems

Energy Technology Data Exchange (ETDEWEB)

Carey, G.F.; Young, D.M.

1993-12-31

The program outlined here is directed to research on methods, algorithms, and software for distributed parallel supercomputers. Of particular interest are finite element methods and finite difference methods together with sparse iterative solution schemes for scientific and engineering computations of very large-scale systems. Both linear and nonlinear problems will be investigated. In the nonlinear case, applications with bifurcation to multiple solutions will be considered using continuation strategies. The parallelizable numerical methods of particular interest are a family of partitioning schemes embracing domain decomposition, element-by-element strategies, and multi-level techniques. The methods will be further developed incorporating parallel iterative solution algorithms with associated preconditioners in parallel computer software. The schemes will be implemented on distributed memory parallel architectures such as the CRAY MPP, Intel Paragon, the NCUBE3, and the Connection Machine. We will also consider other new architectures such as the Kendall-Square (KSQ) and proposed machines such as the TERA. The applications will focus on large-scale three-dimensional nonlinear flow and reservoir problems with strong convective transport contributions. These are legitimate grand challenge class computational fluid dynamics (CFD) problems of significant practical interest to DOE. The methods developed and algorithms will, however, be of wider interest.

Seeking Space Aliens and the Strong Approximation Property: A (disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid

Science.gov (United States)

Southworth, Benjamin Scott

linear systems arises often in the modeling of biological and physical phenomenon, data analysis through graphs and networks, and other scientific applications. This work focuses primarily on linear systems resulting from the discretization of partial differential equations (PDEs). Because solving linear systems is the bottleneck of many large simulation codes, there is a rich field of research in developing "fast" solvers, with the ultimate goal being a method that solves an n x n linear system in O(n) operations. One of the most effective classes of solvers is algebraic multigrid (AMG), which is a multilevel iterative method based on projecting the problem into progressively smaller spaces, and scales like O(n) or O(nlog n) for certain classes of problems. The field of AMG is well-developed for symmetric positive definite matrices, and is typically most effective on linear systems resulting from the discretization of scalar elliptic PDEs, such as the heat equation. Systems of PDEs can add additional difficulties, but the underlying linear algebraic theory is consistent and, in many cases, an elliptic system of PDEs can be handled well by AMG with appropriate modifications of the solver. Solving general, nonsymmetric linear systems remains the wild west of AMG (and other fast solvers), lacking significant results in convergence theory as well as robust methods. Here, we develop new theoretical motivation and practical variations of AMG to solve nonsymmetric linear systems, often resulting from the discretization of hyperbolic PDEs. In particular, multilevel convergence of AMG for nonsymmetric systems is proven for the first time. A new nonsymmetric AMG solver is also developed based on an approximate ideal restriction, referred to as AIR, which is able to solve advection-dominated, hyperbolic-type problems that are outside the scope of existing AMG solvers and other fast iterative methods. AIR demonstrates scalable convergence on unstructured meshes, in multiple

Domain decomposition method for nonconforming finite element approximations of anisotropic elliptic problems on nonmatching grids

Energy Technology Data Exchange (ETDEWEB)

Maliassov, S.Y. [Texas A& M Univ., College Station, TX (United States)

1996-12-31

An approach to the construction of an iterative method for solving systems of linear algebraic equations arising from nonconforming finite element discretizations with nonmatching grids for second order elliptic boundary value problems with anisotropic coefficients is considered. The technique suggested is based on decomposition of the original domain into nonoverlapping subdomains. The elliptic problem is presented in the macro-hybrid form with Lagrange multipliers at the interfaces between subdomains. A block diagonal preconditioner is proposed which is spectrally equivalent to the original saddle point matrix and has the optimal order of arithmetical complexity. The preconditioner includes blocks for preconditioning subdomain and interface problems. It is shown that constants of spectral equivalence axe independent of values of coefficients and mesh step size.

Preconditioner Updates for Solving Sequences of Linear Systems in Matrix-Free Environment

Czech Academy of Sciences Publication Activity Database

Duintjer Tebbens, Jurjen; Tůma, Miroslav

2010-01-01

Roč. 17, č. 6 (2010), s. 997-1019 ISSN 1070-5325 R&D Projects: GA AV ČR IAA100300802; GA AV ČR KJB100300703 Grant - others:GA AV ČR(CZ) M100300902 Institutional research plan: CEZ:AV0Z10300504 Source of funding: I - inštitucionálna podpora na rozvoj VO Keywords : preconditioned iterative methods * matrix-free environment * factorization updates * inexact Newton-Krylov methods * incomplete factorizations Subject RIV: BA - General Mathematics Impact factor: 1.163, year: 2010

Improving Euler computations at low Mach numbers

NARCIS (Netherlands)

Koren, B.; Leer, van B.; Deconinck, H.; Koren, B.

1997-01-01

The paper consists of two parts, both dealing with conditioning techniques for lowMach-number Euler-flow computations, in which a multigrid technique is applied. In the first part, for subsonic flows and upwind-discretized, linearized 1-D Euler equations, the smoothing behavior of

Improving Euler computations at low Mach numbers

NARCIS (Netherlands)

Koren, B.

1996-01-01

This paper consists of two parts, both dealing with conditioning techniques for low-Mach-number Euler-flow computations, in which a multigrid technique is applied. In the first part, for subsonic flows and upwind-discretized linearized 1-D Euler equations, the smoothing behavior of

A non-linear multigrid method for the steady Euler equations

NARCIS (Netherlands)

Hemker, P.W.; Koren, B.; Dervieux, A.; Leer, van B.; Periaux, J.; Rizzi, A.

1989-01-01

Higher-order accurate Euler-flow solutions are presented for some airfoil test cases. Second-order accurate solutions are computed by an Iterative Defect Correction process. For two test cases even higher accuracy is obtained by the additional use of a ~xtrapolation technique. Finite volume

Using the Multiplicative Schwarz Alternating Algorithm (MSAA) for Solving the Large Linear System of Equations Related to Global Gravity Field Recovery up to Degree and Order 120

Science.gov (United States)

Safari, A.; Sharifi, M. A.; Amjadiparvar, B.

2010-05-01

The GRACE mission has substantiated the low-low satellite-to-satellite tracking (LL-SST) concept. The LL-SST configuration can be combined with the previously realized high-low SST concept in the CHAMP mission to provide a much higher accuracy. The line of sight (LOS) acceleration difference between the GRACE satellite pair is the mostly used observable for mapping the global gravity field of the Earth in terms of spherical harmonic coefficients. In this paper, mathematical formulae for LOS acceleration difference observations have been derived and the corresponding linear system of equations has been set up for spherical harmonic up to degree and order 120. The total number of unknowns is 14641. Such a linear equation system can be solved with iterative solvers or direct solvers. However, the runtime of direct methods or that of iterative solvers without a suitable preconditioner increases tremendously. This is the reason why we need a more sophisticated method to solve the linear system of problems with a large number of unknowns. Multiplicative variant of the Schwarz alternating algorithm is a domain decomposition method, which allows it to split the normal matrix of the system into several smaller overlaped submatrices. In each iteration step the multiplicative variant of the Schwarz alternating algorithm solves linear systems with the matrices obtained from the splitting successively. It reduces both runtime and memory requirements drastically. In this paper we propose the Multiplicative Schwarz Alternating Algorithm (MSAA) for solving the large linear system of gravity field recovery. The proposed algorithm has been tested on the International Association of Geodesy (IAG)-simulated data of the GRACE mission. The achieved results indicate the validity and efficiency of the proposed algorithm in solving the linear system of equations from accuracy and runtime points of view. Keywords: Gravity field recovery, Multiplicative Schwarz Alternating Algorithm, Low

Kronecker Products on Preconditioning

KAUST Repository

Gao, Longfei

2013-08-01

Numerical techniques for linear systems arising from discretization of partial differential equations are nowadays essential for understanding the physical world. Among these techniques, iterative methods and the accompanying preconditioning techniques have become increasingly popular due to their great potential on large scale computation. In this work, we present preconditioning techniques for linear systems built with tensor product basis functions. Efficient algorithms are designed for various problems by exploiting the Kronecker product structure in the matrices, inherited from tensor product basis functions. Specifically, we design preconditioners for mass matrices to remove the complexity from the basis functions used in isogeometric analysis, obtaining numerical performance independent of mesh size, polynomial order and continuity order; we also present a compound iteration preconditioner for stiffness matrices in two dimensions, obtaining fast convergence speed; lastly, for the Helmholtz problem, we present a strategy to `hide\\' its indefiniteness from Krylov subspace methods by eliminating the part of initial error that corresponds to those negative generalized eigenvalues. For all three cases, the Kronecker product structure in the matrices is exploited to achieve high computational efficiency.

Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation

NARCIS (Netherlands)

P.W. Hemker (Piet); M.H. van Raalte (Marc)

2002-01-01

textabstractIn this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the

A Direct Elliptic Solver Based on Hierarchically Low-Rank Schur Complements

KAUST Repository

Chávez, Gustavo

2017-03-17

A parallel fast direct solver for rank-compressible block tridiagonal linear systems is presented. Algorithmic synergies between Cyclic Reduction and Hierarchical matrix arithmetic operations result in a solver with O(Nlog2N) arithmetic complexity and O(NlogN) memory footprint. We provide a baseline for performance and applicability by comparing with well-known implementations of the $$\\\\mathcal{H}$$ -LU factorization and algebraic multigrid within a shared-memory parallel environment that leverages the concurrency features of the method. Numerical experiments reveal that this method is comparable with other fast direct solvers based on Hierarchical Matrices such as $$\\\\mathcal{H}$$ -LU and that it can tackle problems where algebraic multigrid fails to converge.

Weighted least squares phase unwrapping based on the wavelet transform

Science.gov (United States)

Chen, Jiafeng; Chen, Haiqin; Yang, Zhengang; Ren, Haixia

2007-01-01

The weighted least squares phase unwrapping algorithm is a robust and accurate method to solve phase unwrapping problem. This method usually leads to a large sparse linear equation system. Gauss-Seidel relaxation iterative method is usually used to solve this large linear equation. However, this method is not practical due to its extremely slow convergence. The multigrid method is an efficient algorithm to improve convergence rate. However, this method needs an additional weight restriction operator which is very complicated. For this reason, the multiresolution analysis method based on the wavelet transform is proposed. By applying the wavelet transform, the original system is decomposed into its coarse and fine resolution levels and an equivalent equation system with better convergence condition can be obtained. Fast convergence in separate coarse resolution levels speeds up the overall system convergence rate. The simulated experiment shows that the proposed method converges faster and provides better result than the multigrid method.

Guide to NavyFOAM V1.0

Science.gov (United States)

2011-04-01

solvers such as point-implicit Gauss - Seidel or algebraic multi-grid (AMG) methods. Velocity coupling to ensure mass conservation (continuity) is...state convergence is assumed when the forces (pressure and viscous) on the body change by a negligible amount from one iteration to the next. Figure...linear limited X.X with X.X< 1 will not converge to the same, more accurate, solution as Gauss linear corrected). Subdictionary: interpolationSchemes

A Numerical Study of Scalable Cardiac Electro-Mechanical Solvers on HPC Architectures

Directory of Open Access Journals (Sweden)

Piero Colli Franzone

2018-04-01

Full Text Available We introduce and study some scalable domain decomposition preconditioners for cardiac electro-mechanical 3D simulations on parallel HPC (High Performance Computing architectures. The electro-mechanical model of the cardiac tissue is composed of four coupled sub-models: (1 the static finite elasticity equations for the transversely isotropic deformation of the cardiac tissue; (2 the active tension model describing the dynamics of the intracellular calcium, cross-bridge binding and myofilament tension; (3 the anisotropic Bidomain model describing the evolution of the intra- and extra-cellular potentials in the deforming cardiac tissue; and (4 the ionic membrane model describing the dynamics of ionic currents, gating variables, ionic concentrations and stretch-activated channels. This strongly coupled electro-mechanical model is discretized in time with a splitting semi-implicit technique and in space with isoparametric finite elements. The resulting scalable parallel solver is based on Multilevel Additive Schwarz preconditioners for the solution of the Bidomain system and on BDDC preconditioned Newton-Krylov solvers for the non-linear finite elasticity system. The results of several 3D parallel simulations show the scalability of both linear and non-linear solvers and their application to the study of both physiological excitation-contraction cardiac dynamics and re-entrant waves in the presence of different mechano-electrical feedbacks.

A Robust Incomplete Factorization Preconditioner for Positive Definite Matrices

Czech Academy of Sciences Publication Activity Database

Benzi, M.; Tůma, Miroslav

2003-01-01

Roč. 10, - (2003), s. 385-400 ISSN 1070-5325 R&D Projects: GA AV ČR IAA2030801; GA AV ČR IAA1030103 Institutional research plan: AV0Z1030915 Keywords : sparse linear systems * positive definite matrices * preconditioned conjugate gradient s * incomplete factorization * A-orthogonalization * SAINV Subject RIV: BA - General Mathematics Impact factor: 1.042, year: 2003

Parallel conjugate gradient algorithms for manipulator dynamic simulation

Science.gov (United States)

Fijany, Amir; Scheld, Robert E.

1989-01-01

Parallel conjugate gradient algorithms for the computation of multibody dynamics are developed for the specialized case of a robot manipulator. For an n-dimensional positive-definite linear system, the Classical Conjugate Gradient (CCG) algorithms are guaranteed to converge in n iterations, each with a computation cost of O(n); this leads to a total computational cost of O(n sq) on a serial processor. A conjugate gradient algorithms is presented that provide greater efficiency using a preconditioner, which reduces the number of iterations required, and by exploiting parallelism, which reduces the cost of each iteration. Two Preconditioned Conjugate Gradient (PCG) algorithms are proposed which respectively use a diagonal and a tridiagonal matrix, composed of the diagonal and tridiagonal elements of the mass matrix, as preconditioners. Parallel algorithms are developed to compute the preconditioners and their inversions in O(log sub 2 n) steps using n processors. A parallel algorithm is also presented which, on the same architecture, achieves the computational time of O(log sub 2 n) for each iteration. Simulation results for a seven degree-of-freedom manipulator are presented. Variants of the proposed algorithms are also developed which can be efficiently implemented on the Robot Mathematics Processor (RMP).

Solving groundwater flow problems by conjugate-gradient methods and the strongly implicit procedure

Science.gov (United States)

Hill, Mary C.

1990-01-01

The performance of the preconditioned conjugate-gradient method with three preconditioners is compared with the strongly implicit procedure (SIP) using a scalar computer. The preconditioners considered are the incomplete Cholesky (ICCG) and the modified incomplete Cholesky (MICCG), which require the same computer storage as SIP as programmed for a problem with a symmetric matrix, and a polynomial preconditioner (POLCG), which requires less computer storage than SIP. Although POLCG is usually used on vector computers, it is included here because of its small storage requirements. In this paper, published comparisons of the solvers are evaluated, all four solvers are compared for the first time, and new test cases are presented to provide a more complete basis by which the solvers can be judged for typical groundwater flow problems. Based on nine test cases, the following conclusions are reached: (1) SIP is actually as efficient as ICCG for some of the published, linear, two-dimensional test cases that were reportedly solved much more efficiently by ICCG; (2) SIP is more efficient than other published comparisons would indicate when common convergence criteria are used; and (3) for problems that are three-dimensional, nonlinear, or both, and for which common convergence criteria are used, SIP is often more efficient than ICCG, and is sometimes more efficient than MICCG.

Domain decomposition methods and deflated Krylov subspace iterations

NARCIS (Netherlands)

Nabben, R.; Vuik, C.

2006-01-01

The balancing Neumann-Neumann (BNN) and the additive coarse grid correction (BPS) preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which

Progress in 3D Space-charge Calculations in the GPT Code

NARCIS (Netherlands)

Pöplau, G.; Rienen, van U.; Loos, de M.J.; Geer, van der S.B.

2004-01-01

The mesh-based 3D space-charge routine in the GPT (General Particle Tracer, Pulsar Physics) code scales linearly with the number of particles in terms of CPU time and allows a million particles to be tracked on a normal PC. The crucial ingredient of the routine is a non-equidistant multi-grid

Higher-order differencing method with a multigrid approach for the solution of the incompressible flow equations at high Reynolds numbers

International Nuclear Information System (INIS)

Tzanos, C.P.

1992-01-01

A higher-order differencing method was recently proposed for the convection-diffusion equation, which even with a coarse mesh gives oscillation-free solutions that are far more accurate than those of the upwind scheme. In this paper, the performance of this method is investigated in conjunction with the performance of different iterative solvers for the solution of the Navier-Stokes equations in the vorticity-streamfunction formulation for incompressible flow at high Reynolds numbers. Flow in a square cavity with a moving lid was chosen as a model problem. Solvers that performed well at low Re numbers either failed to converge or had a computationally prohibitive convergence rate at high Re numbers. The additive correction method of Settari and Aziz and an iterative incomplete lower and upper (ILU) solver were used in a multigrid approach that performed well in the whole range of Re numbers considered (from 1000 to 10,000) and for uniform as well as nonuniform grids. At high Re numbers, point or line Gauss-Seidel solvers converged with uniform grids, but failed to converge with nonuniform grids

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

KAUST Repository

Farrell, Patricio; Pestana, Jennifer

2015-01-01

. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported RBFs and a multiscale technique. But the condition number and sparsity

Numerical method for the eigenvalue problem and the singular equation by using the multi-grid method and application to ordinary differential equation

International Nuclear Information System (INIS)

Kanki, Takashi; Uyama, Tadao; Tokuda, Shinji.

1995-07-01

In the numerical method to compute the matching data which are necessary for resistive MHD stability analyses, it is required to solve the eigenvalue problem and the associated singular equation. An iterative method is developed to solve the eigenvalue problem and the singular equation. In this method, the eigenvalue problem is replaced with an equivalent nonlinear equation and a singular equation is derived from Newton's method for the nonlinear equation. The multi-grid method (MGM), a high speed iterative method, can be applied to this method. The convergence of the eigenvalue and the eigenvector, and the CPU time in this method are investigated for a model equation. It is confirmed from the numerical results that this method is effective for solving the eigenvalue problem and the singular equation with numerical stability and high accuracy. It is shown by improving the MGM that the CPU time for this method is 50 times shorter than that of the direct method. (author)

Physics-Based Preconditioning of a Compressible Flow Solver for Large-Scale Simulations of Additive Manufacturing Processes

Science.gov (United States)

Weston, Brian; Nourgaliev, Robert; Delplanque, Jean-Pierre

2017-11-01

We present a new block-based Schur complement preconditioner for simulating all-speed compressible flow with phase change. The conservation equations are discretized with a reconstructed Discontinuous Galerkin method and integrated in time with fully implicit time discretization schemes. The resulting set of non-linear equations is converged using a robust Newton-Krylov framework. Due to the stiffness of the underlying physics associated with stiff acoustic waves and viscous material strength effects, we solve for the primitive-variables (pressure, velocity, and temperature). To enable convergence of the highly ill-conditioned linearized systems, we develop a physics-based preconditioner, utilizing approximate block factorization techniques to reduce the fully-coupled 3×3 system to a pair of reduced 2×2 systems. We demonstrate that our preconditioned Newton-Krylov framework converges on very stiff multi-physics problems, corresponding to large CFL and Fourier numbers, with excellent algorithmic and parallel scalability. Results are shown for the classic lid-driven cavity flow problem as well as for 3D laser-induced phase change. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Application of preconditioned conjugate gradient-like methods to reactor kinetics

International Nuclear Information System (INIS)

Yang, D.Y.; Chen, G.S.; Chou, H.P.

1993-01-01

Several conjugate gradient-like (CG-like) methods are applied to solve the nonsymmetric linear systems of equations derived from the time-dependent two-dimensional two-energy-group neutron diffusion equations by a finite difference approximation. The methods are: the generalized conjugate residual method; the generalized conjugate gradient least-square method; the generalized minimal residual method (GMRES); the conjugate gradient square method; and a variant of bi-conjugate gradient method (Bi-CGSTAB). In order to accelerate these methods, six preconditioning techniques are investigated. Two are based on pointwise incomplete factorization: the incomplete LU (ILU) and the modified incomplete LU (MILU) decompositions; two, based on the block tridiagonal structure of the coefficient matrix, are blockwise and modified blockwise incomplete factorizations, BILU and MBILU; two are the alternating-direction implicit and symmetric successive overrelaxation (SSOR) preconditioners, derived from the basic iterative schemes. Comparisons are made by using CG-like methods combined with different preconditioners to solve a sequence of time-step reactor transient problems. Numerical tests indicate that preconditioned BI-CGSTAB with the preconditioner MBILU requires less CPU time and fewer iterations than other methods. The preconditioned CG-like methods are less sensitive to the time-step size used than the Chebyshev semi-iteration method and line SOR method. The indication is that the CGS, Bi-CGSTAB and GMRES methods are, on average, better than the other methods in reactor kinetics computation and that a good preconditioner is more important than the choice of CG-like methods. The MILU decomposition based on the conventional row-sum criterion has difficulty yielding a convergent solution and an improved version is introduced. (author)

Diablo 2.0: A modern DNS/LES code for the incompressible NSE leveraging new time-stepping and multigrid algorithms

Science.gov (United States)

Cavaglieri, Daniele; Bewley, Thomas; Mashayek, Ali

2015-11-01

We present a new code, Diablo 2.0, for the simulation of the incompressible NSE in channel and duct flows with strong grid stretching near walls. The code leverages the fractional step approach with a few twists. New low-storage IMEX (implicit-explicit) Runge-Kutta time-marching schemes are tested which are superior to the traditional and widely-used CN/RKW3 (Crank-Nicolson/Runge-Kutta-Wray) approach; the new schemes tested are L-stable in their implicit component, and offer improved overall order of accuracy and stability with, remarkably, similar computational cost and storage requirements. For duct flow simulations, our new code also introduces a new smoother for the multigrid solver for the pressure Poisson equation. The classic approach, involving alternating-direction zebra relaxation, is replaced by a new scheme, dubbed tweed relaxation, which achieves the same convergence rate with roughly half the computational cost. The code is then tested on the simulation of a shear flow instability in a duct, a classic problem in fluid mechanics which has been the object of extensive numerical modelling for its role as a canonical pathway to energetic turbulence in several fields of science and engineering.

Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals

Energy Technology Data Exchange (ETDEWEB)

Zuehlsdorff, T. J., E-mail: tjz21@cam.ac.uk; Payne, M. C. [Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE (United Kingdom); Hine, N. D. M. [Department of Physics, University of Warwick, Coventry CV4 7AL (United Kingdom); Haynes, P. D. [Department of Materials, Imperial College London, Exhibition Road, London SW7 2AZ (United Kingdom); Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ (United Kingdom); Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, Exhibition Road, London SW7 2AZ (United Kingdom)

2015-11-28

We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.

M-step preconditioned conjugate gradient methods

Science.gov (United States)

Adams, L.

1983-01-01

Preconditioned conjugate gradient methods for solving sparse symmetric and positive finite systems of linear equations are described. Necessary and sufficient conditions are given for when these preconditioners can be used and an analysis of their effectiveness is given. Efficient computer implementations of these methods are discussed and results on the CYBER 203 and the Finite Element Machine under construction at NASA Langley Research Center are included.

On the Parallel Elliptic Single/Multigrid Solutions about Aligned and Nonaligned Bodies Using the Virtual Machine for Multiprocessors

Directory of Open Access Journals (Sweden)

A. Averbuch

1994-01-01

Full Text Available Parallel elliptic single/multigrid solutions around an aligned and nonaligned body are presented and implemented on two multi-user and single-user shared memory multiprocessors (Sequent Symmetry and MOS and on a distributed memory multiprocessor (a Transputer network. Our parallel implementation uses the Virtual Machine for Muli-Processors (VMMP, a software package that provides a coherent set of services for explicitly parallel application programs running on diverse multiple instruction multiple data (MIMD multiprocessors, both shared memory and message passing. VMMP is intended to simplify parallel program writing and to promote portable and efficient programming. Furthermore, it ensures high portability of application programs by implementing the same services on all target multiprocessors. The performance of our algorithm is investigated in detail. It is seen to fit well the above architectures when the number of processors is less than the maximal number of grid points along the axes. In general, the efficiency in the nonaligned case is higher than in the aligned case. Alignment overhead is observed to be up to 200% in the shared-memory case and up to 65% in the message-passing case. We have demonstrated that when using VMMP, the portability of the algorithms is straightforward and efficient.

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Directory of Open Access Journals (Sweden)

Chunye Gong

2015-01-01

Full Text Available We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations are O(N2M, O(NM2, and O(NM(M + N compared with O(MN for the classical partial differential equations with finite difference methods, where M, N are the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator, short memory principle, fast Fourier transform (FFT based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

Preconditioned augmented Lagrangian formulation for nearly incompressible cardiac mechanics.

Science.gov (United States)

Campos, Joventino Oliveira; Dos Santos, Rodrigo Weber; Sundnes, Joakim; Rocha, Bernardo Martins

2018-04-01

Computational modeling of the heart is a subject of substantial medical and scientific interest, which may contribute to increase the understanding of several phenomena associated with cardiac physiological and pathological states. Modeling the mechanics of the heart have led to considerable insights, but it still represents a complex and a demanding computational problem, especially in a strongly coupled electromechanical setting. Passive cardiac tissue is commonly modeled as hyperelastic and is characterized by quasi-incompressible, orthotropic, and nonlinear material behavior. These factors are known to be very challenging for the numerical solution of the model. The near-incompressibility is known to cause numerical issues such as the well-known locking phenomenon and ill-conditioning of the stiffness matrix. In this work, the augmented Lagrangian method is used to handle the nearly incompressible condition. This approach can potentially improve computational performance by reducing the condition number of the stiffness matrix and thereby improving the convergence of iterative solvers. We also improve the performance of iterative solvers by the use of an algebraic multigrid preconditioner. Numerical results of the augmented Lagrangian method combined with a preconditioned iterative solver for a cardiac mechanics benchmark suite are presented to show its improved performance. Copyright © 2017 John Wiley & Sons, Ltd.

Totally parallel multilevel algorithms

Science.gov (United States)

Frederickson, Paul O.

1988-01-01

Four totally parallel algorithms for the solution of a sparse linear system have common characteristics which become quite apparent when they are implemented on a highly parallel hypercube such as the CM2. These four algorithms are Parallel Superconvergent Multigrid (PSMG) of Frederickson and McBryan, Robust Multigrid (RMG) of Hackbusch, the FFT based Spectral Algorithm, and Parallel Cyclic Reduction. In fact, all four can be formulated as particular cases of the same totally parallel multilevel algorithm, which are referred to as TPMA. In certain cases the spectral radius of TPMA is zero, and it is recognized to be a direct algorithm. In many other cases the spectral radius, although not zero, is small enough that a single iteration per timestep keeps the local error within the required tolerance.

Convergence Analysis of the Preconditioned Group Splitting Methods in Boundary Value Problems

Directory of Open Access Journals (Sweden)

Norhashidah Hj. Mohd Ali

2012-01-01

Full Text Available The construction of a specific splitting-type preconditioner in block formulation applied to a class of group relaxation iterative methods derived from the centred and rotated (skewed finite difference approximations has been shown to improve the convergence rates of these methods. In this paper, we present some theoretical convergence analysis on this preconditioner specifically applied to the linear systems resulted from these group iterative schemes in solving an elliptic boundary value problem. We will theoretically show the relationship between the spectral radiuses of the iteration matrices of the preconditioned methods which affects the rate of convergence of these methods. We will also show that the spectral radius of the preconditioned matrices is smaller than that of their unpreconditioned counterparts if the relaxation parameter is in a certain optimum range. Numerical experiments will also be presented to confirm the agreement between the theoretical and the experimental results.

Automatic mesh refinement and local multigrid methods for contact problems: application to the Pellet-Cladding mechanical Interaction

International Nuclear Information System (INIS)

Liu, Hao

2016-01-01

This Ph.D. work takes place within the framework of studies on Pellet-Cladding mechanical Interaction (PCI) which occurs in the fuel rods of pressurized water reactor. This manuscript focuses on automatic mesh refinement to simulate more accurately this phenomena while maintaining acceptable computational time and memory space for industrial calculations. An automatic mesh refinement strategy based on the combination of the Local Defect Correction multigrid method (LDC) with the Zienkiewicz and Zhu a posteriori error estimator is proposed. The estimated error is used to detect the zones to be refined, where the local sub-grids of the LDC method are generated. Several stopping criteria are studied to end the refinement process when the solution is accurate enough or when the refinement does not improve the global solution accuracy anymore. Numerical results for elastic 2D test cases with pressure discontinuity show the efficiency of the proposed strategy. The automatic mesh refinement in case of unilateral contact problems is then considered. The strategy previously introduced can be easily adapted to the multi-body refinement by estimating solution error on each body separately. Post-processing is often necessary to ensure the conformity of the refined areas regarding the contact boundaries. A variety of numerical experiments with elastic contact (with or without friction, with or without an initial gap) confirms the efficiency and adaptability of the proposed strategy. (author) [fr

On Signed Incomplete Cholesky Factorization Preconditioners for Saddle-Point Systems

Czech Academy of Sciences Publication Activity Database

Scott, J.; Tůma, Miroslav

2014-01-01

Roč. 36, č. 6 (2014), A2984-A3010 ISSN 1064-8275 R&D Projects: GA ČR GA13-06684S Grant - others:EPSRC(GB) EP/I013067/1 Program:GA Institutional support: RVO:67985807 Keywords : sparse matrices * sparse linear systems * indefinite symmetric systems * saddle-point systems * iterative solvers * preconditioning * incomplete Cholesky factorization Subject RIV: BA - General Mathematics Impact factor: 1.854, year: 2014

LINEAR2007, Linear-Linear Interpolation of ENDF Format Cross-Sections

International Nuclear Information System (INIS)

2007-01-01

1 - Description of program or function: LINEAR converts evaluated cross sections in the ENDF/B format into a tabular form that is subject to linear-linear interpolation in energy and cross section. The code also thins tables of cross sections already in that form. Codes used subsequently need thus to consider only linear-linear data. IAEA1311/15: This version include the updates up to January 30, 2007. Changes in ENDF/B-VII Format and procedures, as well as the evaluations themselves, make it impossible for versions of the ENDF/B pre-processing codes earlier than PREPRO 2007 (2007 Version) to accurately process current ENDF/B-VII evaluations. The present code can handle all existing ENDF/B-VI evaluations through release 8, which will be the last release of ENDF/B-VI. Modifications from previous versions: - Linear VERS. 2007-1 (JAN. 2007): checked against all ENDF/B-VII; increased page size from 60,000 to 600,000 points 2 - Method of solution: Each section of data is considered separately. Each section of File 3, 23, and 27 data consists of a table of cross section versus energy with any of five interpolation laws. LINEAR will replace each section with a new table of energy versus cross section data in which the interpolation law is always linear in energy and cross section. The histogram (constant cross section between two energies) interpolation law is converted to linear-linear by substituting two points for each initial point. The linear-linear is not altered. For the log-linear, linear-log and log- log laws, the cross section data are converted to linear by an interval halving algorithm. Each interval is divided in half until the value at the middle of the interval can be approximated by linear-linear interpolation to within a given accuracy. The LINEAR program uses a multipoint fractional error thinning algorithm to minimize the size of each cross section table

Computing Low-Rank Approximation of a Dense Matrix on Multicore CPUs with a GPU and Its Application to Solving a Hierarchically Semiseparable Linear System of Equations

Directory of Open Access Journals (Sweden)

Ichitaro Yamazaki

2015-01-01

of their low-rank properties. To compute a low-rank approximation of a dense matrix, in this paper, we study the performance of QR factorization with column pivoting or with restricted pivoting on multicore CPUs with a GPU. We first propose several techniques to reduce the postprocessing time, which is required for restricted pivoting, on a modern CPU. We then examine the potential of using a GPU to accelerate the factorization process with both column and restricted pivoting. Our performance results on two eight-core Intel Sandy Bridge CPUs with one NVIDIA Kepler GPU demonstrate that using the GPU, the factorization time can be reduced by a factor of more than two. In addition, to study the performance of our implementations in practice, we integrate them into a recently developed software StruMF which algebraically exploits such low-rank structures for solving a general sparse linear system of equations. Our performance results for solving Poisson's equations demonstrate that the proposed techniques can significantly reduce the preconditioner construction time of StruMF on the CPUs, and the construction time can be further reduced by 10%–50% using the GPU.

Computational Fluid Dynamics (CFD) Simulations of a Finned Projectile with Microflaps for Flow Control

Science.gov (United States)

2016-04-01

multigrid approach as the means to efficiently solve the linear algebra problem that results in applying an implicit scheme to both steady-state and...projectile, CFD applications , microflaps, optimized control force 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU 18...counterparts, and successive relaxation allows update of cells as information becomes available and thus aids convergence. CFD++ uses an algebraic

On the Eigenvalues and Eigenvectors of Block Triangular Preconditioned Block Matrices

KAUST Repository

Pestana, Jennifer

2014-01-01

Block lower triangular matrices and block upper triangular matrices are popular preconditioners for 2×2 block matrices. In this note we show that a block lower triangular preconditioner gives the same spectrum as a block upper triangular preconditioner and that the eigenvectors of the two preconditioned matrices are related. © 2014 Society for Industrial and Applied Mathematics.

A Combined Preconditioning Strategy for Nonsymmetric Systems

KAUST Repository

Ayuso Dios, Blanca

2014-01-01

We present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. A variable preconditioner, combining the original nonsymmetric one and a weighted least-squares version of it, is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.

Preconditioning for partial differential equation constrained optimization with control constraints

KAUST Repository

Stoll, Martin

2011-10-18

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal-dual active set method. We also consider the Moreau-Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. © 2011 John Wiley & Sons, Ltd.

Low-memory iterative density fitting.

Science.gov (United States)

Grajciar, Lukáš

2015-07-30

A new low-memory modification of the density fitting approximation based on a combination of a continuous fast multipole method (CFMM) and a preconditioned conjugate gradient solver is presented. Iterative conjugate gradient solver uses preconditioners formed from blocks of the Coulomb metric matrix that decrease the number of iterations needed for convergence by up to one order of magnitude. The matrix-vector products needed within the iterative algorithm are calculated using CFMM, which evaluates them with the linear scaling memory requirements only. Compared with the standard density fitting implementation, up to 15-fold reduction of the memory requirements is achieved for the most efficient preconditioner at a cost of only 25% increase in computational time. The potential of the method is demonstrated by performing density functional theory calculations for zeolite fragment with 2592 atoms and 121,248 auxiliary basis functions on a single 12-core CPU workstation. © 2015 Wiley Periodicals, Inc.

Preconditioning for partial differential equation constrained optimization with control constraints

KAUST Repository

Stoll, Martin; Wathen, Andy

2011-01-01

Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal-dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal-dual active set method. We also consider the Moreau-Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. © 2011 John Wiley & Sons, Ltd.

Improving the Efficiency of the Nodal Integral Method With the Portable, Extensible Tool-kit for Scientific Computation

International Nuclear Information System (INIS)

Toreja, Allen J.; Uddin, Rizwan

2002-01-01

An existing implementation of the nodal integral method for the time-dependent convection-diffusion equation is modified to incorporate various PETSc (Portable, Extensible Tool-kit for Scientific Computation) solver and pre-conditioner routines. In the modified implementation, the default iterative Gauss-Seidel solver is replaced with one of the following PETSc iterative linear solver routines: Generalized Minimal Residuals, Stabilized Bi-conjugate Gradients, or Transpose-Free Quasi-Minimal Residuals. For each solver, a Jacobi or a Successive Over-Relaxation pre-conditioner is used. Two sample problems, one with a low Peclet number and one with a high Peclet number, are solved using the new implementation. In all the cases tested, the new implementation with the PETSc solver routines outperforms the original Gauss-Seidel implementation. Moreover, the PETSc Stabilized Bi-conjugate Gradients routine performs the best on the two sample problems leading to CPU times that are less than half the CPU times of the original implementation. (authors)

Cosmic Microwave Background Mapmaking with a Messenger Field

Science.gov (United States)

Huffenberger, Kevin M.; Næss, Sigurd K.

2018-01-01

We apply a messenger field method to solve the linear minimum-variance mapmaking equation in the context of Cosmic Microwave Background (CMB) observations. In simulations, the method produces sky maps that converge significantly faster than those from a conjugate gradient descent algorithm with a diagonal preconditioner, even though the computational cost per iteration is similar. The messenger method recovers large scales in the map better than conjugate gradient descent, and yields a lower overall χ2. In the single, pencil beam approximation, each iteration of the messenger mapmaking procedure produces an unbiased map, and the iterations become more optimal as they proceed. A variant of the method can handle differential data or perform deconvolution mapmaking. The messenger method requires no preconditioner, but a high-quality solution needs a cooling parameter to control the convergence. We study the convergence properties of this new method and discuss how the algorithm is feasible for the large data sets of current and future CMB experiments.

Linearization Method and Linear Complexity

Science.gov (United States)

Tanaka, Hidema

We focus on the relationship between the linearization method and linear complexity and show that the linearization method is another effective technique for calculating linear complexity. We analyze its effectiveness by comparing with the logic circuit method. We compare the relevant conditions and necessary computational cost with those of the Berlekamp-Massey algorithm and the Games-Chan algorithm. The significant property of a linearization method is that it needs no output sequence from a pseudo-random number generator (PRNG) because it calculates linear complexity using the algebraic expression of its algorithm. When a PRNG has n [bit] stages (registers or internal states), the necessary computational cost is smaller than O(2n). On the other hand, the Berlekamp-Massey algorithm needs O(N2) where N(≅2n) denotes period. Since existing methods calculate using the output sequence, an initial value of PRNG influences a resultant value of linear complexity. Therefore, a linear complexity is generally given as an estimate value. On the other hand, a linearization method calculates from an algorithm of PRNG, it can determine the lower bound of linear complexity.

Stochastic Galerkin methods for the steady-state Navier–Stokes equations

Energy Technology Data Exchange (ETDEWEB)

Sousedík, Bedřich, E-mail: sousedik@umbc.edu [Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (United States); Elman, Howard C., E-mail: elman@cs.umd.edu [Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 (United States)

2016-07-01

We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.

Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction.

Science.gov (United States)

Fessler, J A; Booth, S D

1999-01-01

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Parallel assembling and equation solving via graph algorithms with an application to the FE simulation of metal extrusion processes

CERN Document Server

Unterkircher, A

2005-01-01

We propose methods for parallel assembling and iterative equation solving based on graph algorithms. The assembling technique is independent of dimension, element type and model shape. As a parallel solving technique we construct a multiplicative symmetric Schwarz preconditioner for the conjugate gradient method. Both methods have been incorporated into a non-linear FE code to simulate 3D metal extrusion processes. We illustrate the efficiency of these methods on shared memory computers by realistic examples.

A Combined Preconditioning Strategy for Nonsymmetric Systems

Energy Technology Data Exchange (ETDEWEB)

de Dios, B. Ayuso [Univ. of Bologna (Italy). Dept. of Mathematics; King Abdullah Univ. of Science and Technology, Thuwal (Saudi Arabia); Barker, A. T. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Vassilevski, P. S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

2014-11-04

Here, we present and analyze a class of nonsymmetric preconditioners within a normal (weighted least-squares) matrix form for use in GMRES to solve nonsymmetric matrix problems that typically arise in finite element discretizations. An example of the additive Schwarz method applied to nonsymmetric but definite matrices is presented for which the abstract assumptions are verified. Variable preconditioner, which combines the original nonsymmetric one and a weighted least-squares version of it, and it is shown to be convergent and provides a viable strategy for using nonsymmetric preconditioners in practice. Numerical results are included to assess the theory and the performance of the proposed preconditioners.

Higher-order schemes for the Laplace transformation method for parabolic problems

KAUST Repository

Douglas, C.

2011-01-01

In this paper we solve linear parabolic problems using the three stage noble algorithms. First, the time discretization is approximated using the Laplace transformation method, which is both parallel in time (and can be in space, too) and extremely high order convergent. Second, higher-order compact schemes of order four and six are used for the the spatial discretization. Finally, the discretized linear algebraic systems are solved using multigrid to show the actual convergence rate for numerical examples, which are compared to other numerical solution methods. © 2011 Springer-Verlag.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Directory of Open Access Journals (Sweden)

Ben C. Yee

2017-09-01

Full Text Available In this paper, we present a new multilevel in space and energy diffusion (MSED method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1 a grey (one-group diffusion equation used to efficiently converge the fission source and eigenvalue, (2 a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3 a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

Preconditioned conjugate-gradient methods for low-speed flow calculations

Science.gov (United States)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1993-01-01

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations is integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the Lower-Upper Successive Symmetric Over-Relaxation iterative scheme is more efficient than a preconditioner based on Incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional Line Gauss-Seidel Relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver.

Preconditioned Conjugate Gradient methods for low speed flow calculations

Science.gov (United States)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1993-01-01

An investigation is conducted into the viability of using a generalized Conjugate Gradient-like method as an iterative solver to obtain steady-state solutions of very low-speed fluid flow problems. Low-speed flow at Mach 0.1 over a backward-facing step is chosen as a representative test problem. The unsteady form of the two dimensional, compressible Navier-Stokes equations are integrated in time using discrete time-steps. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. The new iterative solver is used to solve a linear system of equations at each step of the time-integration. Preconditioning techniques are used with the new solver to enhance the stability and the convergence rate of the solver and are found to be critical to the overall success of the solver. A study of various preconditioners reveals that a preconditioner based on the lower-upper (L-U)-successive symmetric over-relaxation iterative scheme is more efficient than a preconditioner based on incomplete L-U factorizations of the iteration matrix. The performance of the new preconditioned solver is compared with a conventional line Gauss-Seidel relaxation (LGSR) solver. Overall speed-up factors of 28 (in terms of global time-steps required to converge to a steady-state solution) and 20 (in terms of total CPU time on one processor of a CRAY-YMP) are found in favor of the new preconditioned solver, when compared with the LGSR solver.

Preconditioned conjugate gradient methods for the Navier-Stokes equations

Science.gov (United States)

Ajmani, Kumud; Ng, Wing-Fai; Liou, Meng-Sing

1994-01-01

A preconditioned Krylov subspace method (GMRES) is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux-split formulation. Several preconditioning techniques are investigated to enhance the efficiency and convergence rate of the implicit solver based on the GMRES algorithm. The superiority of the new solver is established by comparisons with a conventional implicit solver, namely line Gauss-Seidel relaxation (LGSR). Computational test results for low-speed (incompressible flow over a backward-facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade), and hypersonic flow (shock-on-shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the Mach 0.1 case, overall speedup factors of up to 17 (in terms of time-steps) and 15 (in terms of CPU time on a CRAY-YMP/8) are found in favor of the preconditioned GMRES solver, when compared with the LGSR solver. The corresponding speedup factors for the transonic flow case are 17 and 23, respectively. The hypersonic flow case shows slightly lower speedup factors of 9 and 13, respectively. The study of preconditioners conducted in this research reveals that a new LUSGS-type preconditioner is much more efficient than a conventional incomplete LU-type preconditioner.

Potential formation in the plasma confinement region of a radio-frequency plugged linear device

International Nuclear Information System (INIS)

Fujita, Hideki; Kumazawa, Ryuhei; Howald, A.M.; Okamura, Shoichi; Sato, Teruyuki; Adati, Keizo; Garner, H.R.; Nishimura, Kiyohiko.

1987-08-01

Plasma potential formation in an open-ended plasma confinement system with RF plugging (the RFC-XX-M device) is investigated. The plasma potential in the central confinement region is measured with a heavy ion beam probe system and potentials at the RF plug section are measured with multi-grid energy analyzers. The measured plasma potential is compared with that deduced from the generalized Pastukhov formula. Results show that the plasma potential develops as an ambipolar potential to equate ion and electron end losses. During RF plugging, electrons are heated by Landau damping, while ions are not heated since adiabatic conditions apply during ion plugging in this experiment. (author)

Preface: Introductory Remarks: Linear Scaling Methods

Science.gov (United States)

Bowler, D. R.; Fattebert, J.-L.; Gillan, M. J.; Haynes, P. D.; Skylaris, C.-K.

2008-07-01

problems, there was another important theme: extending functionality. The search for greater accuracy has given an implementation of density functional designed to model van der Waals interactions accurately as well as local correlation, TDDFT and QMC and GW methods which, while not explicitly O(N), take advantage of localisation. All speakers at the workshop were invited to contribute to this issue, but not all were able to do this. Hence it is useful to give a complete list of the talks presented, with the names of the sessions; however, many talks fell within more than one area. This is an exciting time for linear scaling methods, which are already starting to contribute significantly to important scientific problems. Applications to nanostructures and biomolecules A DFT study on the structural stability of Ge 3D nanostructures on Si(001) using CONQUEST Tsuyoshi Miyazaki, D R Bowler, M J Gillan, T Otsuka and T Ohno Large scale electronic structure calculation theory and several applications Takeo Fujiwara and Takeo Hoshi ONETEP:Linear-scaling DFT with plane waves Chris-Kriton Skylaris, Peter D Haynes, Arash A Mostofi, Mike C Payne Maximally-localised Wannier functions as building blocks for large-scale electronic structure calculations Arash A Mostofi and Nicola Marzari A linear scaling three dimensional fragment method for ab initio calculations Lin-Wang Wang, Zhengji Zhao, Juan Meza Peta-scalable reactive Molecular dynamics simulation of mechanochemical processes Aiichiro Nakano, Rajiv K. Kalia, Ken-ichi Nomura, Fuyuki Shimojo and Priya Vashishta Recent developments and applications of the real-space multigrid (RMG) method Jerzy Bernholc, M Hodak, W Lu, and F Ribeiro Energy minimisation functionals and algorithms CONQUEST: A linear scaling DFT Code David R Bowler, Tsuyoshi Miyazaki, Antonio Torralba, Veronika Brazdova, Milica Todorovic, Takao Otsuka and Mike Gillan Kernel optimisation and the physical significance of optimised local orbitals in the ONETEP code Peter

Linear Algebra and Smarandache Linear Algebra

OpenAIRE

Vasantha, Kandasamy

2003-01-01

The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite p...

PyTrilinos: Recent Advances in the Python Interface to Trilinos

Directory of Open Access Journals (Sweden)

William F. Spotz

2012-01-01

Full Text Available PyTrilinos is a set of Python interfaces to compiled Trilinos packages. This collection supports serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, algebraic and multilevel preconditioners, nonlinear solvers and continuation algorithms, eigensolvers and partitioning algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms and matrix generation. PyTrilinos vector objects are compatible with the popular NumPy Python package. As a Python front end to compiled libraries, PyTrilinos takes advantage of the flexibility and ease of use of Python, and the efficiency of the underlying C++, C and Fortran numerical kernels. This paper covers recent, previously unpublished advances in the PyTrilinos package.

Improving the Stability and Robustness of Incomplete Symmetric Indefinite Factorization Preconditioners

Czech Academy of Sciences Publication Activity Database

Scott, J.; Tůma, Miroslav

2017-01-01

Roč. 24, č. 5 (2017), č. článku e2099. ISSN 1070-5325 Grant - others:GA ČR(CZ) GC17-04150J; GA ČR(CZ) GC17-04150J; EPSRC(GB) EP/I013067/1 Institutional support: RVO:67985807 Keywords : incomplete factorizations * indefinite symmetric systems * iterative solvers * pivoting * preconditioning * sparse linear systems * sparse matrices Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics Impact factor: 1.303, year: 2016

Enhanced nonlinear iterative techniques applied to a non-equilibrium plasma flow

Energy Technology Data Exchange (ETDEWEB)

Knoll, D.A.; McHugh, P.R. [Idaho National Engineering Lab., Idaho Falls, ID (United States)

1996-12-31

We study the application of enhanced nonlinear iterative methods to the steady-state solution of a system of two-dimensional convection-diffusion-reaction partial differential equations that describe the partially-ionized plasma flow in the boundary layer of a tokamak fusion reactor. This system of equations is characterized by multiple time and spatial scales, and contains highly anisotropic transport coefficients due to a strong imposed magnetic field. We use Newton`s method to linearize the nonlinear system of equations resulting from an implicit, finite volume discretization of the governing partial differential equations, on a staggered Cartesian mesh. The resulting linear systems are neither symmetric nor positive definite, and are poorly conditioned. Preconditioned Krylov iterative techniques are employed to solve these linear systems. We investigate both a modified and a matrix-free Newton-Krylov implementation, with the goal of reducing CPU cost associated with the numerical formation of the Jacobian. A combination of a damped iteration, one-way multigrid and a pseudo-transient continuation technique are used to enhance global nonlinear convergence and CPU efficiency. GMRES is employed as the Krylov method with Incomplete Lower-Upper(ILU) factorization preconditioning. The goal is to construct a combination of nonlinear and linear iterative techniques for this complex physical problem that optimizes trade-offs between robustness, CPU time, memory requirements, and code complexity. It is shown that a one-way multigrid implementation provides significant CPU savings for fine grid calculations. Performance comparisons of the modified Newton-Krylov and matrix-free Newton-Krylov algorithms will be presented.

The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements

KAUST Repository

Collier, Nathan; Dalcin, Lisandro; Pardo, David; Calo, Victor M.

2013-01-01

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using Co B-splines, which span traditional finite element spaces, and Cp-1 B-splines, which represent maximum continuity We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most 33p2/8 times more expensive for the more continuous space, although for moderately low p, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high p. Preconditioning options can be up to p3 times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization. © 2013 Society for Industrial and Applied Mathematics.

The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements

KAUST Repository

Collier, Nathan

2013-03-19

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using Co B-splines, which span traditional finite element spaces, and Cp-1 B-splines, which represent maximum continuity We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most 33p2/8 times more expensive for the more continuous space, although for moderately low p, this number is significantly reduced. Moreover, if static condensation is not employed, this number further reduces to at most a value of 8, even for high p. Preconditioning options can be up to p3 times more expensive to set up, although this difference significantly decreases for some popular preconditioners such as incomplete LU factorization. © 2013 Society for Industrial and Applied Mathematics.

Reduction of Linear Programming to Linear Approximation

OpenAIRE

Vaserstein, Leonid N.

2006-01-01

It is well known that every Chebyshev linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to a Chebyshev linear approximation problem.

A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

KAUST Repository

Efendiev, Yalchin R.

2014-03-11

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

Some nonlinear space decomposition algorithms

Energy Technology Data Exchange (ETDEWEB)

Tai, Xue-Cheng; Espedal, M. [Univ. of Bergen (Norway)

1996-12-31

Convergence of a space decomposition method is proved for a general convex programming problem. The space decomposition refers to methods that decompose a space into sums of subspaces, which could be a domain decomposition or a multigrid method for partial differential equations. Two algorithms are proposed. Both can be used for linear as well as nonlinear elliptic problems and they reduce to the standard additive and multiplicative Schwarz methods for linear elliptic problems. Two {open_quotes}hybrid{close_quotes} algorithms are also presented. They converge faster than the additive one and have better parallelism than the multiplicative method. Numerical tests with a two level domain decomposition for linear, nonlinear and interface elliptic problems are presented for the proposed algorithms.

Weighted graph based ordering techniques for preconditioned conjugate gradient methods

Science.gov (United States)

Clift, Simon S.; Tang, Wei-Pai

1994-01-01

We describe the basis of a matrix ordering heuristic for improving the incomplete factorization used in preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from well-known algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners.

Structure formation by a fifth force: N-body versus linear simulations

International Nuclear Information System (INIS)

Li Baojiu; Zhao Hongsheng

2009-01-01

We lay out the frameworks to numerically study the structure formation in both linear and nonlinear regimes in general dark-matter-coupled scalar field models, and give an explicit example where the scalar field serves as a dynamical dark energy. Adopting parameters of the scalar field which yield a realistic cosmic microwave background (CMB) spectrum, we generate the initial conditions for our N-body simulations, which follow the spatial distributions of the dark matter and the scalar field by solving their equations of motion using the multilevel adaptive grid technique. We show that the spatial configuration of the scalar field tracks well the voids and clusters of dark matter. Indeed, the propagation of scalar degree of freedom effectively acts as a fifth force on dark matter particles, whose range and magnitude are determined by the two model parameters (μ,γ), local dark matter density as well as the background value for the scalar field. The model behaves like the ΛCDM paradigm on scales relevant to the CMB spectrum, which are well beyond the probe of the local fifth force and thus not significantly affected by the matter-scalar coupling. On scales comparable or shorter than the range of the local fifth force, the fifth force is perfectly parallel to gravity and their strengths have a fixed ratio 2γ 2 determined by the matter-scalar coupling, provided that the chameleon effect is weak; if on the other hand there is a strong chameleon effect (i.e., the scalar field almost resides at its effective potential minimum everywhere in the space), the fifth force indeed has suppressed effects in high density regions and shows no obvious correlation with gravity, which means that the dark-matter-scalar-field coupling is not simply equivalent to a rescaling of the gravitational constant or the mass of the dark matter particles. We show these spatial distributions and (lack of) correlations at typical redshifts (z=0,1,5.5) in our multigrid million

Structure formation by a fifth force: N-body versus linear simulations

Science.gov (United States)

Li, Baojiu; Zhao, Hongsheng

2009-08-01

We lay out the frameworks to numerically study the structure formation in both linear and nonlinear regimes in general dark-matter-coupled scalar field models, and give an explicit example where the scalar field serves as a dynamical dark energy. Adopting parameters of the scalar field which yield a realistic cosmic microwave background (CMB) spectrum, we generate the initial conditions for our N-body simulations, which follow the spatial distributions of the dark matter and the scalar field by solving their equations of motion using the multilevel adaptive grid technique. We show that the spatial configuration of the scalar field tracks well the voids and clusters of dark matter. Indeed, the propagation of scalar degree of freedom effectively acts as a fifth force on dark matter particles, whose range and magnitude are determined by the two model parameters (μ,γ), local dark matter density as well as the background value for the scalar field. The model behaves like the ΛCDM paradigm on scales relevant to the CMB spectrum, which are well beyond the probe of the local fifth force and thus not significantly affected by the matter-scalar coupling. On scales comparable or shorter than the range of the local fifth force, the fifth force is perfectly parallel to gravity and their strengths have a fixed ratio 2γ2 determined by the matter-scalar coupling, provided that the chameleon effect is weak; if on the other hand there is a strong chameleon effect (i.e., the scalar field almost resides at its effective potential minimum everywhere in the space), the fifth force indeed has suppressed effects in high density regions and shows no obvious correlation with gravity, which means that the dark-matter-scalar-field coupling is not simply equivalent to a rescaling of the gravitational constant or the mass of the dark matter particles. We show these spatial distributions and (lack of) correlations at typical redshifts (z=0,1,5.5) in our multigrid million-particle simulations

A comparison between anisotropic analytical and multigrid superposition dose calculation algorithms in radiotherapy treatment planning

International Nuclear Information System (INIS)

Wu, Vincent W.C.; Tse, Teddy K.H.; Ho, Cola L.M.; Yeung, Eric C.Y.

2013-01-01

Monte Carlo (MC) simulation is currently the most accurate dose calculation algorithm in radiotherapy planning but requires relatively long processing time. Faster model-based algorithms such as the anisotropic analytical algorithm (AAA) by the Eclipse treatment planning system and multigrid superposition (MGS) by the XiO treatment planning system are 2 commonly used algorithms. This study compared AAA and MGS against MC, as the gold standard, on brain, nasopharynx, lung, and prostate cancer patients. Computed tomography of 6 patients of each cancer type was used. The same hypothetical treatment plan using the same machine and treatment prescription was computed for each case by each planning system using their respective dose calculation algorithm. The doses at reference points including (1) soft tissues only, (2) bones only, (3) air cavities only, (4) soft tissue-bone boundary (Soft/Bone), (5) soft tissue-air boundary (Soft/Air), and (6) bone-air boundary (Bone/Air), were measured and compared using the mean absolute percentage error (MAPE), which was a function of the percentage dose deviations from MC. Besides, the computation time of each treatment plan was recorded and compared. The MAPEs of MGS were significantly lower than AAA in all types of cancers (p<0.001). With regards to body density combinations, the MAPE of AAA ranged from 1.8% (soft tissue) to 4.9% (Bone/Air), whereas that of MGS from 1.6% (air cavities) to 2.9% (Soft/Bone). The MAPEs of MGS (2.6%±2.1) were significantly lower than that of AAA (3.7%±2.5) in all tissue density combinations (p<0.001). The mean computation time of AAA for all treatment plans was significantly lower than that of the MGS (p<0.001). Both AAA and MGS algorithms demonstrated dose deviations of less than 4.0% in most clinical cases and their performance was better in homogeneous tissues than at tissue boundaries. In general, MGS demonstrated relatively smaller dose deviations than AAA but required longer computation time

Multilevel domain decomposition for electronic structure calculations

International Nuclear Information System (INIS)

Barrault, M.; Cances, E.; Hager, W.W.; Le Bris, C.

2007-01-01

We introduce a new multilevel domain decomposition method (MDD) for electronic structure calculations within semi-empirical and density functional theory (DFT) frameworks. This method iterates between local fine solvers and global coarse solvers, in the spirit of domain decomposition methods. Using this approach, calculations have been successfully performed on several linear polymer chains containing up to 40,000 atoms and 200,000 atomic orbitals. Both the computational cost and the memory requirement scale linearly with the number of atoms. Additional speed-up can easily be obtained by parallelization. We show that this domain decomposition method outperforms the density matrix minimization (DMM) method for poor initial guesses. Our method provides an efficient preconditioner for DMM and other linear scaling methods, variational in nature, such as the orbital minimization (OM) procedure

Linear versus non-linear supersymmetry, in general

Energy Technology Data Exchange (ETDEWEB)

Ferrara, Sergio [Theoretical Physics Department, CERN,CH-1211 Geneva 23 (Switzerland); INFN - Laboratori Nazionali di Frascati,Via Enrico Fermi 40, I-00044 Frascati (Italy); Department of Physics and Astronomy, UniversityC.L.A.,Los Angeles, CA 90095-1547 (United States); Kallosh, Renata [SITP and Department of Physics, Stanford University,Stanford, California 94305 (United States); Proeyen, Antoine Van [Institute for Theoretical Physics, Katholieke Universiteit Leuven,Celestijnenlaan 200D, B-3001 Leuven (Belgium); Wrase, Timm [Institute for Theoretical Physics, Technische Universität Wien,Wiedner Hauptstr. 8-10, A-1040 Vienna (Austria)

2016-04-12

We study superconformal and supergravity models with constrained superfields. The underlying version of such models with all unconstrained superfields and linearly realized supersymmetry is presented here, in addition to the physical multiplets there are Lagrange multiplier (LM) superfields. Once the equations of motion for the LM superfields are solved, some of the physical superfields become constrained. The linear supersymmetry of the original models becomes non-linearly realized, its exact form can be deduced from the original linear supersymmetry. Known examples of constrained superfields are shown to require the following LM’s: chiral superfields, linear superfields, general complex superfields, some of them are multiplets with a spin.

Linear versus non-linear supersymmetry, in general

International Nuclear Information System (INIS)

Ferrara, Sergio; Kallosh, Renata; Proeyen, Antoine Van; Wrase, Timm

2016-01-01

We study superconformal and supergravity models with constrained superfields. The underlying version of such models with all unconstrained superfields and linearly realized supersymmetry is presented here, in addition to the physical multiplets there are Lagrange multiplier (LM) superfields. Once the equations of motion for the LM superfields are solved, some of the physical superfields become constrained. The linear supersymmetry of the original models becomes non-linearly realized, its exact form can be deduced from the original linear supersymmetry. Known examples of constrained superfields are shown to require the following LM’s: chiral superfields, linear superfields, general complex superfields, some of them are multiplets with a spin.

Particle-in-cell Simulations with Kinetic Electrons

International Nuclear Information System (INIS)

Lewandowski, J.L.V.

2004-01-01

A new scheme, based on an exact separation between adiabatic and nonadiabatic electron responses, for particle-in-cell (PIC) simulations of drift-type modes is presented. The (linear and nonlinear) elliptic equations for the scalar fields are solved using a multi-grid solver. The new scheme yields linear growth rates in excellent agreement with theory and it is shown to conserve energy well into the nonlinear regime. It is also demonstrated that simulations with few electrons are reliable and accurate, suggesting that large-scale, PIC simulations with electron dynamics in toroidal geometry (e.g., tokamaks and stellarators plasmas) are within reach of present-day massively parallel supercomputers

Local multiplicative Schwarz algorithms for convection-diffusion equations

Science.gov (United States)

Cai, Xiao-Chuan; Sarkis, Marcus

1995-01-01

We develop a new class of overlapping Schwarz type algorithms for solving scalar convection-diffusion equations discretized by finite element or finite difference methods. The preconditioners consist of two components, namely, the usual two-level additive Schwarz preconditioner and the sum of some quadratic terms constructed by using products of ordered neighboring subdomain preconditioners. The ordering of the subdomain preconditioners is determined by considering the direction of the flow. We prove that the algorithms are optimal in the sense that the convergence rates are independent of the mesh size, as well as the number of subdomains. We show by numerical examples that the new algorithms are less sensitive to the direction of the flow than either the classical multiplicative Schwarz algorithms, and converge faster than the additive Schwarz algorithms. Thus, the new algorithms are more suitable for fluid flow applications than the classical additive or multiplicative Schwarz algorithms.

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

Energy Technology Data Exchange (ETDEWEB)

Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States); Salgado, Abner J., E-mail: asalgad1@utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States); Wang, Cheng, E-mail: cwang1@umassd.edu [Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747 (United States); Wise, Steven M., E-mail: swise1@utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States)

2017-04-01

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

Foundations of linear and generalized linear models

CERN Document Server

Agresti, Alan

2015-01-01

A valuable overview of the most important ideas and results in statistical analysis Written by a highly-experienced author, Foundations of Linear and Generalized Linear Models is a clear and comprehensive guide to the key concepts and results of linear statistical models. The book presents a broad, in-depth overview of the most commonly used statistical models by discussing the theory underlying the models, R software applications, and examples with crafted models to elucidate key ideas and promote practical model building. The book begins by illustrating the fundamentals of linear models,

A new scripting library for modeling flow and transport in fractured rock with channel networks

Science.gov (United States)

Dessirier, Benoît; Tsang, Chin-Fu; Niemi, Auli

2018-02-01

Deep crystalline bedrock formations are targeted to host spent nuclear fuel owing to their overall low permeability. They are however highly heterogeneous and only a few preferential paths pertaining to a small set of dominant rock fractures usually carry most of the flow or mass fluxes, a behavior known as channeling that needs to be accounted for in the performance assessment of repositories. Channel network models have been developed and used to investigate the effect of channeling. They are usually simpler than discrete fracture networks based on rock fracture mappings and rely on idealized full or sparsely populated lattices of channels. This study reexamines the fundamental parameter structure required to describe a channel network in terms of groundwater flow and solute transport, leading to an extended description suitable for unstructured arbitrary networks of channels. An implementation of this formalism in a Python scripting library is presented and released along with this article. A new algebraic multigrid preconditioner delivers a significant speedup in the flow solution step compared to previous channel network codes. 3D visualization is readily available for verification and interpretation of the results by exporting the results to an open and free dedicated software. The new code is applied to three example cases to verify its results on full uncorrelated lattices of channels, sparsely populated percolation lattices and to exemplify the use of unstructured networks to accommodate knowledge on local rock fractures.

On the linear programming bound for linear Lee codes.

Science.gov (United States)

Astola, Helena; Tabus, Ioan

2016-01-01

Based on an invariance-type property of the Lee-compositions of a linear Lee code, additional equality constraints can be introduced to the linear programming problem of linear Lee codes. In this paper, we formulate this property in terms of an action of the multiplicative group of the field [Formula: see text] on the set of Lee-compositions. We show some useful properties of certain sums of Lee-numbers, which are the eigenvalues of the Lee association scheme, appearing in the linear programming problem of linear Lee codes. Using the additional equality constraints, we formulate the linear programming problem of linear Lee codes in a very compact form, leading to a fast execution, which allows to efficiently compute the bounds for large parameter values of the linear codes.

On the Robustness and Prospects of Adaptive BDDC Methods for Finite Element Discretizations of Elliptic PDEs with High-Contrast Coefficients

KAUST Repository

Zampini, Stefano

2016-06-02

Balancing Domain Decomposition by Constraints (BDDC) methods have proven to be powerful preconditioners for large and sparse linear systems arising from the finite element discretization of elliptic PDEs. Condition number bounds can be theoretically established that are independent of the number of subdomains of the decomposition. The core of the methods resides in the design of a larger and partially discontinuous finite element space that allows for fast application of the preconditioner, where Cholesky factorizations of the subdomain finite element problems are additively combined with a coarse, global solver. Multilevel and highly-scalable algorithms can be obtained by replacing the coarse Cholesky solver with a coarse BDDC preconditioner. BDDC methods have the remarkable ability to control the condition number, since the coarse space of the preconditioner can be adaptively enriched at the cost of solving local eigenproblems. The proper identification of these eigenproblems extends the robustness of the methods to any heterogeneity in the distribution of the coefficients of the PDEs, not only when the coefficients jumps align with the subdomain boundaries or when the high contrast regions are confined to lie in the interior of the subdomains. The specific adaptive technique considered in this paper does not depend upon any interaction of discretization and partition; it relies purely on algebraic operations. Coarse space adaptation in BDDC methods has attractive algorithmic properties, since the technique enhances the concurrency and the arithmetic intensity of the preconditioning step of the sparse implicit solver with the aim of controlling the number of iterations of the Krylov method in a black-box fashion, thus reducing the number of global synchronization steps and matrix vector multiplications needed by the iterative solver; data movement and memory bound kernels in the solve phase can be thus limited at the expense of extra local ops during the setup of

High-Order Calderón Preconditioned Time Domain Integral Equation Solvers

KAUST Repository

Valdes, Felipe; Ghaffari-Miab, Mohsen; Andriulli, Francesco P.; Cools, Kristof; Michielssen,

2013-01-01

Two high-order accurate Calderón preconditioned time domain electric field integral equation (TDEFIE) solvers are presented. In contrast to existing Calderón preconditioned time domain solvers, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of fully-localized high-order div-and quasi curl-conforming (DQCC) basis functions. Numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly, regardless of the mesh density and of the order of the current expansion. © 1963-2012 IEEE.

High-Order Calderón Preconditioned Time Domain Integral Equation Solvers

KAUST Repository

Valdes, Felipe

2013-05-01

Two high-order accurate Calderón preconditioned time domain electric field integral equation (TDEFIE) solvers are presented. In contrast to existing Calderón preconditioned time domain solvers, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of fully-localized high-order div-and quasi curl-conforming (DQCC) basis functions. Numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly, regardless of the mesh density and of the order of the current expansion. © 1963-2012 IEEE.

Preconditioning for Allen–Cahn variational inequalities with non-local constraints

KAUST Repository

Blank, Luise

2012-06-01

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach. © 2012 Elsevier Inc.

An iteration for indefinite and non-symmetric systems and its application to the Navier-Stokes equations

Energy Technology Data Exchange (ETDEWEB)

Wathen, A. [Oxford Univ. (United Kingdom); Golub, G. [Stanford Univ., CA (United States)

1996-12-31

A simple fixed point linearisation of the Navier-Stokes equations leads to the Oseen problem which after appropriate discretisation yields large sparse linear systems with coefficient matrices of the form (A B{sup T} B -C). Here A is non-symmetric but its symmetric part is positive definite, and C is symmetric and positive semi-definite. Such systems arise in other situations. In this talk we will describe and present some analysis for an iteration based on an indefinite and symmetric preconditioner of the form (D B{sup T} B -C).

Linear algebra

CERN Document Server

Shilov, Georgi E

1977-01-01

Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers.

Real-time simulation of contact and cutting of heterogeneous soft-tissues.

Science.gov (United States)

Courtecuisse, Hadrien; Allard, Jérémie; Kerfriden, Pierre; Bordas, Stéphane P A; Cotin, Stéphane; Duriez, Christian

2014-02-01

This paper presents a numerical method for interactive (real-time) simulations, which considerably improves the accuracy of the response of heterogeneous soft-tissue models undergoing contact, cutting and other topological changes. We provide an integrated methodology able to deal both with the ill-conditioning issues associated with material heterogeneities, contact boundary conditions which are one of the main sources of inaccuracies, and cutting which is one of the most challenging issues in interactive simulations. Our approach is based on an implicit time integration of a non-linear finite element model. To enable real-time computations, we propose a new preconditioning technique, based on an asynchronous update at low frequency. The preconditioner is not only used to improve the computation of the deformation of the tissues, but also to simulate the contact response of homogeneous and heterogeneous bodies with the same accuracy. We also address the problem of cutting the heterogeneous structures and propose a method to update the preconditioner according to the topological modifications. Finally, we apply our approach to three challenging demonstrators: (i) a simulation of cataract surgery (ii) a simulation of laparoscopic hepatectomy (iii) a brain tumor surgery. Copyright © 2013 Elsevier B.V. All rights reserved.

Application of Conjugate Gradient methods to tidal simulation

Science.gov (United States)

Barragy, E.; Carey, G.F.; Walters, R.A.

1993-01-01

A harmonic decomposition technique is applied to the shallow water equations to yield a complex, nonsymmetric, nonlinear, Helmholtz type problem for the sea surface and an accompanying complex, nonlinear diagonal problem for the velocities. The equation for the sea surface is linearized using successive approximation and then discretized with linear, triangular finite elements. The study focuses on applying iterative methods to solve the resulting complex linear systems. The comparative evaluation includes both standard iterative methods for the real subsystems and complex versions of the well known Bi-Conjugate Gradient and Bi-Conjugate Gradient Squared methods. Several Incomplete LU type preconditioners are discussed, and the effects of node ordering, rejection strategy, domain geometry and Coriolis parameter (affecting asymmetry) are investigated. Implementation details for the complex case are discussed. Performance studies are presented and comparisons made with a frontal solver. ?? 1993.

Multilevel Methods for Elliptic Problems with Highly Varying Coefficients on Nonaligned Coarse Grids

Energy Technology Data Exchange (ETDEWEB)

Scheichl, Robert [Univ. of Bath (United Kingdom). Dept. of Mathematical Sciences; Vassilevski, Panayot S. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Zikatanov, Ludmil T. [Pennsylvania State Univ., University Park, PA (United States). Dept. of Mathematics

2012-06-21

We generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. One assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincaré-type inequalities. Finally, numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

Linear and non-linear optics of condensed matter

International Nuclear Information System (INIS)

McLean, T.P.

1977-01-01

Part I - Linear optics: 1. General introduction. 2. Frequency dependence of epsilon(ω, k vector). 3. Wave-vector dependence of epsilon(ω, k vector). 4. Tensor character of epsilon(ω, k vector). Part II - Non-linear optics: 5. Introduction. 6. A classical theory of non-linear response in one dimension. 7. The generalization to three dimensions. 8. General properties of the polarizability tensors. 9. The phase-matching condition. 10. Propagation in a non-linear dielectric. 11. Second harmonic generation. 12. Coupling of three waves. 13. Materials and their non-linearities. 14. Processes involving energy exchange with the medium. 15. Two-photon absorption. 16. Stimulated Raman effect. 17. Electro-optic effects. 18. Limitations of the approach presented here. (author)

Choosing order of operations to accelerate strip structure analysis in parameter range

Science.gov (United States)

Kuksenko, S. P.; Akhunov, R. R.; Gazizov, T. R.

2018-05-01

The paper considers the issue of using iteration methods in solving the sequence of linear algebraic systems obtained in quasistatic analysis of strip structures with the method of moments. Using the analysis of 4 strip structures, the authors have proved that additional acceleration (up to 2.21 times) of the iterative process can be obtained during the process of solving linear systems repeatedly by means of choosing a proper order of operations and a preconditioner. The obtained results can be used to accelerate the process of computer-aided design of various strip structures. The choice of the order of operations to accelerate the process is quite simple, universal and could be used not only for strip structure analysis but also for a wide range of computational problems.

Perturbation of eigenvalues of preconditioned Navier-Stokes operators

Energy Technology Data Exchange (ETDEWEB)

Elman, H.C. [Univ. of Maryland, College Park, MD (United States)

1996-12-31

We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.

Linearity and Non-linearity of Photorefractive effect in Materials ...

African Journals Online (AJOL)

In this paper we have studied the Linearity and Non-linearity of Photorefractive effect in materials using the band transport model. For low light beam intensities the change in the refractive index is proportional to the electric field for linear optics while for non- linear optics the change in refractive index is directly proportional ...

Application of preconditioned GMRES to the numerical solution of the neutron transport equation

International Nuclear Information System (INIS)

Patton, B.W.; Holloway, J.P.

2002-01-01

The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU(τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU(τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU(τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements

Special set linear algebra and special set fuzzy linear algebra

OpenAIRE

Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.

2009-01-01

The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...

Copper Mountain conference on iterative methods: Proceedings: Volume 2

Energy Technology Data Exchange (ETDEWEB)

NONE

1996-10-01

This volume (the second of two) contains information presented during the last two days of the Copper Mountain Conference on Iterative Methods held April 9-13, 1996 at Copper Mountain, Colorado. Topics of the sessions held these two days include domain decomposition, Krylov methods, computational fluid dynamics, Markov chains, sparse and parallel basic linear algebra subprograms, multigrid methods, applications of iterative methods, equation systems with multiple right-hand sides, projection methods, and the Helmholtz equation. Selected papers indexed separately for the Energy Science and Technology Database.

Trapping and accumulation of positrons from a pulsed beam produced by a linear accelerator for gravitational interaction of antimatter study

International Nuclear Information System (INIS)

Grandemange, Pierre

2013-01-01

The Gravitational Behaviour of Anti-hydrogen at Rest experiment - GBAR - is designed to perform a direct measurement of the weak equivalence principle on antimatter by measuring the acceleration (g-bar) of anti-hydrogen atoms in free fall. Its originality is to produce H-bar + ions and use sympathetic cooling to achieve μK temperature. H-bar + ions are produced by the reactions: p-bar + Ps → H-bar + e - , and H-bar + Ps → H-bar + + e - , where p-bar is an antiproton, Ps stands for positronium (the bound-state of a positron and an electron), H-bar is the anti-hydrogen and H-bar + the anti-ion associated. To produce enough Ps atoms, 2*10 10 positrons must be impinged on a porous SiO 2 target within 100 ns. Such an intense flux requires the accumulation (collection and cooling) of the positrons in a particle trap. This thesis describes the injector being commissioned at CEA Saclay for GBAR. It consists of a Penning-Malmberg trap (moved from RIKEN) fed by a slow positron beam. A 4.3 MeV linear accelerator shooting electrons on a tungsten target produces the pulsed positron beam, which is moderated by a multi-grid tungsten moderator. The slow positron flux is 10 4 e + /pulse, or 2*10 6 e + /s at 200 Hz. This work presents the first ever accumulation of low-energy positrons produced by an accelerator (rather than a radioactive source) and their cooling by a prepared reservoir of 2*10 10 cold electrons. (author) [fr

Algorithms for parallel and vector computations

Science.gov (United States)

Ortega, James M.

1995-01-01

This is a final report on work performed under NASA grant NAG-1-1112-FOP during the period March, 1990 through February 1995. Four major topics are covered: (1) solution of nonlinear poisson-type equations; (2) parallel reduced system conjugate gradient method; (3) orderings for conjugate gradient preconditioners, and (4) SOR as a preconditioner.

Linear Algebraic Method for Non-Linear Map Analysis

International Nuclear Information System (INIS)

Yu, L.; Nash, B.

2009-01-01

We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.

Large capacity water and air bath calorimeters

International Nuclear Information System (INIS)

James, S.J.; Kasperski, P.W.; Renz, D.P.; Wetzel, J.R.

1993-01-01

EG and G Mound Applied Technologies has developed an 11 in. x 17 in. sample size water bath and an 11 in. x 17 in. sample size air bath calorimeter which both function under servo control mode of operation. The water bath calorimeter has four air bath preconditioners to increase sample throughput and the air bath calorimeter has two air bath preconditioners. The large capacity calorimeters and preconditioners were unique to Mound design which brought about unique design challenges. Both large capacity systems calculate the optimum set temperature for each preconditioner which is available to the operator. Each system is controlled by a personal computer under DOS which allows the operator to download data to commercial software packages when the calorimeter is idle. Qualification testing yielded a one standard deviation of 0.6% for 0.2W to 3.0W Pu-238 heat standard range in the water bath calorimeter and a one standard deviation of 0.3% for the 6.0W to 20.0W Pu-238 heat standard range in the air bath calorimeter

Migration of vectorized iterative solvers to distributed memory architectures

Energy Technology Data Exchange (ETDEWEB)

Pommerell, C. [AT& T Bell Labs., Murray Hill, NJ (United States); Ruehl, R. [CSCS-ETH, Manno (Switzerland)

1994-12-31

Both necessity and opportunity motivate the use of high-performance computers for iterative linear solvers. Necessity results from the size of the problems being solved-smaller problems are often better handled by direct methods. Opportunity arises from the formulation of the iterative methods in terms of simple linear algebra operations, even if this {open_quote}natural{close_quotes} parallelism is not easy to exploit in irregularly structured sparse matrices and with good preconditioners. As a result, high-performance implementations of iterative solvers have attracted a lot of interest in recent years. Most efforts are geared to vectorize or parallelize the dominating operation-structured or unstructured sparse matrix-vector multiplication, or to increase locality and parallelism by reformulating the algorithm-reducing global synchronization in inner products or local data exchange in preconditioners. Target architectures for iterative solvers currently include mostly vector supercomputers and architectures with one or few optimized (e.g., super-scalar and/or super-pipelined RISC) processors and hierarchical memory systems. More recently, parallel computers with physically distributed memory and a better price/performance ratio have been offered by vendors as a very interesting alternative to vector supercomputers. However, programming comfort on such distributed memory parallel processors (DMPPs) still lags behind. Here the authors are concerned with iterative solvers and their changing computing environment. In particular, they are considering migration from traditional vector supercomputers to DMPPs. Application requirements force one to use flexible and portable libraries. They want to extend the portability of iterative solvers rather than reimplementing everything for each new machine, or even for each new architecture.

Computational Science Research in Support of Petascale Electromagnetic Modeling

International Nuclear Information System (INIS)

Lee, L.-Q.

2008-01-01

Computational science research components were vital parts of the SciDAC-1 accelerator project and are continuing to play a critical role in newly-funded SciDAC-2 accelerator project, the Community Petascale Project for Accelerator Science and Simulation (ComPASS). Recent advances and achievements in the area of computational science research in support of petascale electromagnetic modeling for accelerator design analysis are presented, which include shape determination of superconducting RF cavities, mesh-based multilevel preconditioner in solving highly-indefinite linear systems, moving window using h- or p- refinement for time-domain short-range wakefield calculations, and improved scalable application I/O

Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

KAUST Repository

Giraldi, Loic; Nouy, Anthony

2017-01-01

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

Weakly intrusive low-rank approximation method for nonlinear parameter-dependent equations

KAUST Repository

Giraldi, Loic

2017-06-30

This paper presents a weakly intrusive strategy for computing a low-rank approximation of the solution of a system of nonlinear parameter-dependent equations. The proposed strategy relies on a Newton-like iterative solver which only requires evaluations of the residual of the parameter-dependent equation and of a preconditioner (such as the differential of the residual) for instances of the parameters independently. The algorithm provides an approximation of the set of solutions associated with a possibly large number of instances of the parameters, with a computational complexity which can be orders of magnitude lower than when using the same Newton-like solver for all instances of the parameters. The reduction of complexity requires efficient strategies for obtaining low-rank approximations of the residual, of the preconditioner, and of the increment at each iteration of the algorithm. For the approximation of the residual and the preconditioner, weakly intrusive variants of the empirical interpolation method are introduced, which require evaluations of entries of the residual and the preconditioner. Then, an approximation of the increment is obtained by using a greedy algorithm for low-rank approximation, and a low-rank approximation of the iterate is finally obtained by using a truncated singular value decomposition. When the preconditioner is the differential of the residual, the proposed algorithm is interpreted as an inexact Newton solver for which a detailed convergence analysis is provided. Numerical examples illustrate the efficiency of the method.

Advanced statistics: linear regression, part I: simple linear regression.

Science.gov (United States)

Marill, Keith A

2004-01-01

Simple linear regression is a mathematical technique used to model the relationship between a single independent predictor variable and a single dependent outcome variable. In this, the first of a two-part series exploring concepts in linear regression analysis, the four fundamental assumptions and the mechanics of simple linear regression are reviewed. The most common technique used to derive the regression line, the method of least squares, is described. The reader will be acquainted with other important concepts in simple linear regression, including: variable transformations, dummy variables, relationship to inference testing, and leverage. Simplified clinical examples with small datasets and graphic models are used to illustrate the points. This will provide a foundation for the second article in this series: a discussion of multiple linear regression, in which there are multiple predictor variables.

Genetic design of interpolated non-linear controllers for linear plants

International Nuclear Information System (INIS)

Ajlouni, N.

2000-01-01

The techniques of genetic algorithms are proposed as a means of designing non-linear PID control systems. It is shown that the use of genetic algorithms for this purpose results in highly effective non-linear PID control systems. These results are illustrated by using genetic algorithms to design a non-linear PID control system and contrasting the results with an optimally tuned linear PID controller. (author)

Similarities and Differences Between Warped Linear Prediction and Laguerre Linear Prediction

NARCIS (Netherlands)

Brinker, Albertus C. den; Krishnamoorthi, Harish; Verbitskiy, Evgeny A.

2011-01-01

Linear prediction has been successfully applied in many speech and audio processing systems. This paper presents the similarities and differences between two classes of linear prediction schemes, namely, Warped Linear Prediction (WLP) and Laguerre Linear Prediction (LLP). It is shown that both

In-beam test of the Boron-10 Multi-Grid neutron detector at the IN6 time-of-flight spectrometer at the ILL

Energy Technology Data Exchange (ETDEWEB)

Birch, J; Hultman, L; Höglund, C [Linköping University, Thin Film Physics Division, IFM, SE-581 83 Linköping (Sweden); Buffet, J-C; Clergeau, J-F; Correa, J; Van Esch, P; Ferraton, M; Guerard, B; Halbwachs, J; Khaplanov, A; Koza, M; Piscitelli, F; Zbiri, M [Institute Laue Langevin, Rue Jules Horowitz, FR-38000 Grenoble (France); Hall-Wilton, R [European Spallation Source ESS AB, P.O Box 176, SE-221 00 Lund (Sweden)

2014-07-24

A neutron detector concept based on solid layers of boron carbide enriched in {sup 10}B has been in development for the last few years as an alternative for {sup 3}He by collaboration between the ILL, ESS and Linköping University. This Multi-Grid detector uses layers of aluminum substrates coated with {sup 10}B{sub 4}C on both sides that are traversed by the incoming neutrons. Detection is achieved using a gas counter readout principle. By segmenting the substrate and using multiple anode wires, the detector is made inherently position sensitive. This development is aimed primarily at neutron scattering instruments with large detector areas, such as time-of-flight chopper spectrometers. The most recent prototype has been built to be interchangeable with the {sup 3}He detectors of IN6 at ILL. The {sup 10}B detector has an active area of 32 x 48cm{sup 2}. It was installed at the IN6 instrument and operated for several weeks, collecting data in parallel with the regularly scheduled experiments, thus providing the first side-by-side comparison with the conventional {sup 3}He detectors. Results include an efficiency comparison, assessment of the in-detector scattering contribution, sensitivity to gamma-rays and the signal-to-noise ratio in time-of-flight spectra. The good expected performance has been confirmed with the exception of an unexpected background count rate. This has been identified as natural alpha activity in aluminum. New convertor substrates are under study to eliminate this source of background.

Spectral Analysis of Large Finite Element Problems by Optimization Methods

Directory of Open Access Journals (Sweden)

Luca Bergamaschi

1994-01-01

Full Text Available Recently an efficient method for the solution of the partial symmetric eigenproblem (DACG, deflated-accelerated conjugate gradient was developed, based on the conjugate gradient (CG minimization of successive Rayleigh quotients over deflated subspaces of decreasing size. In this article four different choices of the coefficient βk required at each DACG iteration for the computation of the new search direction Pk are discussed. The “optimal” choice is the one that yields the same asymptotic convergence rate as the CG scheme applied to the solution of linear systems. Numerical results point out that the optimal βk leads to a very cost effective algorithm in terms of CPU time in all the sample problems presented. Various preconditioners are also analyzed. It is found that DACG using the optimal βk and (LLT−1 as a preconditioner, L being the incomplete Cholesky factor of A, proves a very promising method for the partial eigensolution. It appears to be superior to the Lanczos method in the evaluation of the 40 leftmost eigenpairs of five finite element problems, and particularly for the largest problem, with size equal to 4560, for which the speed gain turns out to fall between 2.5 and 6.0, depending on the eigenpair level.

Finite-dimensional linear algebra

CERN Document Server

Gockenbach, Mark S

2010-01-01

Some Problems Posed on Vector SpacesLinear equationsBest approximationDiagonalizationSummaryFields and Vector SpacesFields Vector spaces Subspaces Linear combinations and spanning sets Linear independence Basis and dimension Properties of bases Polynomial interpolation and the Lagrange basis Continuous piecewise polynomial functionsLinear OperatorsLinear operatorsMore properties of linear operatorsIsomorphic vector spaces Linear operator equations Existence and uniqueness of solutions The fundamental theorem; inverse operatorsGaussian elimination Newton's method Linear ordinary differential eq

Linear algebra

CERN Document Server

Said-Houari, Belkacem

2017-01-01

This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing t...

Linear gate

International Nuclear Information System (INIS)

Suwono.

1978-01-01

A linear gate providing a variable gate duration from 0,40μsec to 4μsec was developed. The electronic circuity consists of a linear circuit and an enable circuit. The input signal can be either unipolar or bipolar. If the input signal is bipolar, the negative portion will be filtered. The operation of the linear gate is controlled by the application of a positive enable pulse. (author)

GPCG - generalized preconditioned CG method and its use with non-linear and non-symmetric displacement decomposition preconditioners

Czech Academy of Sciences Publication Activity Database

Blaheta, Radim

2002-01-01

Roč. 9, 6/7 (2002), s. 525-550 ISSN 1070-5325 Grant - others:INCO Copernicus(XE) KIT977006 Institutional research plan: CEZ:AV0Z3086906 Keywords : elasticity * displacement decomposition Subject RIV: BA - General Mathematics Impact factor: 0.706, year: 2002

A node-centered local refinement algorithm for poisson's equation in complex geometries

International Nuclear Information System (INIS)

McCorquodale, Peter; Colella, Phillip; Grote, David P.; Vay, Jean-Luc

2004-01-01

This paper presents a method for solving Poisson's equation with Dirichlet boundary conditions on an irregular bounded three-dimensional region. The method uses a nodal-point discretization and adaptive mesh refinement (AMR) on Cartesian grids, and the AMR multigrid solver of Almgren. The discrete Laplacian operator at internal boundaries comes from either linear or quadratic (Shortley-Weller) extrapolation, and the two methods are compared. It is shown that either way, solution error is second order in the mesh spacing. Error in the gradient of the solution is first order with linear extrapolation, but second order with Shortley-Weller. Examples are given with comparison with the exact solution. The method is also applied to a heavy-ion fusion accelerator problem, showing the advantage of adaptivity

The linear programming bound for binary linear codes

NARCIS (Netherlands)

Brouwer, A.E.

1993-01-01

Combining Delsarte's (1973) linear programming bound with the information that certain weights cannot occur, new upper bounds for dmin (n,k), the maximum possible minimum distance of a binary linear code with given word length n and dimension k, are derived.

An efficient flexible-order model for 3D nonlinear water waves

Science.gov (United States)

Engsig-Karup, A. P.; Bingham, H. B.; Lindberg, O.

2009-04-01

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211-228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss-Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.

Risk assessment of salt contamination of groundwater under uncertain aquifer properties

KAUST Repository

Litvinenko, Alexander

2017-10-01

One of the central topics in hydrogeology and environmental science is the investigation of salinity-driven groundwater flow in heterogeneous porous media. Our goals are to model and to predict pollution of water resources. We simulate a density driven groundwater flow with uncertain porosity and permeability. This strongly non-linear model describes the unstable transport of salt water with building ‘fingers’-shaped patterns. The computation requires a very fine unstructured mesh and, therefore, high computational resources. We run the highly-parallel multigrid solver, based on ug4, on supercomputer Shaheen II. A MPI-based parallelization is done in the geometrical as well as in the stochastic spaces. Every scenario is computed on 32 cores and requires a mesh with ~8M grid points and 1500 or more time steps. 200 scenarios are computed concurrently. The total number of cores in parallel computation is 200x32=6400. The main goal of this work is to estimate propagation of uncertainties through the model, to investigate sensitivity of the solution to the input uncertain parameters. Additionally, we demonstrate how the multigrid ug4-based solver can be applied as a black-box in the uncertainty quantification framework.

An efficient flexible-order model for 3D nonlinear water waves

International Nuclear Information System (INIS)

Engsig-Karup, A.P.; Bingham, H.B.; Lindberg, O.

2009-01-01

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211-228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss-Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature

Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations

Directory of Open Access Journals (Sweden)

Han Guo

2012-01-01

Full Text Available Hierarchical (H- matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE- based computational electromagnetics, H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve H-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of H-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.

Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties

DEFF Research Database (Denmark)

Christensen, Max la Cour; Vassilevski, Panayot S.; Villa, Umberto

2017-01-01

discretizations on general unstructured grids for a large class of nonlinear partial differential equations, including saddle point problems. The approximation properties of the coarse spaces ensure that our FAS approach for general unstructured meshes leads to optimal mesh-independent convergence rates similar...... to those achieved by geometric FAS on a nested hierarchy of refined meshes. In the numerical results, Newton’s method and Picard iterations with state-of-the-art inner linear solvers are compared to our FAS algorithm for the solution of a nonlinear saddle point problem arising from porous media flow...

Handbook on linear motor application

International Nuclear Information System (INIS)

1988-10-01

This book guides the application for Linear motor. It lists classification and speciality of Linear Motor, terms of linear-induction motor, principle of the Motor, types on one-side linear-induction motor, bilateral linear-induction motor, linear-DC Motor on basic of the motor, linear-DC Motor for moving-coil type, linear-DC motor for permanent-magnet moving type, linear-DC motor for electricity non-utility type, linear-pulse motor for variable motor, linear-pulse motor for permanent magneto type, linear-vibration actuator, linear-vibration actuator for moving-coil type, linear synchronous motor, linear electromagnetic motor, linear electromagnetic solenoid, technical organization and magnetic levitation and linear motor and sensor.

Reconstruction of Typhoon Structure Using 3-Dimensional Doppler Radar Radial Velocity Data with the Multigrid Analysis: A Case Study in an Idealized Simulation Context

Directory of Open Access Journals (Sweden)

Hongli Fu

2016-01-01

Full Text Available Extracting multiple-scale observational information is critical for accurately reconstructing the structure of mesoscale circulation systems such as typhoon. The Space and Time Mesoscale Analysis System (STMAS with multigrid data assimilation developed in Earth System Research Laboratory (ESRL in National Oceanic and Atmospheric Administration (NOAA has addressed this issue. Previous studies have shown the capability of STMAS to retrieve multiscale information in 2-dimensional Doppler radar radial velocity observations. This study explores the application of 3-dimensional (3D Doppler radar radial velocities with STMAS for reconstructing a 3D typhoon structure. As for the first step, here, we use an idealized simulation framework. A two-scale simulated “typhoon” field is constructed and referred to as “truth,” from which randomly distributed conventional wind data and 3D Doppler radar radial wind data are generated. These data are used to reconstruct the synthetic 3D “typhoon” structure by the STMAS and the traditional 3D variational (3D-Var analysis. The degree by which the “truth” 3D typhoon structure is recovered is an assessment of the impact of the data type or analysis scheme being evaluated. We also examine the effects of weak constraint and strong constraint on STMAS analyses. Results show that while the STMAS is superior to the traditional 3D-Var for reconstructing the 3D typhoon structure, the strong constraint STMAS can produce better analyses on both horizontal and vertical velocities.

Solving non-linear Horn clauses using a linear Horn clause solver

DEFF Research Database (Denmark)

Kafle, Bishoksan; Gallagher, John Patrick; Ganty, Pierre

2016-01-01

In this paper we show that checking satisfiability of a set of non-linear Horn clauses (also called a non-linear Horn clause program) can be achieved using a solver for linear Horn clauses. We achieve this by interleaving a program transformation with a satisfiability checker for linear Horn...... clauses (also called a solver for linear Horn clauses). The program transformation is based on the notion of tree dimension, which we apply to a set of non-linear clauses, yielding a set whose derivation trees have bounded dimension. Such a set of clauses can be linearised. The main algorithm...... dimension. We constructed a prototype implementation of this approach and performed some experiments on a set of verification problems, which shows some promise....

Linear and non-linear simulation of joints contact surface using ...

African Journals Online (AJOL)

The joint modelling including non-linear effects needs accurate and precise study of their behaviors. When joints are under the dynamic loading, micro, macro- slip happens in contact surface which is non-linear reason of the joint contact surface. The non-linear effects of joint contact surface on total behavior of structure are ...

Linear shaped charge

Energy Technology Data Exchange (ETDEWEB)

Peterson, David; Stofleth, Jerome H.; Saul, Venner W.

2017-07-11

Linear shaped charges are described herein. In a general embodiment, the linear shaped charge has an explosive with an elongated arrowhead-shaped profile. The linear shaped charge also has and an elongated v-shaped liner that is inset into a recess of the explosive. Another linear shaped charge includes an explosive that is shaped as a star-shaped prism. Liners are inset into crevices of the explosive, where the explosive acts as a tamper.

An efficient method for model refinement in diffuse optical tomography

Science.gov (United States)

Zirak, A. R.; Khademi, M.

2007-11-01

Diffuse optical tomography (DOT) is a non-linear, ill-posed, boundary value and optimization problem which necessitates regularization. Also, Bayesian methods are suitable owing to measurements data are sparse and correlated. In such problems which are solved with iterative methods, for stabilization and better convergence, the solution space must be small. These constraints subject to extensive and overdetermined system of equations which model retrieving criteria specially total least squares (TLS) must to refine model error. Using TLS is limited to linear systems which is not achievable when applying traditional Bayesian methods. This paper presents an efficient method for model refinement using regularized total least squares (RTLS) for treating on linearized DOT problem, having maximum a posteriori (MAP) estimator and Tikhonov regulator. This is done with combination Bayesian and regularization tools as preconditioner matrices, applying them to equations and then using RTLS to the resulting linear equations. The preconditioning matrixes are guided by patient specific information as well as a priori knowledge gained from the training set. Simulation results illustrate that proposed method improves the image reconstruction performance and localize the abnormally well.

NITSOL: A Newton iterative solver for nonlinear systems

Energy Technology Data Exchange (ETDEWEB)

Pernice, M. [Univ. of Utah, Salt Lake City, UT (United States); Walker, H.F. [Utah State Univ., Logan, UT (United States)

1996-12-31

Newton iterative methods, also known as truncated Newton methods, are implementations of Newton`s method in which the linear systems that characterize Newton steps are solved approximately using iterative linear algebra methods. Here, we outline a well-developed Newton iterative algorithm together with a Fortran implementation called NITSOL. The basic algorithm is an inexact Newton method globalized by backtracking, in which each initial trial step is determined by applying an iterative linear solver until an inexact Newton criterion is satisfied. In the implementation, the user can specify inexact Newton criteria in several ways and select an iterative linear solver from among several popular {open_quotes}transpose-free{close_quotes} Krylov subspace methods. Jacobian-vector products used by the Krylov solver can be either evaluated analytically with a user-supplied routine or approximated using finite differences of function values. A flexible interface permits a wide variety of preconditioning strategies and allows the user to define a preconditioner and optionally update it periodically. We give details of these and other features and demonstrate the performance of the implementation on a representative set of test problems.

A Newton-Krylov method with an approximate analytical Jacobian for implicit solution of Navier-Stokes equations on staggered overset-curvilinear grids with immersed boundaries.

Science.gov (United States)

Asgharzadeh, Hafez; Borazjani, Iman

2017-02-15

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

Science.gov (United States)

Asgharzadeh, Hafez; Borazjani, Iman

2016-01-01

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier-Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for nonlinear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton-Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier-Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor-Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge-Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the

Heat Transfer Measurements for a Film Cooled Turbine Vane Cascade

Science.gov (United States)

2008-05-01

Performance of Turbine Airfoils. Journal of Turbomachinery 1998, 120, 522– 529. Arnone, A.; Liou, M.-S.; Povinelli , L. A. Multigrid Calculation of... Povinelli , L. A. Development of an Explicit Multiblock/Multigrid Flow Solver for Viscous Flows in Complex Geometries; AIAA-93-2380; 1993. Wilcox, D. C

Linear algebra

CERN Document Server

Stoll, R R

1968-01-01

Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand