Frequency response as a surrogate eigenvalue problem in topology optimization

DEFF Research Database (Denmark)

Andreassen, Erik; Ferrari, Federico; Sigmund, Ole

2018-01-01

This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity as...

Efficient solutions to the NDA-NCA low-order eigenvalue problem

International Nuclear Information System (INIS)

Willert, J. A.; Kelley, C. T.

2013-01-01

Recent algorithmic advances combine moment-based acceleration and Jacobian-Free Newton-Krylov (JFNK) methods to accelerate the computation of the dominant eigenvalue in a k-eigenvalue calculation. In particular, NDA-NCA [1], builds a sequence of low-order (LO) diffusion-based eigenvalue problems in which the solution converges to the true eigenvalue solution. Within NDA-NCA, the solution to the LO k-eigenvalue problem is computed by solving a system of nonlinear equation using some variant of Newton's method. We show that we can speed up the solution to the LO problem dramatically by abandoning the JFNK method and exploiting the structure of the Jacobian matrix. (authors)

Eigenvalue solutions in finite element thermal transient problems

International Nuclear Information System (INIS)

Stoker, J.R.

1975-01-01

The eigenvalue economiser concept can be useful in solving large finite element transient heat flow problems in which the boundary heat transfer coefficients are constant. The usual economiser theory is equivalent to applying a unit thermal 'force' to each of a small sub-set of nodes on the finite element mesh, and then calculating sets of resulting steady state temperatures. Subsequently it is assumed that the required transient temperature distributions can be approximated by a linear combination of this comparatively small set of master temperatures. The accuracy of a reduced eigenvalue calculation depends upon a good choice of master nodes, which presupposes at least a little knowledge about what sort of shape is expected in the unknown temperature distributions. There are some instances, however, where a reasonably good idea exists of the required shapes, permitting a modification to the economiser process which leads to greater economy in the number of master temperatures. The suggested new approach is to use manually prescribed temperature distributions as the master distributions, rather than using temperatures resulting from unit thermal forces. Thus, with a little pre-knowledge one may write down a set of master distributions which, as a linear combination, can represent the required solution over the range of interest to a reasonable engineering accuracy, and using the minimum number of variables. The proposed modified eigenvalue economiser technique then uses the master distributions in an automatic way to arrive at the required solution. The technique is illustrated by some simple finite element examples

A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

International Nuclear Information System (INIS)

Hwang, F-N; Wei, Z-H; Huang, T-M; Wang Weichung

2010-01-01

We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETSc's efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schroedinger's equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0 GHz processors.

A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer.

Science.gov (United States)

Hwang, Seong Jae; Collins, Maxwell D; Ravi, Sathya N; Ithapu, Vamsi K; Adluru, Nagesh; Johnson, Sterling C; Singh, Vikas

2015-12-01

Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.

The nonconforming virtual element method for eigenvalue problems

Energy Technology Data Exchange (ETDEWEB)

Gardini, Francesca [Univ. of Pavia (Italy). Dept. of Mathematics; Manzini, Gianmarco [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Vacca, Giuseppe [Univ. of Milano-Bicocca, Milan (Italy). Dept. of Mathematics and Applications

2018-02-05

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L²-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

Transmission eigenvalues

Science.gov (United States)

Cakoni, Fioralba; Haddar, Houssem

2013-10-01

associated transmission eigenfunctions. The three papers by respectively Robbiano [11], Blasten and Päivärinta [12], and Lakshtanov and Vainberg [13] provide new complementary results on the existence of transmission eigenvalues for the scalar problem under weak assumptions on the (possibly complex valued) refractive index that mainly stipulates that the contrast does not change sign on the boundary. It is interesting here to see three different new methods to obtain these results. On the other hand, the paper by Bonnet-Ben Dhia and Chesnel [14] addresses the Fredholm properties of the interior transmission problem when the contrast changes sign on the boundary, exhibiting cases where this property fails. Using more standard approaches, the existence and structure of transmission eigenvalues are analyzed in the paper by Delbary [15] for the case of frequency dependent materials in the context of Maxwell's equations, whereas the paper by Vesalainen [16] initiates the study of the transmission eigenvalue problem in unbounded domains by considering the transmission eigenvalues for Schrödinger equation with non-compactly supported potential. The paper by Monk and Selgas [17] addresses the case where the dielectric is mounted on a perfect conductor and provides some numerical examples of the localization of associated eigenvalues using the linear sampling method. A series of papers then addresses the question of localization of transmission eigenvalues and the associated inverse spectral problem for spherically stratified media. More specifically, the paper by Colton and Leung [18] provides new results on complex transmission eigenvalues and a new proof for uniqueness of a solution to the inverse spectral problem, whereas the paper by Sylvester [19] provides sharp results on how to locate all the transmission eigenvalues associated with angular independent eigenfunctions when the index of refraction is constant. The paper by Gintides and Pallikarakis [20] investigates an

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

Science.gov (United States)

Wu, Sheng-Jhih; Chu, Moody T.

2017-08-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

International Nuclear Information System (INIS)

Wu, Sheng-Jhih; Chu, Moody T

2017-01-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing–Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations. (paper)

Generalization of the Fourier Convergence Analysis in the Neutron Diffusion Eigenvalue Problem

International Nuclear Information System (INIS)

Lee, Hyun Chul; Noh, Jae Man; Joo, Hyung Kook

2005-01-01

Fourier error analysis has been a standard technique for the stability and convergence analysis of linear and nonlinear iterative methods. Lee et al proposed new 2- D/1-D coupling methods and demonstrated several advantages of the new methods by performing a Fourier convergence analysis of the methods as well as two existing methods for a fixed source problem. We demonstrated the Fourier convergence analysis of one of the 2-D/1-D coupling methods applied to a neutron diffusion eigenvalue problem. However, the technique cannot be used directly to analyze the convergence of the other 2-D/1-D coupling methods since some algorithm-specific features were used in our previous study. In this paper we generalized the Fourier convergence analysis technique proposed and analyzed the convergence of the 2-D/1-D coupling methods applied to a neutron diffusion Eigenvalue problem using the generalized technique

NESTLE: Few-group neutron diffusion equation solver utilizing the nodal expansion method for eigenvalue, adjoint, fixed-source steady-state and transient problems

International Nuclear Information System (INIS)

Turinsky, P.J.; Al-Chalabi, R.M.K.; Engrand, P.; Sarsour, H.N.; Faure, F.X.; Guo, W.

1994-06-01

NESTLE is a FORTRAN77 code that solves the few-group neutron diffusion equation utilizing the Nodal Expansion Method (NEM). NESTLE can solve the eigenvalue (criticality); eigenvalue adjoint; external fixed-source steady-state; or external fixed-source. or eigenvalue initiated transient problems. The code name NESTLE originates from the multi-problem solution capability, abbreviating Nodal Eigenvalue, Steady-state, Transient, Le core Evaluator. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent. Two or four energy groups can be utilized, with all energy groups being thermal groups (i.e. upscatter exits) if desired. Core geometries modelled include Cartesian and Hexagonal. Three, two and one dimensional models can be utilized with various symmetries. The non-linear iterative strategy associated with the NEM method is employed. An advantage of the non-linear iterative strategy is that NSTLE can be utilized to solve either the nodal or Finite Difference Method representation of the few-group neutron diffusion equation

The eigenvalue problem in phase space.

Science.gov (United States)

Cohen, Leon

2018-06-30

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

Jacobi-Davidson methods for generalized MHD-eigenvalue problems

NARCIS (Netherlands)

J.G.L. Booten; D.R. Fokkema; G.L.G. Sleijpen; H.A. van der Vorst (Henk)

1995-01-01

textabstractA Jacobi-Davidson algorithm for computing selected eigenvalues and associated eigenvectors of the generalized eigenvalue problem $Ax = lambda Bx$ is presented. In this paper the emphasis is put on the case where one of the matrices, say the B-matrix, is Hermitian positive definite. The

Approximative analytic eigenvalues for orbital excitations in the case of a coulomb potential plus linear and quadratic radial terms

International Nuclear Information System (INIS)

Rekab, S.; Zenine, N.

2006-01-01

We consider the three dimensional non relativistic eigenvalue problem in the case of a Coulomb potential plus linear and quadratic radial terms. In the framework of the Rayleigh-Schrodinger Perturbation Theory, using a specific choice of the unperturbed Hamiltonian, we obtain approximate analytic expressions for the eigenvalues of orbital excitations. The implications and the range of validity of the obtained analytic expression are discussed

EISPACK-J: subprogram package for solving eigenvalue problems

International Nuclear Information System (INIS)

Fujimura, Toichiro; Tsutsui, Tsuneo

1979-05-01

EISPACK-J, a subprogram package for solving eigenvalue problems, has been developed and subprograms with a variety of functions have been prepared. These subprograms can solve standard problems of complex matrices, general problems of real matrices and special problems in which only the required eigenvalues and eigenvectors are calculated. They are compared to existing subprograms, showing their features through benchmark tests. Many test problems, including realistic scale problems, are provided for the benchmark tests. Discussions are made on computer core storage and computing time required for each subprogram, and accuracy of the solution. The results show that the subprograms of EISPACK-J, based on Householder, QR and inverse iteration methods, are the best in computing time and accuracy. (author)

Deflation of Eigenvalues for GMRES in Lattice QCD

International Nuclear Information System (INIS)

Morgan, Ronald B.; Wilcox, Walter

2002-01-01

Versions of GMRES with deflation of eigenvalues are applied to lattice QCD problems. Approximate eigenvectors corresponding to the smallest eigenvalues are generated at the same time that linear equations are solved. The eigenvectors improve convergence for the linear equations, and they help solve other right-hand sides

Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

Directory of Open Access Journals (Sweden)

Manfred Möller

2013-01-01

Full Text Available Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameter λ and which has separable boundary conditions depending linearly on λ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

Lagrangian Differentiation, Integration and Eigenvalues Problems

International Nuclear Information System (INIS)

Durand, L.

1983-01-01

Calogero recently proposed a new and very powerful method for the solution of Sturm-Liouville eigenvalue problems based on Lagrangian differentiation. In this paper, some results of a numerical investigation of Calogero's method for physical interesting problems are presented. It is then shown that one can 'invert' his differentiation technique to obtain a flexible, factorially convergent Lagrangian integration scheme which should be useful in a variety of problems, e.g. solution of integral equations

Nonlinear Eigenvalue Problems in Elliptic Variational Inequalities: a local study

International Nuclear Information System (INIS)

Conrad, F.; Brauner, C.; Issard-Roch, F.; Nicolaenko, B.

1985-01-01

The authors consider a class of Nonlinear Eigenvalue Problems (N.L.E.P.) associated with Elliptic Variational Inequalities (E.V.I.). First the authors introduce the main tools for a local study of branches of solutions; the authors extend the linearization process required in the case of equations. Next the authors prove the existence of arcs of solutions close to regular vs singular points, and determine their local behavior up to the first order. Finally, the authors discuss the connection between their regularity condition and some stability concept. 37 references, 6 figures

Instability of the cored barotropic disc: the linear eigenvalue formulation

Science.gov (United States)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

A non-self-adjoint quadratic eigenvalue problem describing a fluid-solid interaction Part II : analysis of convergence

NARCIS (Netherlands)

Bourne, D.P.; Elman, H.; Osborn, J.E.

2009-01-01

This paper is the second part of a two-part paper treating a non-self-adjoint quadratic eigenvalue problem for the linear stability of solutions to the Taylor-Couette problem for flow of a viscous liquid in a deformable cylinder, with the cylinder modelled as a membrane. The first part formulated

2nd International Workshop on Eigenvalue Problems : Algorithms, Software and Applications in Petascale Computing

CERN Document Server

Zhang, Shao-Liang; Imamura, Toshiyuki; Yamamoto, Yusaku; Kuramashi, Yoshinobu; Hoshi, Takeo

2017-01-01

This book provides state-of-the-art and interdisciplinary topics on solving matrix eigenvalue problems, particularly by using recent petascale and upcoming post-petascale supercomputers. It gathers selected topics presented at the International Workshops on Eigenvalue Problems: Algorithms; Software and Applications, in Petascale Computing (EPASA2014 and EPASA2015), which brought together leading researchers working on the numerical solution of matrix eigenvalue problems to discuss and exchange ideas – and in so doing helped to create a community for researchers in eigenvalue problems. The topics presented in the book, including novel numerical algorithms, high-performance implementation techniques, software developments and sample applications, will contribute to various fields that involve solving large-scale eigenvalue problems.

Solving complex band structure problems with the FEAST eigenvalue algorithm

Science.gov (United States)

Laux, S. E.

2012-08-01

With straightforward extension, the FEAST eigenvalue algorithm [Polizzi, Phys. Rev. B 79, 115112 (2009)] is capable of solving the generalized eigenvalue problems representing traveling-wave problems—as exemplified by the complex band-structure problem—even though the matrices involved are complex, non-Hermitian, and singular, and hence outside the originally stated range of applicability of the algorithm. The obtained eigenvalues/eigenvectors, however, contain spurious solutions which must be detected and removed. The efficiency and parallel structure of the original algorithm are unaltered. The complex band structures of Si layers of varying thicknesses and InAs nanowires of varying radii are computed as test problems.

Estimates for lower order eigenvalues of a clamped plate problem

OpenAIRE

Cheng, Qing-Ming; Huang, Guangyue; Wei, Guoxin

2009-01-01

For a bounded domain $\\Omega$ in a complete Riemannian manifold $M^n$, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.

Schiffer's Conjecture, Interior Transmission Eigenvalues and Invisibility Cloaking: Singular Problem vs. Nonsingular Problem

OpenAIRE

Liu, Hongyu

2012-01-01

In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.

Multi-level nonlinear diffusion acceleration method for multigroup transport k-Eigenvalue problems

International Nuclear Information System (INIS)

Anistratov, Dmitriy Y.

2011-01-01

The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in 1D slab geometry. The proposed method is defined by a multi-level system of equations that includes multigroup and effective one-group low-order NDA equations. The Eigenvalue is evaluated in the exact projected solution space of smallest dimensionality, namely, by solving the effective one- group eigenvalue transport problem. Numerical results that illustrate performance of the new algorithm are demonstrated. (author)

Computing the eigenvalues and eigenvectors of a fuzzy matrix

Directory of Open Access Journals (Sweden)

A. Kumar

2012-08-01

Full Text Available Computation of fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix is a challenging problem. Determining the maximal and minimal symmetric solution can help to find the eigenvalues. So, we try to compute these eigenvalues by determining the maximal and minimal symmetric solution of the fully fuzzy linear system $widetilde{A}widetilde{X}= widetilde{lambda} widetilde{X}.$

On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations

Directory of Open Access Journals (Sweden)

Heinrich Voss

2016-06-01

Full Text Available Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.

Ab initio nuclear structure - the large sparse matrix eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Vary, James P; Maris, Pieter [Department of Physics, Iowa State University, Ames, IA, 50011 (United States); Ng, Esmond; Yang, Chao [Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (United States); Sosonkina, Masha, E-mail: jvary@iastate.ed [Scalable Computing Laboratory, Ames Laboratory, Iowa State University, Ames, IA, 50011 (United States)

2009-07-01

The structure and reactions of light nuclei represent fundamental and formidable challenges for microscopic theory based on realistic strong interaction potentials. Several ab initio methods have now emerged that provide nearly exact solutions for some nuclear properties. The ab initio no core shell model (NCSM) and the no core full configuration (NCFC) method, frame this quantum many-particle problem as a large sparse matrix eigenvalue problem where one evaluates the Hamiltonian matrix in a basis space consisting of many-fermion Slater determinants and then solves for a set of the lowest eigenvalues and their associated eigenvectors. The resulting eigenvectors are employed to evaluate a set of experimental quantities to test the underlying potential. For fundamental problems of interest, the matrix dimension often exceeds 10{sup 10} and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving this large sparse matrix eigenvalue problem. We also outline the challenges that lie ahead for achieving further breakthroughs in fundamental nuclear theory using these ab initio approaches.

Ab initio nuclear structure - the large sparse matrix eigenvalue problem

International Nuclear Information System (INIS)

Vary, James P; Maris, Pieter; Ng, Esmond; Yang, Chao; Sosonkina, Masha

2009-01-01

The structure and reactions of light nuclei represent fundamental and formidable challenges for microscopic theory based on realistic strong interaction potentials. Several ab initio methods have now emerged that provide nearly exact solutions for some nuclear properties. The ab initio no core shell model (NCSM) and the no core full configuration (NCFC) method, frame this quantum many-particle problem as a large sparse matrix eigenvalue problem where one evaluates the Hamiltonian matrix in a basis space consisting of many-fermion Slater determinants and then solves for a set of the lowest eigenvalues and their associated eigenvectors. The resulting eigenvectors are employed to evaluate a set of experimental quantities to test the underlying potential. For fundamental problems of interest, the matrix dimension often exceeds 10 10 and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving this large sparse matrix eigenvalue problem. We also outline the challenges that lie ahead for achieving further breakthroughs in fundamental nuclear theory using these ab initio approaches.

Eigenstructure of of singular systems. Perturbation analysis of simple eigenvalues

OpenAIRE

García Planas, María Isabel; Tarragona Romero, Sonia

2014-01-01

The problem to study small perturbations of simple eigenvalues with a change of parameters is of general interest in applied mathematics. After to introduce a systematic way to know if an eigenvalue of a singular system is simple or not, the aim of this work is to study the behavior of a simple eigenvalue of singular linear system family

Accurate high-lying eigenvalues of Schroedinger and Sturm-Liouville problems

International Nuclear Information System (INIS)

Vanden Berghe, G.; Van Daele, M.; De Meyer, H.

1994-01-01

A modified difference and a Numerov-like scheme have been introduced in a shooting algorithm for the determination of the (higher-lying) eigenvalues of Schroedinger equations and Sturm-Liouville problems. Some numerical experiments are introduced. Time measurements have been performed. The proposed algorithms are compared with other previously introduced shooting schemes. The structure of the eigenvalue error is discussed. ((orig.))

A note on quasilinear elliptic eigenvalue problems

Directory of Open Access Journals (Sweden)

Gianni Arioli

1999-11-01

Full Text Available We study an eigenvalue problem by a non-smooth critical point theory. Under general assumptions, we prove the existence of at least one solution as a minimum of a constrained energy functional. We apply some results on critical point theory with symmetry to provide a multiplicity result.

Hybrid subgroup decomposition method for solving fine-group eigenvalue transport problems

International Nuclear Information System (INIS)

Yasseri, Saam; Rahnema, Farzad

2014-01-01

Highlights: • An acceleration technique for solving fine-group eigenvalue transport problems. • Coarse-group quasi transport theory to solve coarse-group eigenvalue transport problems. • Consistent and inconsistent formulations for coarse-group quasi transport theory. • Computational efficiency amplified by a factor of 2 using hybrid SGD for 1D BWR problem. - Abstract: In this paper, a new hybrid method for solving fine-group eigenvalue transport problems is developed. This method extends the subgroup decomposition method to efficiently couple a new coarse-group quasi transport theory with a set of fixed-source transport decomposition sweeps to obtain the fine-group transport solution. The advantages of the quasi transport theory are its high accuracy, straight-forward implementation and numerical stability. The hybrid method is analyzed for a 1D benchmark problem characteristic of boiling water reactors (BWR). It is shown that the method reproduces the fine-group transport solution with high accuracy while increasing the computational efficiency up to 12 times compared to direct fine-group transport calculations

Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

Science.gov (United States)

Costiner, Sorin; Ta'asan, Shlomo

1995-07-01

Algorithms for nonlinear eigenvalue problems (EP's) often require solving self-consistently a large number of EP's. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP's obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schrödinger Poisson EP's. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schrödinger-Poisson EP in two and three dimensions, presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per fine level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained.

On a quadratic inverse eigenvalue problem

International Nuclear Information System (INIS)

Cai, Yunfeng; Xu, Shufang

2009-01-01

This paper concerns the quadratic inverse eigenvalue problem (QIEP) of constructing real symmetric matrices M, C and K of size n × n, with M nonsingular, so that the quadratic matrix polynomial Q(λ) ≡ λ 2 M + λC + K has a completely prescribed set of eigenvalues and eigenvectors. It is shown via construction that the QIEP has a solution if and only if r 0, where r and δ are computable from the prescribed spectral data. A necessary and sufficient condition for the existence of a solution to the QIEP with M being positive definite is also established in a constructive way. Furthermore, two algorithms are developed: one is to solve the QIEP; another is to find a particular solution to the QIEP with the leading coefficient matrix being positive definite, which also provides us an approach to a simultaneous reduction of real symmetric matrix triple (M, C, K) by real congruence. Numerical results show that the two algorithms are feasible and numerically reliable

Use of exact albedo conditions in numerical methods for one-dimensional one-speed discrete ordinates eigenvalue problems

International Nuclear Information System (INIS)

Abreu, M.P. de

1994-01-01

The use of exact albedo boundary conditions in numerical methods applied to one-dimensional one-speed discrete ordinates (S n ) eigenvalue problems for nuclear reactor global calculations is described. An albedo operator that treats the reflector region around a nuclear reactor core implicitly is described and exactly was derived. To illustrate the method's efficiency and accuracy, it was used conventional linear diamond method with the albedo option to solve typical model problems. (author)

Photonic band structure calculations using nonlinear eigenvalue techniques

International Nuclear Information System (INIS)

Spence, Alastair; Poulton, Chris

2005-01-01

This paper considers the numerical computation of the photonic band structure of periodic materials such as photonic crystals. This calculation involves the solution of a Hermitian nonlinear eigenvalue problem. Numerical methods for nonlinear eigenvalue problems are usually based on Newton's method or are extensions of techniques for the standard eigenvalue problem. We present a new variation on existing methods which has its derivation in methods for bifurcation problems, where bordered matrices are used to compute critical points in singular systems. This new approach has several advantages over the current methods. First, in our numerical calculations the new variation is more robust than existing techniques, having a larger domain of convergence. Second, the linear systems remain Hermitian and are nonsingular as the method converges. Third, the approach provides an elegant and efficient way of both thinking about the problem and organising the computer solution so that only one linear system needs to be factorised at each stage in the solution process. Finally, first- and higher-order derivatives are calculated as a natural extension of the basic method, and this has advantages in the electromagnetic problem discussed here, where the band structure is plotted as a set of paths in the (ω,k) plane

Heuristic geometric ''eigenvalue universality'' in a one-dimensional neutron transport problem with anisotropic scattering

International Nuclear Information System (INIS)

Goncalves, G.A.; Vilhena, M.T. de; Bodmann, B.E.J.

2010-01-01

In the present work we propose a heuristic construction of a transport equation for neutrons with anisotropic scattering considering only the radial cylinder dimension. The eigenvalues of the solutions of the equation correspond to the positive values for the one dimensional case. The central idea of the procedure is the application of the S N method for the discretisation of the angular variable followed by the application of the zero order Hankel transformation. The basis the construction of the scattering terms in form of an integro-differential equation for stationary transport resides in the hypothesis that the eigenvalues that compose the elementary solutions are independent of geometry for a homogeneous medium. We compare the solutions for the cartesian one dimensional problem for an infinite cylinder with azimuthal symmetry and linear anisotropic scattering for two cases. (orig.)

Methods for computing SN eigenvalues and eigenvectors of slab geometry transport problems

International Nuclear Information System (INIS)

Yavuz, Musa

1998-01-01

We discuss computational methods for computing the eigenvalues and eigenvectors of single energy-group neutral particle transport (S N ) problems in homogeneous slab geometry, with an arbitrary scattering anisotropy of order L. These eigensolutions are important when exact (or very accurate) solutions are desired for coarse spatial cell problems demanding rapid execution times. Three methods, one of which is 'new', are presented for determining the eigenvalues and eigenvectors of such S N problems. In the first method, separation of variables is directly applied to the S N equations. In the second method, common characteristics of the S N and P N-1 equations are used. In the new method, the eigenvalues and eigenvectors can be computed provided that the cell-interface Green's functions (transmission and reflection factors) are known. Numerical results for S 4 test problems are given to compare the new method with the existing methods

Methods for computing SN eigenvalues and eigenvectors of slab geometry transport problems

International Nuclear Information System (INIS)

Yavuz, M.

1997-01-01

We discuss computational methods for computing the eigenvalues and eigenvectors of single energy-group neutral particle transport (S N ) problems in homogeneous slab geometry, with an arbitrary scattering anisotropy of order L. These eigensolutions are important when exact (or very accurate) solutions are desired for coarse spatial cell problems demanding rapid execution times. Three methods, one of which is 'new', are presented for determining the eigenvalues and eigenvectors of such S N problems. In the first method, separation of variables is directly applied to the S N equations. In the second method, common characteristics of the S N and P N-1 equations are used. In the new method, the eigenvalues and eigenvectors can be computed provided that the cell-interface Green's functions (transmission and reflection factors) are known. Numerical results for S 4 test problems are given to compare the new method with the existing methods. (author)

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.

Preconditioned Krylov subspace methods for eigenvalue problems

Energy Technology Data Exchange (ETDEWEB)

Wu, Kesheng; Saad, Y.; Stathopoulos, A. [Univ. of Minnesota, Minneapolis, MN (United States)

1996-12-31

Lanczos algorithm is a commonly used method for finding a few extreme eigenvalues of symmetric matrices. It is effective if the wanted eigenvalues have large relative separations. If separations are small, several alternatives are often used, including the shift-invert Lanczos method, the preconditioned Lanczos method, and Davidson method. The shift-invert Lanczos method requires direct factorization of the matrix, which is often impractical if the matrix is large. In these cases preconditioned schemes are preferred. Many applications require solution of hundreds or thousands of eigenvalues of large sparse matrices, which pose serious challenges for both iterative eigenvalue solver and preconditioner. In this paper we will explore several preconditioned eigenvalue solvers and identify the ones suited for finding large number of eigenvalues. Methods discussed in this paper make up the core of a preconditioned eigenvalue toolkit under construction.

On the Solution of the Eigenvalue Assignment Problem for Discrete-Time Systems

Directory of Open Access Journals (Sweden)

El-Sayed M. E. Mostafa

2017-01-01

Full Text Available The output feedback eigenvalue assignment problem for discrete-time systems is considered. The problem is formulated first as an unconstrained minimization problem, where a three-term nonlinear conjugate gradient method is proposed to find a local solution. In addition, a cut to the objective function is included, yielding an inequality constrained minimization problem, where a logarithmic barrier method is proposed for finding the local solution. The conjugate gradient method is further extended to tackle the eigenvalue assignment problem for the two cases of decentralized control systems and control systems with time delay. The performance of the methods is illustrated through various test examples.

Discontinuous Sturm-Liouville Problems with Eigenvalue Dependent Boundary Condition

Energy Technology Data Exchange (ETDEWEB)

Amirov, R. Kh., E-mail: emirov@cumhuriyet.edu.tr; Ozkan, A. S., E-mail: sozkan@cumhuriyet.edu.tr [Cumhuriyet University, Department of Mathematics Faculty of Art and Science (Turkey)

2014-12-15

In this study, an inverse problem for Sturm-Liouville differential operators with discontinuities is studied when an eigenparameter appears not only in the differential equation but it also appears in the boundary condition. Uniqueness theorems of inverse problems according to the Prüfer angle, the Weyl function and two different eigenvalues sets are proved.

Prolongation structure and linear eigenvalue equations for Einstein-Maxwell fields

International Nuclear Information System (INIS)

Kramer, D.; Neugebauer, G.

1981-01-01

The Einstein-Maxwell equations for stationary axisymmetric exterior fields are shown to be the integrability conditions of a set of linear eigenvalue equations for pseudopotentials. Using the method of Wahlquist and Estabrook (J. Math Phys.; 16:1 (1975)) it is shown that the prolongation structure of the Einstein-Maxwell equations contains the SU(2,1) Lie algebra. A new mapping of known solutions to other solutions has been found. (author)

Parallel Symmetric Eigenvalue Problem Solvers

Science.gov (United States)

2015-05-01

Research” and the use of copyright material. Approved by Major Professor(s): Approved by: Head of the Departmental Graduate Program Date Alicia Marie... matrix . . . . . . . . . . . . . . . . . 106 8.15 Sparsity patterns for the Nastran benchmark of order 1.5 million . . . . 108 8.16 Sparsity patterns...magnitude eigenvalues of a given matrix pencil (A,B) along with their associated eigenvectors. Computing the smallest eigenvalues is more difficult

Elementary Baecklund transformations for a discrete Ablowitz-Ladik eigenvalue problem

International Nuclear Information System (INIS)

Rourke, David E

2004-01-01

Elementary Baecklund transformations (BTs) are described for a discretization of the Zakharov-Shabat eigenvalue problem (a special case of the Ablowitz-Ladik eigenvalue problem). Elementary BTs allow the process of adding bound states to a system (i.e., the add-one-soliton BT) to be 'factorized' to solving two simpler sub-problems. They are used to determine the effect on the scattering data when bound states are added. They are shown to provide a method of calculating discrete solitons-this is achieved by constructing a lattice of intermediate potentials, with the parameters used in the calculation of the lattice simply related to the soliton scattering data. When the potentials, S n , T n , in the system are related by S n = -T n , they enable simple derivations to be obtained of the add-one-soliton BT and the nonlinear superposition formula

Parallelization of mathematical library for generalized eigenvalue problem for real band matrices

International Nuclear Information System (INIS)

Tanaka, Yasuhisa.

1997-05-01

This research has focused on a parallelization of the mathematical library for a generalized eigenvalue problem for real band matrices on IBM SP and Hitachi SR2201. The origin of the library is LASO (Lanczos Algorithm with Selective Orthogonalization), which was developed on the basis of Block Lanczos method for standard eigenvalue problem for real band matrices at Texas University. We adopted D.O.F. (Degree Of Freedom) decomposition method for a parallelization of this library, and evaluated its parallel performance. (author)

The numerical analysis of eigenvalue problem solutions in multigroup neutron diffusion theory

International Nuclear Information System (INIS)

Woznicki, Z.I.

1995-01-01

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iterations within global iterations. Particular iterative strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 35 figs, 16 tabs

Numerical Investigations on Several Stabilized Finite Element Methods for the Stokes Eigenvalue Problem

Directory of Open Access Journals (Sweden)

Pengzhan Huang

2011-01-01

Full Text Available Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and it is a favorite method among these methods for the Stokes eigenvalue problem.

Solving eigenvalue problems on curved surfaces using the Closest Point Method

KAUST Repository

Macdonald, Colin B.

2011-06-01

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. © 2011 Elsevier Inc.

A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

Directory of Open Access Journals (Sweden)

Muhammed I. Syam

2017-11-01

Full Text Available This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\\alpha $-order Sturm-Liouville problem where $\\frac{1}{2}< \\alpha \\leq 1$. In this paper, we implement the reproducing kernel method RKM to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Solving the generalized symmetric eigenvalue problem using tile algorithms on multicore architectures

KAUST Repository

Ltaief, Hatem

2012-01-01

This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric eigenvalue problem by computing the Cholesky factorization of the right hand side symmetric definite positive matrix (first stage), and applying the inverse of the freshly computed triangular Cholesky factors to the original dense symmetric matrix of the problem (second stage). Calculating the eigenpairs of the resulting problem is then equivalent to the eigenpairs of the original problem. The computation proceeds by reducing the updated dense symmetric matrix to symmetric band form (third stage). The band structure is further reduced by applying a bulge chasing procedure, which annihilates the extra off-diagonal entries using orthogonal transformations (fourth stage). More details on the third and fourth stage can be found in Haidar et al. [Accepted at SC\\'11, November 2011]. The eigenvalues are then calculated from the tridiagonal form using the standard LAPACK QR algorithm (i.e., DTSEQR routine), while the complex and challenging eigenvector computations will be addressed in a companion paper. The tasks from the various stages can concurrently run in an out-of-order fashion. The data dependencies are cautiously tracked by the dynamic runtime system environment QUARK, which ensures the dependencies are not violated for numerical correctness purposes. The obtained tile four-stage generalized symmetric eigenvalue solver significantly outperforms the state-of-the-art numerical libraries (up to 21-fold speed up against multithreaded LAPACK with optimized multithreaded MKL BLAS and up to 4-fold speed up against the corresponding routine from the commercial numerical software Intel MKL) on four sockets twelve cores AMD system with a 24000×24000 matrix size. © 2012 The authors and IOS Press. All rights reserved.

Fundaments of transport equation splitting and the eigenvalue problem

International Nuclear Information System (INIS)

Stancic, V.

2000-01-01

In order to remove some singularities concerning the boundary conditions of one dimensional transport equation, a split form of transport equation describing the forward i.e. μ≥0, and a backward μ<0 directed neutrons is being proposed here. The eigenvalue problem has also been considered here (author)

Eigenvalues of the -Laplacian and disconjugacy criteria

Directory of Open Access Journals (Sweden)

Pinasco Juan P

2006-01-01

Full Text Available We derive oscillation and nonoscillation criteria for the one-dimensional -Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.

The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory

International Nuclear Information System (INIS)

Woznicki, Z.I.

1994-01-01

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs

The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory

Energy Technology Data Exchange (ETDEWEB)

Woznicki, Z I [Institute of Atomic Energy, Otwock-Swierk (Poland)

1994-12-31

The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs.

Solving the generalized symmetric eigenvalue problem using tile algorithms on multicore architectures

KAUST Repository

Ltaief, Hatem; Luszczek, Piotr R.; Haidar, Azzam; Dongarra, Jack

2012-01-01

This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric

Linear finite element method for one-dimensional diffusion problems

Energy Technology Data Exchange (ETDEWEB)

Brandao, Michele A.; Dominguez, Dany S.; Iglesias, Susana M., E-mail: micheleabrandao@gmail.com, E-mail: dany@labbi.uesc.br, E-mail: smiglesias@uesc.br [Universidade Estadual de Santa Cruz (LCC/DCET/UESC), Ilheus, BA (Brazil). Departamento de Ciencias Exatas e Tecnologicas. Laboratorio de Computacao Cientifica

2011-07-01

We describe in this paper the fundamentals of Linear Finite Element Method (LFEM) applied to one-speed diffusion problems in slab geometry. We present the mathematical formulation to solve eigenvalue and fixed source problems. First, we discretized a calculus domain using a finite set of elements. At this point, we obtain the spatial balance equations for zero order and first order spatial moments inside each element. Then, we introduce the linear auxiliary equations to approximate neutron flux and current inside the element and architect a numerical scheme to obtain the solution. We offer numerical results for fixed source typical model problems to illustrate the method's accuracy for coarse-mesh calculations in homogeneous and heterogeneous domains. Also, we compare the accuracy and computational performance of LFEM formulation with conventional Finite Difference Method (FDM). (author)

The eigenvalue problem for a singular quasilinear elliptic equation

Directory of Open Access Journals (Sweden)

Benjin Xuan

2004-02-01

Full Text Available We show that many results about the eigenvalues and eigenfunctions of a quasilinear elliptic equation in the non-singular case can be extended to the singular case. Among these results, we have the first eigenvalue is associated to a $C^{1,alpha}(Omega$ eigenfunction which is positive and unique (up to a multiplicative constant, that is, the first eigenvalue is simple. Moreover the first eigenvalue is isolated and is the unique positive eigenvalue associated to a non-negative eigenfunction. We also prove some variational properties of the second eigenvalue.

Multi-level methods for solving multigroup transport eigenvalue problems in 1D slab geometry

International Nuclear Information System (INIS)

Anistratov, D. Y.; Gol'din, V. Y.

2009-01-01

A methodology for solving eigenvalue problems for the multigroup neutron transport equation in 1D slab geometry is presented. In this paper we formulate and compare different variants of nonlinear multi-level iteration methods. They are defined by means of multigroup and effective one-group low-order quasi diffusion (LOQD) equations. We analyze the effects of utilization of the effective one-group LOQD problem for estimating the eigenvalue. We present numerical results to demonstrate the performance of the iteration algorithms in different types of reactor-physics problems. (authors)

Hardy inequality, compact embeddings and properties of certain eigenvalue problems

Czech Academy of Sciences Publication Activity Database

Drábek, P.; Kufner, Alois

2017-01-01

Roč. 49, č. 1 (2017), s. 5-17 ISSN 0049-4704 Institutional support: RVO:67985840 Keywords : BD-property * compact embeddings * degenerate and singular eigenvalue problem Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics https://www.openstarts.units.it/handle/10077/16201

AMDLIBF, IBM 360 Subroutine Library, Eigenvalues, Eigenvectors, Matrix Inversion

International Nuclear Information System (INIS)

Wang, Jesse Y.

1980-01-01

Description of problem or function: AMDLIBF is a subset of the IBM 360 Subroutine Library at the Applied Mathematics Division at Argonne. This subset includes library category F: Identification/Description: F152S F SYMINV: Invert sym. matrices, solve lin. systems; F154S A DOTP: Double plus precision accum. inner prod.; F156S F RAYCOR: Rayleigh corrections for eigenvalues; F161S F XTRADP: A fast extended precision inner product; F162S A XTRADP: Inner product of two DP real vectors; F202S F1 EIGEN: Eigen-system for real symmetric matrix; F203S F: Driver for F202S; F248S F RITZIT: Largest eigenvalue and vec. real sym. matrix; F261S F EIGINV: Inverse eigenvalue problem; F313S F CQZHES: Reduce cmplx matrices to upper Hess and tri; F314S F CQZVAL: Reduce complex matrix to upper Hess. form; F315S F CQZVEC: Eigenvectors of cmplx upper triang. syst.; F316S F CGG: Driver for complex general Eigen-problem; F402S F MATINV: Matrix inversion and sol. of linear eqns.; F403S F: Driver for F402S; F452S F CHOLLU,CHOLEQ: Sym. decomp. of pos. def. band matrices; F453S F MATINC: Inversion of complex matrices; F454S F CROUT: Solution of simultaneous linear equations; F455S F CROUTC: Sol. of simultaneous complex linear eqns.; F456S F1 DIAG: Integer preserving Gaussian elimination

Towards an ideal preconditioner for linearized Navier-Stokes problems

Energy Technology Data Exchange (ETDEWEB)

Murphy, M.F. [Univ. of Bristol (United Kingdom)

1996-12-31

Discretizing certain linearizations of the steady-state Navier-Stokes equations gives rise to nonsymmetric linear systems with indefinite symmetric part. We show that for such systems there exists a block diagonal preconditioner which gives convergence in three GMRES steps, independent of the mesh size and viscosity parameter (Reynolds number). While this {open_quotes}ideal{close_quotes} preconditioner is too expensive to be used in practice, it provides a useful insight into the problem. We then consider various approximations to the ideal preconditioner, and describe the eigenvalues of the preconditioned systems. Finally, we compare these preconditioners numerically, and present our conclusions.

Singular perturbation of simple eigenvalues

International Nuclear Information System (INIS)

Greenlee, W.M.

1976-01-01

Two operator theoretic theorems which generalize those of asymptotic regular perturbation theory and which apply to singular perturbation problems are proved. Application of these theorems to concrete problems is involved, but the perturbation expansions for eigenvalues and eigenvectors are developed in terms of solutions of linear operator equations. The method of correctors, as well as traditional boundary layer techniques, can be used to apply these theorems. The current formulation should be applicable to highly singular ''hard core'' potential perturbations of the radial equation of quantum mechanics. The theorems are applied to a comparatively simple model problem whose analysis is basic to that of the quantum mechanical problem

Variational methods for eigenvalue problems an introduction to the weinstein method of intermediate problems

CERN Document Server

Gould, S H

1966-01-01

The first edition of this book gave a systematic exposition of the Weinstein method of calculating lower bounds of eigenvalues by means of intermediate problems. This second edition presents new developments in the framework of the material contained in the first edition, which is retained in somewhat modified form.

Colpitts, Eigenvalues and Chaos

DEFF Research Database (Denmark)

Lindberg, Erik

1997-01-01

It is possible to obtain insight in the chaotic nature of a nonlinear oscillator by means of a study of the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator. The movements of the eigenvalues as functions of time are found. The instantaneous power in t...

Eigenfunctions and Eigenvalues for a Scalar Riemann-Hilbert Problem Associated to Inverse Scattering

Science.gov (United States)

Pelinovsky, Dmitry E.; Sulem, Catherine

A complete set of eigenfunctions is introduced within the Riemann-Hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations. In particular, we consider the time-independent and time-dependent Schrödinger problems which are related to the KdV and KPI equations possessing solitons and lumps, respectively. Non-standard scalar products, orthogonality and completeness relations are derived for these problems. The complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations. We classify two different types of bifurcations of new eigenvalues and analyze their characteristic features. One type corresponds to thresholdless generation of solitons in the KdV equation, while the other predicts a threshold for generation of lumps in the KPI equation.

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.

Solution of the multigroup neutron diffusion Eigenvalue problem in slab geometry by modified power method

Energy Technology Data Exchange (ETDEWEB)

Zanette, Rodrigo [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pós-Graduação em Matemática Aplicada; Petersen, Claudio Z.; Tavares, Matheus G., E-mail: rodrigozanette@hotmail.com, E-mail: claudiopetersen@yahoo.com.br, E-mail: matheus.gulartetavares@gmail.com [Universidade Federal de Pelotas (UFPEL), RS (Brazil). Programa de Pós-Graduação em Modelagem Matemática

2017-07-01

We describe in this work the application of the modified power method for solve the multigroup neutron diffusion eigenvalue problem in slab geometry considering two-dimensions for nuclear reactor global calculations. It is well known that criticality calculations can often be best approached by solving eigenvalue problems. The criticality in nuclear reactors physics plays a relevant role since establishes the ratio between the numbers of neutrons generated in successive fission reactions. In order to solve the eigenvalue problem, a modified power method is used to obtain the dominant eigenvalue (effective multiplication factor (K{sub eff})) and its corresponding eigenfunction (scalar neutron flux), which is non-negative in every domain, that is, physically relevant. The innovation of this work is solving the neutron diffusion equation in analytical form for each new iteration of the power method. For solve this problem we propose to apply the Finite Fourier Sine Transform on one of the spatial variables obtaining a transformed problem which is resolved by well-established methods for ordinary differential equations. The inverse Fourier transform is used to reconstruct the solution for the original problem. It is known that the power method is an iterative source method in which is updated by the neutron flux expression of previous iteration. Thus, for each new iteration, the neutron flux expression becomes larger and more complex due to analytical solution what makes propose that it be reconstructed through an polynomial interpolation. The methodology is implemented to solve a homogeneous problem and the results are compared with works presents in the literature. (author)

Dynamic Eigenvalue Problem of Concrete Slab Road Surface

Science.gov (United States)

Pawlak, Urszula; Szczecina, Michał

2017-10-01

The paper presents an analysis of the dynamic eigenvalue problem of concrete slab road surface. A sample concrete slab was modelled using Autodesk Robot Structural Analysis software and calculated with Finite Element Method. The slab was set on a one-parameter elastic subsoil, for which the modulus of elasticity was separately calculated. The eigen frequencies and eigenvectors (as maximal vertical nodal displacements) were presented. On the basis of the results of calculations, some basic recommendations for designers of concrete road surfaces were offered.

Extending the subspace hybrid method for eigenvalue problems in reactor physics calculation

International Nuclear Information System (INIS)

Zhang, Q.; Abdel-Khalik, H. S.

2013-01-01

This paper presents an innovative hybrid Monte-Carlo-Deterministic method denoted by the SUBSPACE method designed for improving the efficiency of hybrid methods for reactor analysis applications. The SUBSPACE method achieves its high computational efficiency by taking advantage of the existing correlations between desired responses. Recently, significant gains in computational efficiency have been demonstrated using this method for source driven problems. Within this work the mathematical theory behind the SUBSPACE method is introduced and extended to address core wide level k-eigenvalue problems. The method's efficiency is demonstrated based on a three-dimensional quarter-core problem, where responses are sought on the pin cell level. The SUBSPACE method is compared to the FW-CADIS method and is found to be more efficient for the utilized test problem because of the reason that the FW-CADIS method solves a forward eigenvalue problem and an adjoint fixed-source problem while the SUBSPACE method only solves an adjoint fixed-source problem. Based on the favorable results obtained here, we are confident that the applicability of Monte Carlo for large scale reactor analysis could be realized in the near future. (authors)

New exact approaches to the nuclear eigenvalue problem

International Nuclear Information System (INIS)

Andreozzi, F.; Lo Iudice, N.; Porrino, A.; Knapp, F.; Kvasil, J.

2005-01-01

In a recent past some of us have developed a new algorithm for diagonalizing the shell model Hamiltonian which consists of an iterative sequence of diagonalization of sub-matrices of small dimensions. The method, apart from being easy to implement, is robust, yielding always stable numerical solutions, and free of ghost eigenvalues. Subsequently, we have endowed the algorithm with an importance sampling, which leads to a drastic truncation of the shell model space, while keeping the accuracy of the solutions under control. Applications to typical nuclei show that the sampling yields also an extrapolation law to the exact eigenvalues. Complementary to the shell model algorithm is a method we are developing for studying collective and non collective excitations. To this purpose we solve the nuclear eigenvalue problem in a space which is the direct sum of Tamm-Dancoff n-phonon subspaces (n=0,1, ...N). The multiphonon basis is constructed by an iterative equation of motion method, which generates an over complete set of n-phonon states from the (n-1)-phonon basis. The redundancy is removed completely and exactly by a method based on the Choleski decomposition. The full Hamiltonian matrix comes out to have a simple structure and, therefore, can be drastically truncated before diagonalization by the mentioned importance sampling method. The phonon composition of the basis states allows removing naturally and maximally the spurious admixtures induced by the centre of mass motion. An application of the method to 16 O will be given for illustrative purposes. (authors)

Asymptotic eigenvalue estimates for a Robin problem with a large parameter

Czech Academy of Sciences Publication Activity Database

Exner, Pavel; Minakov, A.; Parnovski, L.

2014-01-01

Roč. 71, č. 2 (2014), s. 141-156 ISSN 0032-5155 R&D Projects: GA ČR(CZ) GA14-06818S Institutional support: RVO:61389005 Keywords : Laplacian * Robin problem * eigenvalue asymptotics Subject RIV: BE - Theoretical Physics Impact factor: 0.250, year: 2014

Solving large nonlinear generalized eigenvalue problems from Density Functional Theory calculations in parallel

DEFF Research Database (Denmark)

Bendtsen, Claus; Nielsen, Ole Holm; Hansen, Lars Bruno

2001-01-01

The quantum mechanical ground state of electrons is described by Density Functional Theory, which leads to large minimization problems. An efficient minimization method uses a self-consistent field (SCF) solution of large eigenvalue problems. The iterative Davidson algorithm is often used, and we...

Perturbative stability of the approximate Killing field eigenvalue problem

International Nuclear Information System (INIS)

Beetle, Christopher; Wilder, Shawn

2014-01-01

An approximate Killing field may be defined on a compact, Riemannian geometry by solving an eigenvalue problem for a certain elliptic operator. This paper studies the effect of small perturbations in the Riemannian metric on the resulting vector field. It shows that small metric perturbations, as measured using a Sobolev-type supremum norm on the space of Riemannian geometries on a fixed manifold, yield small perturbations in the approximate Killing field, as measured using a Hilbert-type square integral norm. It also discusses applications to the problem of computing the spin of a generic black hole in general relativity. (paper)

Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

Science.gov (United States)

Lakshtanov, E.; Vainberg, B.

2013-10-01

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on a possible location of the transmission eigenvalues. If the index of refraction \\sqrt{n(x)} is real, then we obtain a result on the existence of infinitely many positive ITEs and the Weyl-type lower bound on its counting function. All the results are obtained under the assumption that n(x) - 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

A subspace preconditioning algorithm for eigenvector/eigenvalue computation

Energy Technology Data Exchange (ETDEWEB)

Bramble, J.H.; Knyazev, A.V.; Pasciak, J.E.

1996-12-31

We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen-spaces of a symmetric positive definite matrix. In our applications, the dimension of a matrix is large and the cost of its inverting is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning. Estimates will be provided which show that the preconditioned method converges linearly and uniformly in the matrix dimension when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.

Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

Directory of Open Access Journals (Sweden)

Winfried Auzinger

2006-01-01

Full Text Available We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.

Bound-state Dirac eigenvalues for scalar potentials

International Nuclear Information System (INIS)

Ram, B.; Arafah, M.

1981-01-01

The Dirac equation is solved with a linear and a quadratic scalar potential using an approach in which the Dirac equation is first transformed to a one-dimensional Schroedinger equation with an effective potential. The WKB method is used to obtain the energy eigenvalues. The eigenvalues for the quadratic scalar potential are real just as they are for the linear potential. The results with the linear potential agree well with those obtained by Critchfield. (author)

Those Do What? Connecting Eigenvectors and Eigenvalues to the Rest of Linear Algebra: Using Visual Enhancements to Help Students Connect Eigenvectors to the Rest of Linear Algebra

Science.gov (United States)

Nyman, Melvin A.; Lapp, Douglas A.; St. John, Dennis; Berry, John S.

2010-01-01

This paper discusses student difficulties in grasping concepts from Linear Algebra--in particular, the connection of eigenvalues and eigenvectors to other important topics in linear algebra. Based on our prior observations from student interviews, we propose technology-enhanced instructional approaches that might positively impact student…

Interior transmission eigenvalues of a rectangle

International Nuclear Information System (INIS)

Sleeman, B D; Stocks, D C

2016-01-01

The problem of scattering of acoustic waves by an inhomogeneous medium is intimately connected with so called inside–outside duality, in which the interior transmission eigenvalue problem plays a fundamental role. Here a study of the interior transmission eigenvalues for rectangular domains of constant refractive index is made. By making a nonstandard use of the classical separation of variables technique both real and complex eigenvalues are determined. (paper)

Tensor eigenvalues and their applications

CERN Document Server

Qi, Liqun; Chen, Yannan

2018-01-01

This book offers an introduction to applications prompted by tensor analysis, especially by the spectral tensor theory developed in recent years. It covers applications of tensor eigenvalues in multilinear systems, exponential data fitting, tensor complementarity problems, and tensor eigenvalue complementarity problems. It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics. This book is a valuable reference resource for researchers and graduate students who are interested in applications of tensor eigenvalues.

Eigenvalue ratio detection based on exact moments of smallest and largest eigenvalues

KAUST Repository

Shakir, Muhammad; Tang, Wuchen; Rao, Anlei; Imran, Muhammad Ali; Alouini, Mohamed-Slim

2011-01-01

Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes always depend on asymptotic assumptions since the close-formed expression of exact eigenvalues ratio distribution is exceptionally complex to compute in practice. In this paper, non-asymptotic spectrum sensing approach to approximate the extreme eigenvalues is introduced. In this context, the Gaussian approximation approach based on exact analytical moments of extreme eigenvalues is presented. In this approach, the extreme eigenvalues are considered as dependent Gaussian random variables such that the joint probability density function (PDF) is approximated by bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. In this context, the definition of Copula is cited to analyze the extent of the dependency between the extreme eigenvalues. Later, the decision threshold based on the ratio of dependent Gaussian extreme eigenvalues is derived. The performance analysis of our newly proposed approach is compared with the already published asymptotic Tracy-Widom approximation approach. © 2011 ICST.

On an Optimal -Control Problem in Coefficients for Linear Elliptic Variational Inequality

Directory of Open Access Journals (Sweden)

Olha P. Kupenko

2013-01-01

Full Text Available We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients in the main part of the elliptic operator as controls in . Since the eigenvalues of such matrices may vanish and be unbounded in , it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called -admissible solutions.

Dependence of the fundamental time eigenvalue of linear transport operator on the system size and other parameters - An application of the Perron-Frobenius theorem

International Nuclear Information System (INIS)

Sahni, D.C.

1991-01-01

Many papers have been devoted to the study of the spectral properties of the linear (neutron) transport equation. Most of the theoretical investigations have concentrated on the existence (or otherwise) of a continuous spectrum, point spectrum, a leading/dominant eigenvalue, and a corresponding positive eigenvector. It is shown that the fundamental time eigenvalue of the linear transport operator increases with the size of the system. This follows from the increase in the largest eigenvalue of a non-negative irreducible matrix whenever any matrix element his increased. This result of matrix analysis is generalized to more general Krein-Rutman operators that leave a cone of vectors invariant

Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices

International Nuclear Information System (INIS)

Kabashima, Yoshiyuki; Takahashi, Hisanao; Watanabe, Osamu

2010-01-01

A methodology to analyze the properties of the first (largest) eigenvalue and its eigenvector is developed for large symmetric random sparse matrices utilizing the cavity method of statistical mechanics. Under a tree approximation, which is plausible for infinitely large systems, in conjunction with the introduction of a Lagrange multiplier for constraining the length of the eigenvector, the eigenvalue problem is reduced to a bunch of optimization problems of a quadratic function of a single variable, and the coefficients of the first and the second order terms of the functions act as cavity fields that are handled in cavity analysis. We show that the first eigenvalue is determined in such a way that the distribution of the cavity fields has a finite value for the second order moment with respect to the cavity fields of the first order coefficient. The validity and utility of the developed methodology are examined by applying it to two analytically solvable and one simple but non-trivial examples in conjunction with numerical justification.

Some remarks on the optimization of eigenvalue problems involving the p-Laplacian

Directory of Open Access Journals (Sweden)

Wacław Pielichowski

2008-01-01

Full Text Available Given a bounded domain \\(\\Omega \\subset \\mathbb{R}^n\\, numbers \\(p \\gt 1\\, \\(\\alpha \\geq 0\\ and \\(A \\in [0,|\\Omega |]\\, consider the optimization problem: find a subset \\(D \\subset \\Omega \\, of measure \\(A\\, for which the first eigenvalue of the operator \\(u\\mapsto -\\text{div} (|\

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)

Parallel algorithms for 2-D cylindrical transport equations of Eigenvalue problem

International Nuclear Information System (INIS)

Wei, J.; Yang, S.

2013-01-01

In this paper, aimed at the neutron transport equations of eigenvalue problem under 2-D cylindrical geometry on unstructured grid, the discrete scheme of Sn discrete ordinate and discontinuous finite is built, and the parallel computation for the scheme is realized on MPI systems. Numerical experiments indicate that the designed parallel algorithm can reach perfect speedup, it has good practicality and scalability. (authors)

Input design for linear dynamic systems using maxmin criteria

DEFF Research Database (Denmark)

Sadegh, Payman; Hansen, Lars H.; Madsen, Henrik

1998-01-01

This paper considers the problem of input design for maximizing the smallest eigenvalue of the information matrix for linear dynamic systems. The optimization of the smallest eigenvalue is of interest in parameter estimation and parameter change detection problems. We describe a simple cutting...

Vertical Slot Convection: A linear study

International Nuclear Information System (INIS)

McAllister, A.; Steinolfson, R.; Tajima, T.

1992-11-01

The linear stability properties of fluid convection in a vertical slot were studied. We use a Fourier-Chebychev decomposition was used to set up the linear eigenvalue problems for the Vertical Slot Convection and Benard problems. The eigenvalues, neutral stability curves, and critical point values of the Grashof number, G, and the wavenumber were determined. Plots of the real and imaginary parts of the eigenvalues as functions of G and α are given for a wide range of the Prandtl number, Pr, and special note is made of the complex mode that becomes linearly unstable above Pr ∼ 12.5. A discussion comparing different special cases facilitates the physical understanding of the VSC equations, especially the interaction of the shear-flow and buoyancy induced physics. Making use of the real and imaginary eigenvalues and the phase properties of the eigenmodes, the eigenmodes were characterized. One finds that the mode structure becomes progressively simpler with increasing Pr, with the greatest complexity in the mid ranges where the terms in the heat equation are of roughly the same size

On the Shape Sensitivity of the First Dirichlet Eigenvalue for Two-Phase Problems

International Nuclear Information System (INIS)

Dambrine, M.; Kateb, D.

2011-01-01

We consider a two-phase problem in thermal conductivity: inclusions filled with a material of conductivity σ 1 are layered in a body of conductivity σ 2 . We address the shape sensitivity of the first eigenvalue associated with Dirichlet boundary conditions when both the boundaries of the inclusions and the body can be modified. We prove a differentiability result and provide the expressions of the first and second order derivatives. We apply the results to the optimal design of an insulated body. We prove the stability of the optimal design thanks to a second order analysis. We also continue the study of an extremal eigenvalue problem for a two-phase conductor in a ball initiated by Conca et al. (Appl. Math. Optim. 60(2):173-184, 2009) and pursued in Conca et al. (CANUM 2008, ESAIM Proc., vol. 27, pp. 311-321, EDP Sci., Les Ulis, 2009).

Oscillators and Eigenvalues

DEFF Research Database (Denmark)

Lindberg, Erik

1997-01-01

In order to obtain insight in the nature of nonlinear oscillators the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator are found and displayed as functions of time. A number of oscillators are studied including Dewey's oscillator (piecewise linear wit...... with negative resistance), Kennedy's Colpitts-oscillator (with and without chaos) and a new 4'th order oscillator with hyper-chaos....

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science.

Science.gov (United States)

Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H

2014-05-28

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem

A parallel algorithm for the non-symmetric eigenvalue problem

International Nuclear Information System (INIS)

Sidani, M.M.

1991-01-01

An algorithm is presented for the solution of the non-symmetric eigenvalue problem. The algorithm is based on a divide-and-conquer procedure that provides initial approximations to the eigenpairs, which are then refined using Newton iterations. Since the smaller subproblems can be solved independently, and since Newton iterations with different initial guesses can be started simultaneously, the algorithm - unlike the standard QR method - is ideal for parallel computers. The author also reports on his investigation of deflation methods designed to obtain further eigenpairs if needed. Numerical results from implementations on a host of parallel machines (distributed and shared-memory) are presented

Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem

KAUST Repository

Raman, Aaswath

2010-02-26

We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.

Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem

KAUST Repository

Raman, Aaswath; Fan, Shanhui

2010-01-01

We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.

Eigenvalue study of a chaotic resonator

Energy Technology Data Exchange (ETDEWEB)

Banova, Todorka [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany); Technische Universitaet Darmstadt, Graduate School of Computational Engineering, Dolivostrasse 15, D-64293 Darmstadt (Germany); Ackermann, Wolfgang; Weiland, Thomas [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany)

2013-07-01

The field of quantum chaos comprises the study of the manifestations of classical chaos in the properties of the corresponding quantum systems. Within this work, we compute the eigenfrequencies that are needed for the level spacing analysis of a microwave resonator with chaotic characteristics. The major challenges posed by our work are: first, the ability of the approaches to tackle the large scale eigenvalue problem and second, the capability to extract many, i.e. order of thousands, eigenfrequencies for the considered cavity. The first proposed approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in time domain of a superconducting cavity and by means of signal-processing techniques extracts the eigenfrequencies. The second approach is based on the finite element method with curvilinear elements, which transforms the continuous eigenvalue problem to a discrete generalized eigenvalue problem. Afterwards, the Lanczos algorithm is used for the solution of the generalized eigenvalue problem. In the poster, a summary of the applied algorithms, as well as, critical implementation details together with the simulation results are provided.

MARG2D code. 1. Eigenvalue problem for two dimensional Newcomb equation

Energy Technology Data Exchange (ETDEWEB)

Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko

1997-10-01

A new method and a code MARG2D have been developed to solve the 2-dimensional Newcomb equation which plays an important role in the magnetohydrodynamic (MHD) stability analysis in an axisymmetric toroidal plasma such as a tokamak. In the present formulation, an eigenvalue problem is posed for the 2-D Newcomb equation, where the weight function (the kinetic energy integral) and the boundary conditions at rational surfaces are chosen so that an eigenfunction correctly behaves as the linear combination of the small solution and the analytical solutions around each of the rational surfaces. Thus, the difficulty on solving the 2-D Newcomb equation has been resolved. By using the MARG2D code, the ideal MHD marginally stable state can be identified for a 2-D toroidal plasma. The code is indispensable on computing the outer-region matching data necessary for the resistive MHD stability analysis. Benchmark with ERATOJ, an ideal MHD stability code, has been carried out and the MARG2D code demonstrates that it indeed identifies both stable and marginally stable states against ideal MHD motion. (author)

Perturbation Theory of Embedded Eigenvalues

DEFF Research Database (Denmark)

Engelmann, Matthias

project gives a general and systematic approach to analytic perturbation theory of embedded eigenvalues. The spectral deformation technique originally developed in the theory of dilation analytic potentials in the context of Schrödinger operators is systematized by the use of Mourre theory. The group...... of dilations is thereby replaced by the unitary group generated y the conjugate operator. This then allows to treat the perturbation problem with the usual Kato theory.......We study problems connected to perturbation theory of embedded eigenvalues in two different setups. The first part deals with second order perturbation theory of mass shells in massive translation invariant Nelson type models. To this end an expansion of the eigenvalues w.r.t. fiber parameter up...

A multilevel, level-set method for optimizing eigenvalues in shape design problems

International Nuclear Information System (INIS)

Haber, E.

2004-01-01

In this paper, we consider optimal design problems that involve shape optimization. The goal is to determine the shape of a certain structure such that it is either as rigid or as soft as possible. To achieve this goal we combine two new ideas for an efficient solution of the problem. First, we replace the eigenvalue problem with an approximation by using inverse iteration. Second, we use a level set method but rather than propagating the front we use constrained optimization methods combined with multilevel continuation techniques. Combining these two ideas we obtain a robust and rapid method for the solution of the optimal design problem

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

Science.gov (United States)

Ito, Kazufumi; Teglas, Russell

1987-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Science.gov (United States)

Plestenjak, Bor; Gheorghiu, Călin I.; Hochstenbach, Michiel E.

2015-10-01

In numerous science and engineering applications a partial differential equation has to be solved on some fairly regular domain that allows the use of the method of separation of variables. In several orthogonal coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem (MEP); important cases include Mathieu's system, Lamé's system, and a system of spheroidal wave functions. Although multiparameter approaches are exploited occasionally to solve such equations numerically, MEPs remain less well known, and the variety of available numerical methods is not wide. The classical approach of discretizing the equations using standard finite differences leads to algebraic MEPs with large matrices, which are difficult to solve efficiently. The aim of this paper is to change this perspective. We show that by combining spectral collocation methods and new efficient numerical methods for algebraic MEPs it is possible to solve such problems both very efficiently and accurately. We improve on several previous results available in the literature, and also present a MATLAB toolbox for solving a wide range of problems.

Existence of solutions for a fourth order eigenvalue problem ] {Existence of solutions for a fourth order eigenvalue problem with variable exponent under Neumann boundary conditions

Directory of Open Access Journals (Sweden)

Khalil Ben Haddouch

2016-04-01

Full Text Available In this work we will study the eigenvalues for a fourth order elliptic equation with $p(x$-growth conditions $\\Delta^2_{p(x} u=\\lambda |u|^{p(x-2} u$, under Neumann boundary conditions, where $p(x$ is a continuous function defined on the bounded domain with $p(x>1$. Through the Ljusternik-Schnireleman theory on $C^1$-manifold, we prove the existence of infinitely many eigenvalue sequences and $\\sup \\Lambda =+\\infty$, where $\\Lambda$ is the set of all eigenvalues.

The eigenvalue problem. Alpha, lambda and gamma modes and its applications

International Nuclear Information System (INIS)

Carreno, A.; Vidal-Ferrandiz, A.; Verdu, G.; Ginestar, D.

2017-01-01

Modal analysis has been efficiently used to study different problems in reactor physics. In this sense, several eigenvalue problems can be defined for neutron transport equation: the λ-modes, the γ-modes and the α-modes. However, for simplicity, the neutron diffusion equation is used as approximation of each one of these equations that they have been discretized by a high order finite elements. The obtained algebraic eigenproblems are large problems and have to be solved using iterative methods. In this work, we analyze two methods. The first one is the Krylov-Schur method and the second one is the modified block Newton method. The comparison of modes and the performance of these methods have been studied in two benchmark problems, a homogeneous 3D reactor and the 3D Langenbuch reactor. (author)

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.

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

Science.gov (United States)

Ito, K.; Teglas, R.

1984-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Numerical method for multigroup one-dimensional SN eigenvalue problems with no spatial truncation error

International Nuclear Information System (INIS)

Abreu, M.P.; Filho, H.A.; Barros, R.C.

1993-01-01

The authors describe a new nodal method for multigroup slab-geometry discrete ordinates S N eigenvalue problems that is completely free from all spatial truncation errors. The unknowns in the method are the node-edge angular fluxes, the node-average angular fluxes, and the effective multiplication factor k eff . The numerical values obtained for these quantities are exactly those of the dominant analytic solution of the S N eigenvalue problem apart from finite arithmetic considerations. This method is based on the use of the standard balance equation and two nonstandard auxiliary equations. In the nonmultiplying regions, e.g., the reflector, we use the multigroup spectral Green's function (SGF) auxiliary equations. In the fuel regions, we use the multigroup spectral diamond (SD) auxiliary equations. The SD auxiliary equation is an extension of the conventional auxiliary equation used in the diamond difference (DD) method. This hybrid characteristic of the SD-SGF method improves both the numerical stability and the convergence rate

A numerical method to compute interior transmission eigenvalues

International Nuclear Information System (INIS)

Kleefeld, Andreas

2013-01-01

In this paper the numerical calculation of eigenvalues of the interior transmission problem arising in acoustic scattering for constant contrast in three dimensions is considered. From the computational point of view existing methods are very expensive, and are only able to show the existence of such transmission eigenvalues. Furthermore, they have trouble finding them if two or more eigenvalues are situated closely together. We present a new method based on complex-valued contour integrals and the boundary integral equation method which is able to calculate highly accurate transmission eigenvalues. So far, this is the first paper providing such accurate values for various surfaces different from a sphere in three dimensions. Additionally, the computational cost is even lower than those of existing methods. Furthermore, the algorithm is capable of finding complex-valued eigenvalues for which no numerical results have been reported yet. Until now, the proof of existence of such eigenvalues is still open. Finally, highly accurate eigenvalues of the interior Dirichlet problem are provided and might serve as test cases to check newly derived Faber–Krahn type inequalities for larger transmission eigenvalues that are not yet available. (paper)

Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media

Directory of Open Access Journals (Sweden)

Vicenţiu RăDulescu

2005-06-01

Full Text Available We study nonlinear eigenvalue problems of the type −div(a(x∇u=g(λ,x,u in ℝN, where a(x is a degenerate nonnegative weight. We establish the existence of solutions and we obtain information on qualitative properties as multiplicity and location of solutions. Our approach is based on the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality. A specific minimax method is developed without making use of Palais-Smale condition.

The universal eigenvalue bounds of Payne–Pólya–Weinberger, Hile ...

Indian Academy of Sciences (India)

R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22

following universal inequalities for the λi's in the case when n = 2: λk+1 − λk ≤. 2 .... with V ≥ 0 on and eigenvalue problems with a weight (e.g., the fixed ...... [29] Protter M H, Universal inequalities for eigenvalues, Maximum Principles and Eigenvalue. Problems in ... minimal submanifolds, Ann. Scuola Norm. Sup. Pisa Cl.

Principal Eigenvalues of a Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Directory of Open Access Journals (Sweden)

Ruyun Ma

2018-01-01

Full Text Available Let T>2 be an integer and T={1,2,…,T}. We show the existence of the principal eigenvalues of linear periodic eigenvalue problem -Δ2u(j-1+q(ju(j=λg(ju(j,  j∈T, u(0=u(T,  u(1=u(T+1, and we determine the sign of the corresponding eigenfunctions, where λ is a parameter, q(j≥0 and q(j≢0 in T, and the weight function g changes its sign in T. As an application of our spectrum results, we use the global bifurcation theory to study the existence of positive solutions for the corresponding nonlinear problem.

A method for the solution of the RPA eigenvalue

International Nuclear Information System (INIS)

Hoffman, M.J.H.; De Kock, P.R.

1986-01-01

The RPA eigenvalue problem requires the diagonalization of a 2nx2n matrix. In practical calculations, n (the number of particle-hole basis states) can be a few hundred and the diagonalization of such a large non-symmetric matrix may take quite a long time. In this report we firstly discuss sufficient conditions for real and non-zero RPA eigenvalues. The presence of zero or imaginary eigenvalues is related to the relative importance of the groundstate correlations to the total interaction energy. We then rewrite the RPA eigenvalue problem for the cases where these conditions are fulfilled in a form which only requires the diagonalization of two symmetric nxn matrices. The extend to which this method can be applied when zero eigenvalues occur, is also discussed

POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS

Directory of Open Access Journals (Sweden)

FAOUZI HADDOUCHI

2015-11-01

Full Text Available In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP for the following second-order differential equation u''(t + \\lambda a(tf(u(t = 0; 0 0 is a parameter, 0 <\\eta < 1, 0 <\\alpha < 1/{\\eta}. . By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.

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.

Toward a High Performance Tile Divide and Conquer Algorithm for the Dense Symmetric Eigenvalue Problem

KAUST Repository

Haidar, Azzam

2012-01-01

Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel\\'s Math Kernel Library.

A scheme for the evaluation of dominant time-eigenvalues of a nuclear reactor

International Nuclear Information System (INIS)

Modak, R.S.; Gupta, Anurag

2007-01-01

This paper presents a scheme to obtain the fundamental and few dominant solutions of the prompt time eigenvalue problem (referred to as α-eigenvalue problem) for a nuclear reactor using multi-group neutron diffusion theory. The scheme is based on the use of an algorithm called Orthomin(1). This algorithm was originally proposed by Suetomi and Sekimoto [Suetomi, E., Sekimoto, H., 1991. Conjugate gradient like methods and their application to eigenvalue problems for neutron diffusion equations. Ann. Nucl. Energy 18 (4), 205-227] to obtain the fundamental K-eigenvalue (K-effective) of nuclear reactors. Recently, it has been shown that the algorithm can be used to obtain the further dominant K-modes also. Since α-eigenvalue problem is usually more difficult to solve than the K-eigenvalue problem, an attempt has been made here to use Orthomin(1) for its solution. Numerical results are given for realistic 3-D test case

Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

Science.gov (United States)

Huang, Tsung-Ming; Lin, Wen-Wei; Tian, Heng; Chen, Guan-Hua

2018-03-01

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Least Squares Problems with Absolute Quadratic Constraints

Directory of Open Access Journals (Sweden)

R. Schöne

2012-01-01

Full Text Available This paper analyzes linear least squares problems with absolute quadratic constraints. We develop a generalized theory following Bookstein's conic-fitting and Fitzgibbon's direct ellipse-specific fitting. Under simple preconditions, it can be shown that a minimum always exists and can be determined by a generalized eigenvalue problem. This problem is numerically reduced to an eigenvalue problem by multiplications of Givens' rotations. Finally, four applications of this approach are presented.

On Selberg's small eigenvalue conjecture and residual eigenvalues

DEFF Research Database (Denmark)

Risager, Morten S.

2011-01-01

We show that Selberg’s eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbat...

Some algorithms for the solution of the symmetric eigenvalue problem on a multiprocessor electronic computer

International Nuclear Information System (INIS)

Molchanov, I.N.; Khimich, A.N.

1984-01-01

This article shows how a reflection method can be used to find the eigenvalues of a matrix by transforming the matrix to tridiagonal form. The method of conjugate gradients is used to find the smallest eigenvalue and the corresponding eigenvector of symmetric positive-definite band matrices. Topics considered include the computational scheme of the reflection method, the organization of parallel calculations by the reflection method, the computational scheme of the conjugate gradient method, the organization of parallel calculations by the conjugate gradient method, and the effectiveness of parallel algorithms. It is concluded that it is possible to increase the overall effectiveness of the multiprocessor electronic computers by either letting the newly available processors of a new problem operate in the multiprocessor mode, or by improving the coefficient of uniform partition of the original information

Multigrid method applied to the solution of an elliptic, generalized eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Alchalabi, R.M. [BOC Group, Murray Hill, NJ (United States); Turinsky, P.J. [North Carolina State Univ., Raleigh, NC (United States)

1996-12-31

The work presented in this paper is concerned with the development of an efficient MG algorithm for the solution of an elliptic, generalized eigenvalue problem. The application is specifically applied to the multigroup neutron diffusion equation which is discretized by utilizing the Nodal Expansion Method (NEM). The underlying relaxation method is the Power Method, also known as the (Outer-Inner Method). The inner iterations are completed using Multi-color Line SOR, and the outer iterations are accelerated using Chebyshev Semi-iterative Method. Furthermore, the MG algorithm utilizes the consistent homogenization concept to construct the restriction operator, and a form function as a prolongation operator. The MG algorithm was integrated into the reactor neutronic analysis code NESTLE, and numerical results were obtained from solving production type benchmark problems.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

Energy Technology Data Exchange (ETDEWEB)

Vecharynski, Eugene [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Brabec, Jiri [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Shao, Meiyue [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division; Govind, Niranjan [Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Lab.; Yang, Chao [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division

2017-12-01

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.

A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices

Directory of Open Access Journals (Sweden)

Wang Xiaoxiao

2018-04-01

Full Text Available A set in the complex plane which involves n parameters in [0, 1] is given to localize all eigenvalues different from 1 for stochastic matrices. As an application of this set, an upper bound for the moduli of the subdominant eigenvalues of a stochastic matrix is obtained. Lastly, we fix n parameters in [0, 1] to give a new set including all eigenvalues different from 1, which is tighter than those provided by Shen et al. (Linear Algebra Appl. 447 (2014 74-87 and Li et al. (Linear and Multilinear Algebra 63(11 (2015 2159-2170 for estimating the moduli of subdominant eigenvalues.

Cessna Citation X Business Aircraft Eigenvalue Stability – Part2: Flight Envelope Analysis

Directory of Open Access Journals (Sweden)

Yamina BOUGHARI

2017-12-01

Full Text Available Civil aircraft flight control clearance is a time consuming, thus an expensive process in the aerospace industry. This process has to be investigated and proved to be safe for thousands of combinations in terms of speeds, altitudes, gross weights, Xcg / weight configurations and angles of attack. Even in this case, a worst-case condition that could lead to a critical situation might be missed. To address this problem, models that are able to describe an aircraft’s dynamics by taking into account all uncertainties over a region within a flight envelope have been developed using Linear Fractional Representation. In order to investigate the Cessna Citation X aircraft Eigenvalue Stability envelope, the Linear Fractional Representation models are implemented using the speeds and the altitudes as varying parameters. In this paper Part 2, the aircraft longitudinal eigenvalue stability is analyzed in a continuous range of flight envelope with varying parameter of True airspeed and altitude, instead of a single point, like classical methods. This is known as the aeroelastic stability envelope, required for civil aircraft certification as given by the Circular Advisory “Aeroelastic Stability Substantiation of Transport Category Airplanes AC No: 25.629-18”. In this new methodology the analysis is performed in time domain based on Lyapunov stability and solved by convex optimization algorithms by using the linear matrix inequalities to evaluate the eigenvalue stability, which is reduced to search for the negative eigenvalues in a region of flight envelope. It can also be used to study the stability of a system during an arbitrary motion from one point to another in the flight envelope. A whole aircraft analysis results’ for its entire envelope are presented in the form of graphs, thus offering good readability, and making them easily exploitable.

Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem

Directory of Open Access Journals (Sweden)

Xuqing Zhang

2013-01-01

Full Text Available This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001, a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.

An adjoint-based scheme for eigenvalue error improvement

International Nuclear Information System (INIS)

Merton, S.R.; Smedley-Stevenson, R.P.; Pain, C.C.; El-Sheikh, A.H.; Buchan, A.G.

2011-01-01

A scheme for improving the accuracy and reducing the error in eigenvalue calculations is presented. Using a rst order Taylor series expansion of both the eigenvalue solution and the residual of the governing equation, an approximation to the error in the eigenvalue is derived. This is done using a convolution of the equation residual and adjoint solution, which is calculated in-line with the primal solution. A defect correction on the solution is then performed in which the approximation to the error is used to apply a correction to the eigenvalue. The method is shown to dramatically improve convergence of the eigenvalue. The equation for the eigenvalue is shown to simplify when certain normalizations are applied to the eigenvector. Two such normalizations are considered; the rst of these is a fission-source type of normalisation and the second is an eigenvector normalisation. Results are demonstrated on a number of demanding elliptic problems using continuous Galerkin weighted nite elements. Moreover, the correction scheme may also be applied to hyperbolic problems and arbitrary discretization. This is not limited to spatial corrections and may be used throughout the phase space of the discrete equation. The applied correction not only improves fidelity of the calculation, it allows assessment of the reliability of numerical schemes to be made and could be used to guide mesh adaption algorithms or to automate mesh generation schemes. (author)

The BR eigenvalue algorithm

Energy Technology Data Exchange (ETDEWEB)

Geist, G.A. [Oak Ridge National Lab., TN (United States). Computer Science and Mathematics Div.; Howell, G.W. [Florida Inst. of Tech., Melbourne, FL (United States). Dept. of Applied Mathematics; Watkins, D.S. [Washington State Univ., Pullman, WA (United States). Dept. of Pure and Applied Mathematics

1997-11-01

The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.

Semidefinite linear complementarity problems

International Nuclear Information System (INIS)

Eckhardt, U.

1978-04-01

Semidefinite linear complementarity problems arise by discretization of variational inequalities describing e.g. elastic contact problems, free boundary value problems etc. In the present paper linear complementarity problems are introduced and the theory as well as the numerical treatment of them are described. In the special case of semidefinite linear complementarity problems a numerical method is presented which combines the advantages of elimination and iteration methods without suffering from their drawbacks. This new method has very attractive properties since it has a high degree of invariance with respect to the representation of the set of all feasible solutions of a linear complementarity problem by linear inequalities. By means of some practical applications the properties of the new method are demonstrated. (orig.) [de

Construction of local boundary conditions for an eigenvalue problem using micro-local analysis: application to optical waveguide problems

International Nuclear Information System (INIS)

Barucq, Helene; Bekkey, Chokri; Djellouli, Rabia

2004-01-01

We present a general procedure based on the pseudo-differential calculus for deriving artificial boundary conditions for an eigenvalue problem that characterizes the propagation of guided modes in optical waveguides. This new approach allows the construction of local conditions that (a) are independent of the frequency regime, (b) preserve the sparsity pattern of the finite element discretization, and (c) are applicable to arbitrarily shaped convex artificial boundaries. The last feature has the potential for reducing the size of the computational domain. Numerical results are presented to highlight the potential of conditions of order 1/2 and 1, for improving significantly the computational efficiency of finite element methods for the solution of optical waveguide problems

Solving eigenvalue response matrix equations with nonlinear techniques

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2014-01-01

Highlights: • High performance solvers were applied within ERMM for the first time. • Accelerated fixed-point methods were developed that reduce computational times by 2–3. • A nonlinear, Newton-based ERMM led to similar improvement and more robustness. • A 3-D, SN-based ERMM shows how ERMM can apply fine-mesh methods to full-core analysis. - Abstract: This paper presents new algorithms for use in the eigenvalue response matrix method (ERMM) for reactor eigenvalue problems. ERMM spatially decomposes a domain into independent nodes linked via boundary conditions approximated as truncated orthogonal expansions, the coefficients of which are response functions. In its simplest form, ERMM consists of a two-level eigenproblem: an outer Picard iteration updates the k-eigenvalue via balance, while the inner λ-eigenproblem imposes neutron balance between nodes. Efficient methods are developed for solving the inner λ-eigenvalue problem within the outer Picard iteration. Based on results from several diffusion and transport benchmark models, it was found that the Krylov–Schur method applied to the λ-eigenvalue problem reduces Picard solver times (excluding response generation) by a factor of 2–5. Furthermore, alternative methods, including Picard acceleration schemes, Steffensen’s method, and Newton’s method, are developed in this paper. These approaches often yield faster k-convergence and a need for fewer k-dependent response function evaluations, which is important because response generation is often the primary cost for problems using responses computed online (i.e., not from a precomputed database). Accelerated Picard iteration was found to reduce total computational times by 2–3 compared to the unaccelerated case for problems dominated by response generation. In addition, Newton’s method was found to provide nearly the same performance with improved robustness

Stratified source-sampling techniques for Monte Carlo eigenvalue analysis

International Nuclear Information System (INIS)

Mohamed, A.

1998-01-01

In 1995, at a conference on criticality safety, a special session was devoted to the Monte Carlo ''Eigenvalue of the World'' problem. Argonne presented a paper, at that session, in which the anomalies originally observed in that problem were reproduced in a much simplified model-problem configuration, and removed by a version of stratified source-sampling. In this paper, stratified source-sampling techniques are generalized and applied to three different Eigenvalue of the World configurations which take into account real-world statistical noise sources not included in the model problem, but which differ in the amount of neutronic coupling among the constituents of each configuration. It is concluded that, in Monte Carlo eigenvalue analysis of loosely-coupled arrays, the use of stratified source-sampling reduces the probability of encountering an anomalous result over that if conventional source-sampling methods are used. However, this gain in reliability is substantially less than that observed in the model-problem results

An eigenvalue localization set for tensors and its applications.

Science.gov (United States)

Zhao, Jianxing; Sang, Caili

2017-01-01

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al . (Linear Algebra Appl. 481:36-53, 2015) and Huang et al . (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of [Formula: see text]-tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al ., the advantage of our results is that, without considering the selection of nonempty proper subsets S of [Formula: see text], we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of [Formula: see text]-tensors. Finally, numerical examples are given to verify the theoretical results.

Computation of standard deviations in eigenvalue calculations

International Nuclear Information System (INIS)

Gelbard, E.M.; Prael, R.

1990-01-01

In Brissenden and Garlick (1985), the authors propose a modified Monte Carlo method for eigenvalue calculations, designed to decrease particle transport biases in the flux and eigenvalue estimates, and in corresponding estimates of standard deviations. Apparently a very similar method has been used by Soviet Monte Carlo specialists. The proposed method is based on the generation of ''superhistories'', chains of histories run in sequence without intervening renormalization of the fission source. This method appears to have some disadvantages, discussed elsewhere. Earlier numerical experiments suggest that biases in fluxes and eigenvalues are negligibly small, even for very small numbers of histories per generation. Now more recent experiments, run on the CRAY-XMP, tend to confirm these earlier conclusions. The new experiments, discussed in this paper, involve the solution of one-group 1D diffusion theory eigenvalue problems, in difference form, via Monte Carlo. Experiments covered a range of dominance ratios from ∼0.75 to ∼0.985. In all cases flux and eigenvalue biases were substantially smaller than one standard deviation. The conclusion that, in practice, the eigenvalue bias is negligible has strong theoretical support. (author)

Overlapping domain decomposition preconditioners for the generalized Davidson method for the eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Stathopoulos, A.; Fischer, C.F. [Vanderbilt Univ., Nashville, TN (United States); Saad, Y.

1994-12-31

The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.

A robust multilevel simultaneous eigenvalue solver

Science.gov (United States)

Costiner, Sorin; Taasan, Shlomo

1993-01-01

Multilevel (ML) algorithms for eigenvalue problems are often faced with several types of difficulties such as: the mixing of approximated eigenvectors by the solution process, the approximation of incomplete clusters of eigenvectors, the poor representation of solution on coarse levels, and the existence of close or equal eigenvalues. Algorithms that do not treat appropriately these difficulties usually fail, or their performance degrades when facing them. These issues motivated the development of a robust adaptive ML algorithm which treats these difficulties, for the calculation of a few eigenvectors and their corresponding eigenvalues. The main techniques used in the new algorithm include: the adaptive completion and separation of the relevant clusters on different levels, the simultaneous treatment of solutions within each cluster, and the robustness tests which monitor the algorithm's efficiency and convergence. The eigenvectors' separation efficiency is based on a new ML projection technique generalizing the Rayleigh Ritz projection, combined with a technique, the backrotations. These separation techniques, when combined with an FMG formulation, in many cases lead to algorithms of O(qN) complexity, for q eigenvectors of size N on the finest level. Previously developed ML algorithms are less focused on the mentioned difficulties. Moreover, algorithms which employ fine level separation techniques are of O(q(sub 2)N) complexity and usually do not overcome all these difficulties. Computational examples are presented where Schrodinger type eigenvalue problems in 2-D and 3-D, having equal and closely clustered eigenvalues, are solved with the efficiency of the Poisson multigrid solver. A second order approximation is obtained in O(qN) work, where the total computational work is equivalent to only a few fine level relaxations per eigenvector.

INDEFINITE COPOSITIVE MATRICES WITH EXACTLY ONE POSITIVE EIGENVALUE OR EXACTLY ONE NEGATIVE EIGENVALUE

NARCIS (Netherlands)

Jargalsaikhan, Bolor

Checking copositivity of a matrix is a co-NP-complete problem. This paper studies copositive matrices with certain spectral properties. It shows that an indefinite matrix with exactly one positive eigenvalue is copositive if and only if the matrix is nonnegative. Moreover, it shows that finding out

An eigenvalue localization set for tensors and its applications

Directory of Open Access Journals (Sweden)

Jianxing Zhao

2017-03-01

Full Text Available Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015 and Huang et al. (J. Inequal. Appl. 2016:254, 2016. As an application of this set, new bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of N = { 1 , 2 , … , n } $N=\\{1,2,\\ldots,n\\}$ , we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors. Finally, numerical examples are given to verify the theoretical results.

On the distribution of eigenvalues of certain matrix ensembles

International Nuclear Information System (INIS)

Bogomolny, E.; Bohigas, O.; Pato, M.P.

1995-01-01

Invariant random matrix ensembles with weak confinement potentials of the eigenvalues, corresponding to indeterminate moment problems, are investigated. These ensembles are characterized by the fact that the mean density of eigenvalues tends to a continuous function with increasing matrix dimension contrary to the usual cases where it grows indefinitely. It is demonstrated that the standard asymptotic formulae are not applicable in these cases and that the asymptotic distribution of eigenvalues can deviate from the classical ones. (author)

Accounting for Sampling Error in Genetic Eigenvalues Using Random Matrix Theory.

Science.gov (United States)

Sztepanacz, Jacqueline L; Blows, Mark W

2017-07-01

The distribution of genetic variance in multivariate phenotypes is characterized by the empirical spectral distribution of the eigenvalues of the genetic covariance matrix. Empirical estimates of genetic eigenvalues from random effects linear models are known to be overdispersed by sampling error, where large eigenvalues are biased upward, and small eigenvalues are biased downward. The overdispersion of the leading eigenvalues of sample covariance matrices have been demonstrated to conform to the Tracy-Widom (TW) distribution. Here we show that genetic eigenvalues estimated using restricted maximum likelihood (REML) in a multivariate random effects model with an unconstrained genetic covariance structure will also conform to the TW distribution after empirical scaling and centering. However, where estimation procedures using either REML or MCMC impose boundary constraints, the resulting genetic eigenvalues tend not be TW distributed. We show how using confidence intervals from sampling distributions of genetic eigenvalues without reference to the TW distribution is insufficient protection against mistaking sampling error as genetic variance, particularly when eigenvalues are small. By scaling such sampling distributions to the appropriate TW distribution, the critical value of the TW statistic can be used to determine if the magnitude of a genetic eigenvalue exceeds the sampling error for each eigenvalue in the spectral distribution of a given genetic covariance matrix. Copyright © 2017 by the Genetics Society of America.

The numerical analysis of eigenvalue problem solutions in the multigroup diffusion theory

International Nuclear Information System (INIS)

Woznick, Z.I.

1994-01-01

In this paper a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations is described. Usually the solution method is based on the system of inner and outer iterations. The presented matrix formalism allows us to visualize clearly, how the used matrix splitting influences the structure of the matrix in an eigenvalue problem to be solved as well as the independence between inner and outer iterations within global iterations. To keep the page limit, the present version of the paper consists only with first three of five sections given in the original paper under the same title (which will be published soon). (author). 13 refs

Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements

KAUST Repository

Bonito, Andrea; Guermond, Jean-Luc

2011-01-01

We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H-α with α ∈ (1/2, 1). The method is shown to be convergent and spectrally correct. © 2011 American Mathematical Society.

Joint density of eigenvalues in spiked multivariate models.

Science.gov (United States)

Dharmawansa, Prathapasinghe; Johnstone, Iain M

2014-01-01

The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and 'linear' statistics used in testing.

On Optimal Feedback Control for Stationary Linear Systems

International Nuclear Information System (INIS)

Russell, David L.

2010-01-01

We study linear-quadratic optimal control problems for finite dimensional stationary linear systems AX+BU=Z with output Y=CX+DU from the viewpoint of linear feedback solution. We interpret solutions in relation to system robustness with respect to disturbances Z and relate them to nonlinear matrix equations of Riccati type and eigenvalue-eigenvector problems for the corresponding Hamiltonian system. Examples are included along with an indication of extensions to continuous, i.e., infinite dimensional, systems, primarily of elliptic type.

A new localization set for generalized eigenvalues

Directory of Open Access Journals (Sweden)

Jing Gao

2017-05-01

Full Text Available Abstract A new localization set for generalized eigenvalues is obtained. It is shown that the new set is tighter than that in (Numer. Linear Algebra Appl. 16:883-898, 2009. Numerical examples are given to verify the corresponding results.

Eigenvalues of the simplified ideal MHD ballooning equation

International Nuclear Information System (INIS)

Paris, R.B.; Auby, N.; Dagazian, R.Y.

1986-01-01

The investigation of the spectrum of the simplified differential equation describing the variation of the amplitude of the ideal MHD ballooning instability along magnetic field lines constitutes a multiparameter Schroedinger eigenvalue problem. An exact eigenvalue relation for the discrete part of the spectrum is obtained in terms of the oblate spheroidal functions. The dependence of the eigenvalues lambda on the two free parameters γ 2 and μ 2 of the equation is discussed, together with certain analytical approximations in the limits of small and large γ 2 . A brief review of the principal properties of the spheroidal functions is given in an appendix

The Schrodinger Eigenvalue March

Science.gov (United States)

Tannous, C.; Langlois, J.

2011-01-01

A simple numerical method for the determination of Schrodinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric…

Symmetry Adaptation of the Rotation-Vibration Theory for Linear Molecules

Directory of Open Access Journals (Sweden)

Katy L. Chubb

2018-04-01

Full Text Available A numerical application of linear-molecule symmetry properties, described by the D ∞ h point group, is formulated in terms of lower-order symmetry groups D n h with finite n. Character tables and irreducible representation transformation matrices are presented for D n h groups with arbitrary n-values. These groups can subsequently be used in the construction of symmetry-adapted ro-vibrational basis functions for solving the Schrödinger equations of linear molecules. Their implementation into the symmetrisation procedure based on a set of “reduced” vibrational eigenvalue problems with simplified Hamiltonians is used as a practical example. It is shown how the solutions of these eigenvalue problems can also be extended to include the classification of basis-set functions using ℓ, the eigenvalue (in units of ℏ of the vibrational angular momentum operator L ^ z . This facilitates the symmetry adaptation of the basis set functions in terms of the irreducible representations of D n h . 12 C 2 H 2 is used as an example of a linear molecule of D ∞ h point group symmetry to illustrate the symmetrisation procedure of the variational nuclear motion program Theoretical ROVibrational Energies (TROVE.

Application of zero eigenvalue for solving the potential, heat, and wave equations using a sequence of special functions

Directory of Open Access Journals (Sweden)

2006-01-01

Full Text Available In the solution of boundary value problems, usually zero eigenvalue is ignored. This case also happens in calculating the eigenvalues of matrices, so that we would often like to find the nonzero solutions of the linear system A X = λ X when λ ≠ 0 . But λ = 0 implies that det A = 0 for X ≠ 0 and then the rank of matrix A is reduced at least one degree. This comment can similarly be stated for boundary value problems. In other words, if at least one of the eigens of equations related to the main problem is considered zero, then one of the solutions will be specified in advance. By using this note, first we study a class of special functions and then apply it for the potential, heat, and wave equations in spherical coordinate. In this way, some practical examples are also given.

Eigenvalue treatment of cosmological models

International Nuclear Information System (INIS)

Novello, M.; Soares, D.

1976-08-01

From the decomposition of Weyl tensor into its electric and magnetic parts, it is formulated the eigenvalue problem for cosmological models, and is used quasi-maxwellian form of Einstein's equation to propagate it along a time-like congruence. Three related theorems are presented

Complex eigenvalues for neutron transport equation with quadratically anisotropic scattering

International Nuclear Information System (INIS)

Sjoestrand, N.G.

1981-01-01

Complex eigenvalues for the monoenergetic neutron transport equation in the buckling approximation have been calculated for various combinations of linearly and quadratically anisotropic scattering. The results are discussed in terms of the time-dependent case. Tables are given of complex bucklings for real decay constants and of complex decay constants for real bucklings. The results fit nicely into the pattern of real and purely imaginary eigenvalues obtained earlier. (author)

Efficient methods for time-absorption (α) eigenvalue calculations

International Nuclear Information System (INIS)

Hill, T.R.

1983-01-01

The time-absorption eigenvalue (α) calculation is one of the options found in most discrete-ordinates transport codes. Several methods have been developed at Los Alamos to improve the efficiency of this calculation. Two procedures, based on coarse-mesh rebalance, to accelerate the α eigenvalue search are derived. A hybrid scheme to automatically choose the more-effective rebalance method is described. The α rebalance scheme permits some simple modifications to the iteration strategy that eliminates many unnecessary calculations required in the standard search procedure. For several fast supercritical test problems, these methods resulted in convergence with one-fifth the number of iterations required for the conventional eigenvalue search procedure

The Schroedinger equation as a singular perturbation problem

International Nuclear Information System (INIS)

Jager, E.M. de; Kuepper, T.

1978-01-01

Comparisons are made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problem connected with the one dimensional Schroedinger equation in Hilbert space. The difference of the eigenvalues is estimated by applying Weyl's monotonicity principle and the minimum maximum principle. The difference of the eigenfunctions is estimated in L 2 norm and in maximum norm obtained by using simple tools from operator theory in Hilbert spaces. An application concerning perturbations of the Planck ideal linear oscillator is given. (author)

Evaluation of Eigenvalue Routines for Large Scale Applications

Directory of Open Access Journals (Sweden)

V.A. Tischler

1994-01-01

Full Text Available The NASA structural analysis (NASTRAN∗ program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this article is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the block Lanczos aLgorithm. This eigenvalue routine is now availabLe in several mathematics libraries as well as in severaL commerciaL versions of NASTRAN. In addition, the eRA Y library maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The block Lanczos algorithm was found to be extremely efficient, particularly for very large problems.

Depletion GPT-free sensitivity analysis for reactor eigenvalue problems

International Nuclear Information System (INIS)

Kennedy, C.; Abdel-Khalik, H.

2013-01-01

This manuscript introduces a novel approach to solving depletion perturbation theory problems without the need to set up or solve the generalized perturbation theory (GPT) equations. The approach, hereinafter denoted generalized perturbation theory free (GPT-Free), constructs a reduced order model (ROM) using methods based in perturbation theory and computes response sensitivity profiles in a manner that is independent of the number or type of responses, allowing for an efficient computation of sensitivities when many responses are required. Moreover, the reduction error from using the ROM is quantified in the GPT-Free approach by means of a Wilks' order statistics error metric denoted the K-metric. Traditional GPT has been recognized as the most computationally efficient approach for performing sensitivity analyses of models with many input parameters, e.g. when forward sensitivity analyses are computationally intractable. However, most neutronics codes that can solve the fundamental (homogenous) adjoint eigenvalue problem do not have GPT capabilities unless envisioned during code development. The GPT-Free approach addresses this limitation by requiring only the ability to compute the fundamental adjoint. This manuscript demonstrates the GPT-Free approach for depletion reactor calculations performed in SCALE6 using the 7x7 UAM assembly model. A ROM is developed for the assembly over a time horizon of 990 days. The approach both calculates the reduction error over the lifetime of the simulation using the K-metric and benchmarks the obtained sensitivities using sample calculations. (authors)

A numerical study of the eigenvalues in the neutron diffusion theory

International Nuclear Information System (INIS)

Lima Bezerra, J. de.

1982-12-01

A systematic numerical study for the eigenvalue problem in one dimension was carried out. A computer code RED2G was developed to obtain and to discuss a number of numerical solutions concerning eigenvalues problems originating from the discretization of the two groups neutron diffusion equation in one dimension and steady state. The problem of eigenvalues was created from the discretization by the method of finite differences. The solutions were obtained by four different iterative methods, i.e. Power, Wielandt-1, Wielandt-2 and accelerated Power with the Chebyshev polinomials. The numerical results given by the solution of the two test-problems indicate that the RED2G code is fast and efficient in these calculations and the Wielandt-2 method has been found to be the best both in respect of rapidity of calculations as well as programation effort required. (E.G.) [pt

Maximal imaginery eigenvalues in optimal systems

Directory of Open Access Journals (Sweden)

David Di Ruscio

1991-07-01

Full Text Available In this note we present equations that uniquely determine the maximum possible imaginary value of the closed loop eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen, provided a real symmetric solution of the algebraic Riccati equation exists. In addition, the corresponding state weight matrix and the solution to the algebraic Riccati equation are derived for a class of linear systems. A fundamental lemma for the existence of a real symmetric solution to the algebraic Riccati equation is derived for this class of linear systems.

Classical scattering theory of waves from the view point of an eigenvalue problem and application to target identification

International Nuclear Information System (INIS)

Bottcher, C.; Strayer, M.R.; Werby, M.F.

1993-01-01

The Helmholtz-Poincare Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWE's. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can by obtained in matrix form be expanding all relevant terms in partial wave expansions, including a biorthogonal expansion of the Green function. However some freedom of choice in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways to long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermition operator. The methodology will be explained in detail and examples will be presented

Eigenvalues of PT-symmetric oscillators with polynomial potentials

International Nuclear Information System (INIS)

Shin, Kwang C

2005-01-01

We study the eigenvalue problem -u''(z) - [(iz) m + P m-1 (iz)]u(z) λu(z) with the boundary condition that u(z) decays to zero as z tends to infinity along the rays arg z = -π/2 ± 2π/(m+2) in the complex plane, where P m-1 (z) = a 1 z m-1 + a 2 z m-2 + . . . + a m-1 z is a polynomial and integers m ≥ 3. We provide an asymptotic expansion of the eigenvalues λ n as n → +∞, and prove that for each real polynomial P m-1 , the eigenvalues are all real and positive, with only finitely many exceptions

Mathematical methods linear algebra normed spaces distributions integration

CERN Document Server

Korevaar, Jacob

1968-01-01

Mathematical Methods, Volume I: Linear Algebra, Normed Spaces, Distributions, Integration focuses on advanced mathematical tools used in applications and the basic concepts of algebra, normed spaces, integration, and distributions.The publication first offers information on algebraic theory of vector spaces and introduction to functional analysis. Discussions focus on linear transformations and functionals, rectangular matrices, systems of linear equations, eigenvalue problems, use of eigenvectors and generalized eigenvectors in the representation of linear operators, metric and normed vector

On a minimization of the eigenvalues of Schroedinger operator relatively domains

International Nuclear Information System (INIS)

Gasymov, Yu.S.; Niftiev, A.A.

2001-01-01

Minimization of the eigenvalues plays an important role in the operators spectral theory. The problem on the minimization of the eigenvalues of the Schroedinger operator by areas is considered in this work. The algorithm, analogous to the conditional gradient method, is proposed for the numerical solution of this problem in the common case. The result is generalized for the case of the positively determined completely continuous operator [ru

An algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group

International Nuclear Information System (INIS)

Wang, S.J.

1993-04-01

An algebraic approach to the inverse eigenvalue problem for a quantum system with a dynamical group is formulated for the first time. One dimensional problem is treated explicitly in detail for both the finite dimensional and infinite dimensional Hilbert spaces. For the finite dimensional Hilbert space, the su(2) algebraic representation is used; while for the infinite dimensional Hilbert space, the Heisenberg-Weyl algebraic representation is employed. Fourier expansion technique is generalized to the generator space, which is suitable for analysis of irregular spectra. The polynormial operator basis is also used for complement, which is appropriate for analysis of some simple Hamiltonians. The proposed new approach is applied to solve the classical inverse Sturn-Liouville problem and to study the problems of quantum regular and irregular spectra. (orig.)

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

An algebraic substructuring using multiple shifts for eigenvalue computations

International Nuclear Information System (INIS)

Ko, Jin Hwan; Jung, Sung Nam; Byun, Do Young; Bai, Zhaojun

2008-01-01

Algebraic substructuring (AS) is a state-of-the-art method in eigenvalue computations, especially for large-sized problems, but originally it was designed to calculate only the smallest eigenvalues. Recently, an updated version of AS has been introduced to calculate the interior eigenvalues over a specified range by using a shift concept that is referred to as the shifted AS. In this work, we propose a combined method of both AS and the shifted AS by using multiple shifts for solving a considerable number of eigensolutions in a large-sized problem, which is an emerging computational issue of noise or vibration analysis in vehicle design. In addition, we investigated the accuracy of the shifted AS by presenting an error criterion. The proposed method has been applied to the FE model of an automobile body. The combined method yielded a higher efficiency without loss of accuracy in comparison to the original AS

Eigenvalues of Casimir operators for the general linear, the special linear, and the orthosymplectic Lie superalgebras

International Nuclear Information System (INIS)

Scheunert, M.

1982-10-01

The generators of the algebras under consideration can be written in a canonical two-index form and hence the associated standard seuqence of Casimir elements can be constructed. Following the classical approach by Perelomov and Popov, we obtain the eigenvalues of these Casimir elements in an arbitrary highest weight module by calculating the corresponding generating functions. (orig.)

Eigenvalues of Words in Two Positive Definite Letters

OpenAIRE

Hillar, Christopher J; Johnson, Charles R

2005-01-01

The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical physics. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do. In the process, a fundamental question about solvability of symmetric word equations is encoun...

Eigenvalues of the Transferences of Gaussian Optical Systems

Directory of Open Access Journals (Sweden)

W.F. Harris

2005-12-01

Full Text Available The  problem  of  how  to  define  an  average eye leads to the question of what eigenvalues are  possible  for  ray  transferences.  This  paper examines the set of possible eigenvalues in the simplest possible case, that of optical systems consisting  of  elements  that  are  stigmatic  and centred on a common axis.

Solving the RPA eigenvalue equation in real-space

CERN Document Server

Muta, A; Hashimoto, Y; Yabana, K

2002-01-01

We present a computational method to solve the RPA eigenvalue equation employing a uniform grid representation in three-dimensional Cartesian coordinates. The conjugate gradient method is used for this purpose as an interactive method for a generalized eigenvalue problem. No construction of unoccupied orbitals is required in the procedure. We expect this method to be useful for systems lacking spatial symmetry to calculate accurate eigenvalues and transition matrix elements of a few low-lying excitations. Some applications are presented to demonstrate the feasibility of the method, considering the simplified mean-field model as an example of a nuclear physics system and the electronic excitations in molecules with time-dependent density functional theory as an example of an electronic system. (author)

Effective Perron-Frobenius eigenvalue for a correlated random map

Science.gov (United States)

Pool, Roman R.; Cáceres, Manuel O.

2010-09-01

We investigate the evolution of random positive linear maps with various type of disorder by analytic perturbation and direct simulation. Our theoretical result indicates that the statistics of a random linear map can be successfully described for long time by the mean-value vector state. The growth rate can be characterized by an effective Perron-Frobenius eigenvalue that strongly depends on the type of correlation between the elements of the projection matrix. We apply this approach to an age-structured population dynamics model. We show that the asymptotic mean-value vector state characterizes the population growth rate when the age-structured model has random vital parameters. In this case our approach reveals the nontrivial dependence of the effective growth rate with cross correlations. The problem was reduced to the calculation of the smallest positive root of a secular polynomial, which can be obtained by perturbations in terms of Green’s function diagrammatic technique built with noncommutative cumulants for arbitrary n -point correlations.

Relation of deformed nonlinear algebras with linear ones

International Nuclear Information System (INIS)

Nowicki, A; Tkachuk, V M

2014-01-01

The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator. (paper)

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

International Nuclear Information System (INIS)

Conca, Carlos; Mahadevan, Rajesh; Sanz, Leon

2009-01-01

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhaeuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101-117, 1996). Although, the problem in one dimension has a solution (cf. Krein in AMS Transl. Ser. 2(1):163-187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185-220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328-337, 1983)

The linearized pressure Poisson equation for global instability analysis of incompressible flows

Science.gov (United States)

Theofilis, Vassilis

2017-12-01

The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.

A new S-type eigenvalue inclusion set for tensors and its applications.

Science.gov (United States)

Huang, Zheng-Ge; Wang, Li-Gong; Xu, Zhong; Cui, Jing-Jing

2016-01-01

In this paper, a new S -type eigenvalue localization set for a tensor is derived by dividing [Formula: see text] into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H -eigenvalue of strong M -tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

Coarse-mesh rebalancing acceleration for eigenvalue problems

International Nuclear Information System (INIS)

Asaoka, T.; Nakahara, Y.; Miyasaka, S.

1974-01-01

The coarse-mesh rebalance method is adopted for Monte Carlo schemes for aiming at accelerating the convergence of a source iteration process. At every completion of the Monte Carlo game for one batch of neutron histories, the scaling factor for the neutron flux is calculated to achieve the neutron balance in each coarse-mesh zone into which the total system is divided. This rebalance factor is multiplied to the weight of each fission source neutron in the coarse-mesh zone for playing the next Monte Carlo game. The numerical examples have shown that the coarse-mesh rebalance Monte Carlo calculation gives a good estimate of the eigenvalue already after several batches with a negligible extra computer time compared to the standard Monte Carlo. 5 references. (U.S.)

Sensitivity analysis for large-scale problems

Science.gov (United States)

Noor, Ahmed K.; Whitworth, Sandra L.

1987-01-01

The development of efficient techniques for calculating sensitivity derivatives is studied. The objective is to present a computational procedure for calculating sensitivity derivatives as part of performing structural reanalysis for large-scale problems. The scope is limited to framed type structures. Both linear static analysis and free-vibration eigenvalue problems are considered.

A Bootstrap Approach to Eigenvalue Correction

NARCIS (Netherlands)

Hendrikse, A.J.; Spreeuwers, Lieuwe Jan; Veldhuis, Raymond N.J.

2009-01-01

Eigenvalue analysis is an important aspect in many data modeling methods. Unfortunately, the eigenvalues of the sample covariance matrix (sample eigenvalues) are biased estimates of the eigenvalues of the covariance matrix of the data generating process (population eigenvalues). We present a new

Numerical computations of interior transmission eigenvalues for scattering objects with cavities

International Nuclear Information System (INIS)

Peters, Stefan; Kleefeld, Andreas

2016-01-01

In this article we extend the inside-outside duality for acoustic transmission eigenvalue problems by allowing scattering objects that may contain cavities. In this context we provide the functional analytical framework necessary to transfer the techniques that have been used in Kirsch and Lechleiter (2013 Inverse Problems, 29 104011) to derive the inside-outside duality. Additionally, extensive numerical results are presented to show that we are able to successfully detect interior transmission eigenvalues with the inside-outside duality approach for a variety of obstacles with and without cavities in three dimensions. In this context, we also discuss the advantages and disadvantages of the inside-outside duality approach from a numerical point of view. Furthermore we derive the integral equations necessary to extend the algorithm in Kleefeld (2013 Inverse Problems, 29 104012) to compute highly accurate interior transmission eigenvalues for scattering objects with cavities, which we will then use as reference values to examine the accuracy of the inside-outside duality algorithm. (paper)

A theoretical study on a convergence problem of nodal methods

Energy Technology Data Exchange (ETDEWEB)

Shaohong, Z.; Ziyong, L. [Shanghai Jiao Tong Univ., 1954 Hua Shan Road, Shanghai, 200030 (China); Chao, Y. A. [Westinghouse Electric Company, P. O. Box 355, Pittsburgh, PA 15230-0355 (United States)

2006-07-01

The effectiveness of modern nodal methods is largely due to its use of the information from the analytical flux solution inside a homogeneous node. As a result, the nodal coupling coefficients depend explicitly or implicitly on the evolving Eigen-value of a problem during its solution iteration process. This poses an inherently non-linear matrix Eigen-value iteration problem. This paper points out analytically that, whenever the half wave length of an evolving node interior analytic solution becomes smaller than the size of that node, this non-linear iteration problem can become inherently unstable and theoretically can always be non-convergent or converge to higher order harmonics. This phenomenon is confirmed, demonstrated and analyzed via the simplest 1-D problem solved by the simplest analytic nodal method, the Analytic Coarse Mesh Finite Difference (ACMFD, [1]) method. (authors)

Application of the Laplace transform method for the albedo boundary conditions in multigroup neutron diffusion eigenvalue problems in slab geometry

International Nuclear Information System (INIS)

Petersen, Claudio Zen; Vilhena, Marco T.; Barros, Ricardo C.

2009-01-01

In this paper the application of the Laplace transform method is described in order to determine the energy-dependent albedo matrix that is used in the boundary conditions multigroup neutron diffusion eigenvalue problems in slab geometry for nuclear reactor global calculations. In slab geometry, the diffusion albedo substitutes without approximation the baffle-reflector system around the active domain. Numerical results to typical test problems are shown to illustrate the accuracy and the efficiency of the Chebysheff acceleration scheme. (orig.)

Ground eigenvalue and eigenfunction of a spin-weighted spheroidal wave equation in low frequencies

Institute of Scientific and Technical Information of China (English)

Tang Wen-Lin; Tian Gui-Hua

2011-01-01

Spin-weighted spheroidal wave functions play an important role in the study of the linear stability of rotating Kerr black holes and are studied by the perturbation method in supersymmetric quantum mechanics. Their analytic ground eigenvalues and eigenfunctions are obtained by means of a series in low frequency. The ground eigenvalue and eigenfunction for small complex frequencies are numerically determined.

Smallest eigenvalue distribution of the fixed-trace Laguerre beta-ensemble

International Nuclear Information System (INIS)

Chen Yang; Liu Dangzheng; Zhou Dasheng

2010-01-01

In this paper we study the entanglement of the reduced density matrix of a bipartite quantum system in a random pure state. It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed-trace Laguerre ensemble of N x N random matrices. We showed that for finite N the smallest eigenvalue distribution may be expressed in terms of Jack polynomials. Furthermore, based on the exact results, we found a limiting distribution when the smallest eigenvalue is suitably scaled with N followed by a large N limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed-trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large N limit. Consequently, we have solved an open problem: the determination of the smallest eigenvalue distribution of the reduced density matrix-obtained by tracing out the environmental degrees of freedom-for a bipartite quantum system of unequal dimensions.

A comparative study of history-based versus vectorized Monte Carlo methods in the GPU/CUDA environment for a simple neutron eigenvalue problem

International Nuclear Information System (INIS)

Liu, T.; Du, X.; Ji, W.; Xu, G.; Brown, F.B.

2013-01-01

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed. (authors)

A comparative study of history-based versus vectorized Monte Carlo methods in the GPU/CUDA environment for a simple neutron eigenvalue problem

Science.gov (United States)

Liu, Tianyu; Du, Xining; Ji, Wei; Xu, X. George; Brown, Forrest B.

2014-06-01

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.

Improved simple graphical solution for the eigenvalues of the finite square well potential

International Nuclear Information System (INIS)

Burge, E.J.

1985-01-01

The three principal graphical methods for obtaining the energy eigenvalues of the finite square well potential are presented. The forms of the wavefunctions within the well, and the corresponding linear probability densities, are derived directly from the method. A simple extension of the method allows the energy level spectrum to be obtained directly on a linear energy scale. The variations of the energy eigenvalues with well depth and width are separately and jointly displayed, and explicit corresponding functional relationships are derived. Two universal graphs are deduced which allow the rapid appreciation and calculation of the dependence of the energy levels on the depth and width of the well and on the mass of the particle. (author)

Computing with linear equations and matrices

International Nuclear Information System (INIS)

Churchhouse, R.F.

1983-01-01

Systems of linear equations and matrices arise in many disciplines. The equations may accurately represent conditions satisfied by a system or, more likely, provide an approximation to a more complex system of non-linear or differential equations. The system may involve a few or many thousand unknowns and each individual equation may involve few or many of them. Over the past 50 years a vast literature on methods for solving systems of linear equations and the associated problems of finding the inverse or eigenvalues of a matrix has been produced. These lectures cover those methods which have been found to be most useful for dealing with such types of problem. References are given where appropriate and attention is drawn to the possibility of improved methods for use on vector and parallel processors. (orig.)

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

Science.gov (United States)

Tischler, V. A.; Venkayya, Vipperla B.

1993-01-01

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

Generalized Eigenvalues for pairs on heritian matrices

Science.gov (United States)

Rublein, George

1988-01-01

A study was made of certain special cases of a generalized eigenvalue problem. Let A and B be nxn matrics. One may construct a certain polynomial, P(A,B, lambda) which specializes to the characteristic polynomial of B when A equals I. In particular, when B is hermitian, that characteristic polynomial, P(I,B, lambda) has real roots, and one can ask: are the roots of P(A,B, lambda) real when B is hermitian. We consider the case where A is positive definite and show that when N equals 3, the roots are indeed real. The basic tools needed in the proof are Shur's theorem on majorization for eigenvalues of hermitian matrices and the interlacing theorem for the eigenvalues of a positive definite hermitian matrix and one of its principal (n-1)x(n-1) minors. The method of proof first reduces the general problem to one where the diagonal of B has a certain structure: either diag (B) = diag (1,1,1) or diag (1,1,-1), or else the 2 x 2 principal minors of B are all 1. According as B has one of these three structures, we use an appropriate method to replace A by a positive diagonal matrix. Since it can be easily verified that P(D,B, lambda) has real roots, the result follows. For other configurations of B, a scaling and a continuity argument are used to prove the result in general.

Automated finder for the critical condition on the linear stability of fluid motions

International Nuclear Information System (INIS)

Fujimura, Kaoru

1990-03-01

An automated finder routine for the critical condition on the linear stability of fluid motions is proposed. The Newton-Raphson method was utilized for an iteration to solve nonlinear eigenvalue problems appeared in the analysis. The routine was applied to linear stability problem of a free convection between vertical parallel plates with different non-uniform temperatures as well as a plane Poiseuille flow. An efficiency of the finder routine is demonstrated for several parameter sets, numerically. (author)

Topological derivatives of eigenvalues and neural networks in identification of imperfections

International Nuclear Information System (INIS)

Grzanek, M; Nowakowski, A; Sokolowski, J

2008-01-01

Numerical method for identification of imperfections is devised for elliptic spectral problems. The neural networks are employed for numerical solution. The topological derivatives of eigenvalues are used in the learning procedure of the neural networks. The topological derivatives of eigenvalues are determined by the methods of asymptotic analysis in singularly perturbed geometrical domains. The convergence of the numerical method in a probabilistic setting is analysed. The method is presented for the identification of small singular perturbations of the boundary of geometrical domain, however the framework is general and can be used for numerical solutions of inverse problems in the presence of small imperfections in the interior of the domain. Some numerical results are given for elliptic spectral problem in two spatial dimensions.

The regular indefinite linear-quadratic problem with linear endpoint constraints

NARCIS (Netherlands)

Soethoudt, J.M.; Trentelman, H.L.

1989-01-01

This paper deals with the infinite horizon linear-quadratic problem with indefinite cost. Given a linear system, a quadratic cost functional and a subspace of the state space, we consider the problem of minimizing the cost functional over all inputs for which the state trajectory converges to that

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

Science.gov (United States)

Kleefeld, Andreas; Pieronek, Lukas

2018-03-01

We analyze the method of fundamental solutions (MFS) in two different versions with focus on the computation of approximate acoustic interior transmission eigenvalues in 2D for homogeneous media. Our approach is mesh- and integration free, but suffers in general from the ill-conditioning effects of the discretized eigenoperator, which we could then successfully balance using an approved stabilization scheme. Our numerical examples cover many of the common scattering objects and prove to be very competitive in accuracy with the standard methods for PDE-related eigenvalue problems. We finally give an approximation analysis for our framework and provide error estimates, which bound interior transmission eigenvalue deviations in terms of some generalized MFS output.

Menu-Driven Solver Of Linear-Programming Problems

Science.gov (United States)

Viterna, L. A.; Ferencz, D.

1992-01-01

Program assists inexperienced user in formulating linear-programming problems. A Linear Program Solver (ALPS) computer program is full-featured LP analysis program. Solves plain linear-programming problems as well as more-complicated mixed-integer and pure-integer programs. Also contains efficient technique for solution of purely binary linear-programming problems. Written entirely in IBM's APL2/PC software, Version 1.01. Packed program contains licensed material, property of IBM (copyright 1988, all rights reserved).

Collaborative spectrum sensing based on the ratio between largest eigenvalue and Geometric mean of eigenvalues

KAUST Repository

Shakir, Muhammad

2011-12-01

In this paper, we introduce a new detector referred to as Geometric mean detector (GEMD) which is based on the ratio of the largest eigenvalue to the Geometric mean of the eigenvalues for collaborative spectrum sensing. The decision threshold has been derived by employing Gaussian approximation approach. In this approach, the two random variables, i.e. The largest eigenvalue and the Geometric mean of the eigenvalues are considered as independent Gaussian random variables such that their cumulative distribution functions (CDFs) are approximated by a univariate Gaussian distribution function for any number of cooperating secondary users and received samples. The approximation approach is based on the calculation of exact analytical moments of the largest eigenvalue and the Geometric mean of the eigenvalues of the received covariance matrix. The decision threshold has been calculated by exploiting the CDF of the ratio of two Gaussian distributed random variables. In this context, we exchange the analytical moments of the two random variables with the moments of the Gaussian distribution function. The performance of the detector is compared with the performance of the energy detector and eigenvalue ratio detector. Analytical and simulation results show that our newly proposed detector yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, our results based on proposed approximation approach are in perfect agreement with the empirical results. © 2011 IEEE.

A nonlinear eigenvalue problem for self-similar spherical force-free magnetic fields

Energy Technology Data Exchange (ETDEWEB)

Lerche, I. [Institut für Geowissenschaften, Naturwissenschaftliche Fakultät III, Martin-Luther Universität, D-06099 Halle (Germany); Low, B. C. [High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colorado 80307 (United States)

2014-10-15

An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θB{sub φ}=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θB{sub φ}=aA|A|{sup 1/n}, n a positive constant, admits separable fields with A=(A{sub n}(θ))/(r{sup n}) , posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and A{sub n}(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=(H(θ,φ))/(r{sup n+2}) promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4

Two-group k-eigenvalue benchmark calculations for planar geometry transport in a binary stochastic medium

International Nuclear Information System (INIS)

Davis, I.M.; Palmer, T.S.

2005-01-01

Benchmark calculations are performed for neutron transport in a two material (binary) stochastic multiplying medium. Spatial, angular, and energy dependence are included. The problem considered is based on a fuel assembly of a common pressurized water reactor. The mean chord length through the assembly is determined and used as the planar geometry system length. According to assumed or calculated material distributions, this system length is populated with alternating fuel and moderator segments of random size. Neutron flux distributions are numerically computed using a discretized form of the Boltzmann transport equation employing diffusion synthetic acceleration. Average quantities (group fluxes and k-eigenvalue) and variances are calculated from an ensemble of realizations of the mixing statistics. The effects of varying two parameters in the fuel, two different boundary conditions, and three different sets of mixing statistics are assessed. A probability distribution function (PDF) of the k-eigenvalue is generated and compared with previous research. Atomic mix solutions are compared with these benchmark ensemble average flux and k-eigenvalue solutions. Mixing statistics with large standard deviations give the most widely varying ensemble solutions of the flux and k-eigenvalue. The shape of the k-eigenvalue PDF qualitatively agrees with previous work. Its overall shape is independent of variations in fuel cross-sections for the problems considered, but its width is impacted by these variations. Statistical distributions with smaller standard deviations alter the shape of this PDF toward a normal distribution. The atomic mix approximation yields large over-predictions of the ensemble average k-eigenvalue and under-predictions of the flux. Qualitatively correct flux shapes are obtained in some cases. These benchmark calculations indicate that a model which includes higher statistical moments of the mixing statistics is needed for accurate predictions of binary

A spectral nodal method for eigenvalue SN transport problems in two-dimensional rectangular geometry for energy multigroup nuclear reactor global calculations

International Nuclear Information System (INIS)

Silva, Davi Jose M.; Alves Filho, Hermes; Barros, Ricardo C.

2015-01-01

A spectral nodal method is developed for multigroup x,y-geometry discrete ordinates (S N ) eigenvalue problems for nuclear reactor global calculations. This method uses the conventional multigroup SN discretized spatial balance nodal equations with two non-standard auxiliary equations: the spectral diamond (SD) auxiliary equations for the discretization nodes inside the fuel regions, and the spectral Green's function (SGF) auxiliary equations for the non-multiplying regions, such as the baffle and the reactor. This spectral nodal method is derived from the analytical general solution of the SN transverse integrated nodal equations with constant approximations for the transverse leakage terms within each discretization node. The SD and SGF auxiliary equations have parameters, which are determined to preserve the homogeneous and the particular components of these local general solutions. Therefore, we refer to the offered method as the hybrid SD-SGF-Constant Nodal (SD-SGF-CN) method. The S N discretized spatial balance equations, together with the SD and the SGF auxiliary equations form the SD-SGF-CN equations. We solve the SD-SGF-CN equations by using the one-node block inversion inner iterations (NBI), wherein the most recent estimates for the incoming group node-edge average or prescribed boundary conditions are used to evaluate the outgoing group node-edge average fluxes in the directions of the S N transport sweeps, for each estimate of the dominant eigenvalue in the conventional Power outer iterations. We show in numerical calculations that the SD-SGF-CN method is very accurate for coarse-mesh multigroup S N eigenvalue problems, even though the transverse leakage terms are approximated rather simply. (author)

p-Norm SDD tensors and eigenvalue localization

Directory of Open Access Journals (Sweden)

Qilong Liu

2016-07-01

Full Text Available Abstract We present a new class of nonsingular tensors (p-norm strictly diagonally dominant tensors, which is a subclass of strong H $\\mathcal{H}$ -tensors. As applications of the results, we give a new eigenvalue inclusion set, which is tighter than those provided by Li et al. (Linear Multilinear Algebra 64:727-736, 2016 in some case. Based on this set, we give a checkable sufficient condition for the positive (semidefiniteness of an even-order symmetric tensor.

Random eigenvalue problems revisited

Indian Academy of Sciences (India)

statistical distributions; linear stochastic systems. 1. ... dimensional multivariate Gaussian random vector with mean µ ∈ Rm and covariance ... 5, the proposed analytical methods are applied to a three degree-of-freedom system and the ...... The joint pdf ofω1 andω3 is however close to a bivariate Gaussian density function.

Algorithm 589. SICEDR: a FORTRAN subroutine for improving the accuracy of computed matrix eigenvalues

International Nuclear Information System (INIS)

Dongarra, J.J.

1982-01-01

SICEDR is a FORTRAN subroutine for improving the accuracy of a computed real eigenvalue and improving or computing the associated eigenvector. It is first used to generate information during the determination of the eigenvalues by the Schur decomposition technique. In particular, the Schur decomposition technique results in an orthogonal matrix Q and an upper quasi-triangular matrix T, such that A = QTQ/sup T/. Matrices A, Q, and T and the approximate eigenvalue, say lambda, are then used in the improvement phase. SICEDR uses an iterative method similar to iterative improvement for linear systems to improve the accuracy of lambda and improve or compute the eigenvector x in O(n 2 ) work, where n is the order of the matrix A

Two new eigenvalue localization sets for tensors and theirs applications

Directory of Open Access Journals (Sweden)

Zhao Jianxing

2017-10-01

Full Text Available A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324 and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50. As an application, a weaker checkable sufficient condition for the positive (semi-definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1, 187-198. As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.

New algorithms for the symmetric tridiagonal eigenvalue computation

Energy Technology Data Exchange (ETDEWEB)

Pan, V. [City Univ. of New York, Bronx, NY (United States)]|[International Computer Sciences Institute, Berkeley, CA (United States)

1994-12-31

The author presents new algorithms that accelerate the bisection method for the symmetric eigenvalue problem. The algorithms rely on some new techniques, which include acceleration of Newton`s iteration and can also be further applied to acceleration of some other iterative processes, in particular, of iterative algorithms for approximating polynomial zeros.

On Numerical Stability in Large Scale Linear Algebraic Computations

Czech Academy of Sciences Publication Activity Database

Strakoš, Zdeněk; Liesen, J.

2005-01-01

Roč. 85, č. 5 (2005), s. 307-325 ISSN 0044-2267 R&D Projects: GA AV ČR 1ET400300415 Institutional research plan: CEZ:AV0Z10300504 Keywords : linear algebraic systems * eigenvalue problems * convergence * numerical stability * backward error * accuracy * Lanczos method * conjugate gradient method * GMRES method Subject RIV: BA - General Mathematics Impact factor: 0.351, year: 2005

Stochastic Linear Quadratic Optimal Control Problems

International Nuclear Information System (INIS)

Chen, S.; Yong, J.

2001-01-01

This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward-backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well

Perturbed asymptotically linear problems

OpenAIRE

Bartolo, R.; Candela, A. M.; Salvatore, A.

2012-01-01

The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which is just continuous. Also in the case when the problem has not a variational structure, suitable procedures and estimates allow us to prove that the number of distinct crtitical levels of the functional associated to the unperturbed problem is "stable" unde...

Perturbation analysis of linear control problems

International Nuclear Information System (INIS)

Petkov, Petko; Konstantinov, Mihail

2017-01-01

The paper presents a brief overview of the technique of splitting operators, proposed by the authors and intended for perturbation analysis of control problems involving unitary and orthogonal matrices. Combined with the technique of Lyapunov majorants and the implementation of the Banach and Schauder fixed point principles, it allows to obtain rigorous non-local perturbation bounds for a set of sensitivity analysis problems. Among them are the reduction of linear systems into orthogonal canonical forms, the feedback synthesis problem and pole assignment problem in particular, as well as other important problems in control theory and linear algebra. Key words: perturbation analysis, canonical forms, feedback synthesis

A Problem-Centered Approach to Canonical Matrix Forms

Science.gov (United States)

Sylvestre, Jeremy

2014-01-01

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP : Part 2

International Nuclear Information System (INIS)

Chang, Jonghwa

2014-01-01

Today, we can use a computer cluster consist of a few hundreds CPUs with reasonable budget. Such computer system enables us to do detailed modeling of reactor core. The detailed modeling will improve the safety and the economics of a nuclear reactor by eliminating un-necessary conservatism or missing consideration. To take advantage of such a cluster computer, efficient parallel algorithms must be developed. Mechanical structure analysis community has studied the domain decomposition method to solve the stress-strain equation using the finite element methods. One of the most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multi-group diffusion problem in previous study. In this study, we report the result of recent modification to handle the three-dimensional subdomain partitioning, and the sub-domain multi-group problem. Modified FETI-DP algorithm has been successfully applied for the eigenvalue problem of multi-group neutron diffusion equation. The overall CPU time is decreasing as number of sub-domains (partitions) is increasing. However, there may be a limit in decrement due to increment of the number of primal points will increase the CPU time spent by the solution of the global equation. Even distribution of computational load (criterion a) is important to achieve fast computation. The subdomain partition can be effectively performed using suitable graph theory partition package such as MeTIS

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP : Part 2

Energy Technology Data Exchange (ETDEWEB)

Chang, Jonghwa [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)

2014-10-15

Today, we can use a computer cluster consist of a few hundreds CPUs with reasonable budget. Such computer system enables us to do detailed modeling of reactor core. The detailed modeling will improve the safety and the economics of a nuclear reactor by eliminating un-necessary conservatism or missing consideration. To take advantage of such a cluster computer, efficient parallel algorithms must be developed. Mechanical structure analysis community has studied the domain decomposition method to solve the stress-strain equation using the finite element methods. One of the most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multi-group diffusion problem in previous study. In this study, we report the result of recent modification to handle the three-dimensional subdomain partitioning, and the sub-domain multi-group problem. Modified FETI-DP algorithm has been successfully applied for the eigenvalue problem of multi-group neutron diffusion equation. The overall CPU time is decreasing as number of sub-domains (partitions) is increasing. However, there may be a limit in decrement due to increment of the number of primal points will increase the CPU time spent by the solution of the global equation. Even distribution of computational load (criterion a) is important to achieve fast computation. The subdomain partition can be effectively performed using suitable graph theory partition package such as MeTIS.

Scattering of massless lumps and non-local charges in the two-dimensional classical non-linear sigma-model

International Nuclear Information System (INIS)

Luescher, M.; Pohlmeyer, K.

1977-09-01

Finite energy solutions of the field equations of the non-linear sigma-model are shown to decay asymptotically into massless lumps. By means of a linear eigenvalue problem connected with the field equations we then find an infinite set of dynamical conserved charges. They, however, do not provide sufficient information to decode the complicated scattering of lumps. (orig.) [de

Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue

Directory of Open Access Journals (Sweden)

Shu-Guang Guo

2016-05-01

Full Text Available Abstract A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013 and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013 determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.

An Approach for Solving Linear Fractional Programming Problems

OpenAIRE

Andrew Oyakhobo Odior

2012-01-01

Linear fractional programming problems are useful tools in production planning, financial and corporate planning, health care and hospital planning and as such have attracted considerable research interest. The paper presents a new approach for solving a fractional linear programming problem in which the objective function is a linear fractional function, while the constraint functions are in the form of linear inequalities. The approach adopted is based mainly upon solving the problem algebr...

Linear algebra

CERN Document Server

Berberian, Sterling K

2014-01-01

Introductory treatment covers basic theory of vector spaces and linear maps - dimension, determinants, eigenvalues, and eigenvectors - plus more advanced topics such as the study of canonical forms for matrices. 1992 edition.

Escape rate from strange sets as an eigenvalue

International Nuclear Information System (INIS)

Tel, T.

1986-06-01

A new method is applied for calculating the escape rate from chaotic repellers or semi-attractors, based on the eigenvalue problem of the master equation of discrete dynamical systems. The corresponding eigenfunction is found to be smooth along unstable directions and to be, in general, a fractal measure. Examples of one and two dimensional maps are investigated. (author)

The discontinuous finite element method for solving Eigenvalue problems of transport equations

International Nuclear Information System (INIS)

Yang, Shulin; Wang, Ruihong

2011-01-01

In this paper, the multigroup transport equations for solving the eigenvalues λ and K_e_f_f under two dimensional cylindrical coordinate are discussed. Aimed at the equations, the discretizing way combining discontinuous finite element method (DFE) with discrete ordinate method (SN) is developed, and the iterative algorithms and steps are studied. The numerical results show that the algorithms are efficient. (author)

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

Energy Technology Data Exchange (ETDEWEB)

Pieper, Andreas [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Kreutzer, Moritz [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany); Alvermann, Andreas, E-mail: alvermann@physik.uni-greifswald.de [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Galgon, Martin [Bergische Universität Wuppertal (Germany); Fehske, Holger [Ernst-Moritz-Arndt-Universität Greifswald (Germany); Hager, Georg [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany); Lang, Bruno [Bergische Universität Wuppertal (Germany); Wellein, Gerhard [Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

2016-11-15

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 10{sup 2} innermost eigenpairs of a topological insulator matrix with dimension 10{sup 9} derived from quantum physics applications.

Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type

Directory of Open Access Journals (Sweden)

Serguei I. Iakovlev

2013-01-01

Full Text Available It is shown that any \\(\\mu \\in \\mathbb{C}\\ is an infinite multiplicity eigenvalue of the Steklov smoothing operator \\(S_h\\ acting on the space \\(L^1_{loc}(\\mathbb{R}\\. For \\(\\mu \

Normal mode analysis for linear resistive magnetohydrodynamics

International Nuclear Information System (INIS)

Kerner, W.; Lerbinger, K.; Gruber, R.; Tsunematsu, T.

1984-10-01

The compressible, resistive MHD equations are linearized around an equilibrium with cylindrical symmetry and solved numerically as a complex eigenvalue problem. This normal mode code allows to solve for very small resistivity eta proportional 10 -10 . The scaling of growthrates and layer width agrees very well with analytical theory. Especially, both the influence of current and pressure on the instabilities is studied in detail; the effect of resistivity on the ideally unstable internal kink is analyzed. (orig.)

The eigenvalues of the SN transport matrix

International Nuclear Information System (INIS)

Ourique, L.E.; Vilhena, M.T. de

2005-01-01

In a recent paper, we analyze the dependence of the eigenvalues of the S N matrix transport, associated with the system of linear differential equations that corresponds to the S N approximations of the transport equation [1]. By considering a control parameter, we have shown that there exist some bifurcation points. This means that the solutions of S N approximations change from oscillatory to non-oscillatory behavior, a different approach of the study by [2]. Nowadays, the one-dimensional transport equation and related problems have been a source of new techniques for solving particular cases as well the development of analytical methods that search aspects of existence and uniqueness of the solutions [3], [4]. In this work, we generalize the results shown in [1], searching for a model of the distribution of the bifurcation points of the S N matrix transport, studying the one-dimensional case in a slab, with anisotropic differential cross section of order 3. The result indicates that the bifurcation points obey a certain rule of distribution. Beside that, the condition number of the matrix transport increases too much in the neighborhood of these points, as we have seen in [1]. (author)

The Para-Bose oscillator in a finite linear space

International Nuclear Information System (INIS)

Campos, R.G.

1987-01-01

The harmonic oscillator whose canonical variables satisfy the generalized commutation relations proposed by Wigner is studied in a finite linear space of dimension N by elementary methods. The eigenvalue problems of the Hamiltonian and position operators are worked out and it is found that, when N tends to infinity, the H-eigenvectors tend to the two solutions obtained by Ohnuki Kamefuchi evaluated in the X eigenpoints as N is odd or even. Beside this, the P-representative in the finite X-basis resembles the form that it has in the continuous case and the X-eigenvalues satisfy a minimal property. In this context, some properties of the associated Laguerre polynomials and their zeros (some of them already studied) are derived

A spectral nodal method for eigenvalue S{sub N} transport problems in two-dimensional rectangular geometry for energy multigroup nuclear reactor global calculations

Energy Technology Data Exchange (ETDEWEB)

Silva, Davi Jose M.; Alves Filho, Hermes; Barros, Ricardo C., E-mail: davijmsilva@yahoo.com.br, E-mail: halves@iprj.uerj.br, E-mail: rcbarros@pq.cnpq.br [Universidade do Estado do Rio de Janeiro (UERJ), Nova Friburgo, RJ (Brazil). Programa de Pos-Graduacao em Modelagem Computacional

2015-07-01

A spectral nodal method is developed for multigroup x,y-geometry discrete ordinates (S{sub N}) eigenvalue problems for nuclear reactor global calculations. This method uses the conventional multigroup SN discretized spatial balance nodal equations with two non-standard auxiliary equations: the spectral diamond (SD) auxiliary equations for the discretization nodes inside the fuel regions, and the spectral Green's function (SGF) auxiliary equations for the non-multiplying regions, such as the baffle and the reactor. This spectral nodal method is derived from the analytical general solution of the SN transverse integrated nodal equations with constant approximations for the transverse leakage terms within each discretization node. The SD and SGF auxiliary equations have parameters, which are determined to preserve the homogeneous and the particular components of these local general solutions. Therefore, we refer to the offered method as the hybrid SD-SGF-Constant Nodal (SD-SGF-CN) method. The S{sub N} discretized spatial balance equations, together with the SD and the SGF auxiliary equations form the SD-SGF-CN equations. We solve the SD-SGF-CN equations by using the one-node block inversion inner iterations (NBI), wherein the most recent estimates for the incoming group node-edge average or prescribed boundary conditions are used to evaluate the outgoing group node-edge average fluxes in the directions of the S{sub N} transport sweeps, for each estimate of the dominant eigenvalue in the conventional Power outer iterations. We show in numerical calculations that the SD-SGF-CN method is very accurate for coarse-mesh multigroup S{sub N} eigenvalue problems, even though the transverse leakage terms are approximated rather simply. (author)

A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential

Directory of Open Access Journals (Sweden)

Craig Haile

2000-01-01

Full Text Available We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^alpha$, $k ge 0$, $k in mathbb{R}, alpha ge 2$ on a bounded domain $Omega$ in $mathbb{R}^n$, $(n ge 2$ with Dirichlet boundary conditions. In particular, we compare the eigenvalue $lambda_2(Omega$ of the operator $-Delta + k |x|^alpha $ on $Omega$ with the eigenvalue $lambda_2(S_1$ of the same operator $-Delta +kr^alpha$ on a ball $S_1$, where $S_1$ has radius such that the first eigenvalues are the same ($lambda_1(Omega = lambda_1(S_1$. The main result is to show $lambda_2(Omega le lambda_2(S_1$. We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain $Omega$ with Dirichlet boundary conditions.

A Schur Method for Designing LQ-optimal Systems with Prescribed Eigenvalues

Directory of Open Access Journals (Sweden)

David Di Ruscio

1990-01-01

Full Text Available In this paper a new algorithm for solving the LQ-optimal pole placement problem is presented. The method studied is a variant of the classical eigenvector approach and instead uses a set of Schur vectors, thereby gaining substantial numerical advantages. An important task in this method is the LQ-optimal pole placement problem for a second order (sub system. The paper presents a detailed analytical solution to this problem. This part is not only important for solving the general n-dimensional problem but also provides an understanding of the behaviour of an optimal system: The paper shows that in some cases it is an infinite number; in others a finite number, and in still others, non state weighting matrices Q that give the system a set of prescribed eigenvalues. Equations are presented that uniquely determine these state weight matrices as a function of the new prescribed eigcnvalues. From this result we have been able to derive the maximum possible imaginary part of the eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen.

Applied linear algebra and matrix analysis

CERN Document Server

Shores, Thomas S

2018-01-01

In its second edition, this textbook offers a fresh approach to matrix and linear algebra. Its blend of theory, computational exercises, and analytical writing projects is designed to highlight the interplay between these aspects of an application. This approach places special emphasis on linear algebra as an experimental science that provides tools for solving concrete problems. The second edition’s revised text discusses applications of linear algebra like graph theory and network modeling methods used in Google’s PageRank algorithm. Other new materials include modeling examples of diffusive processes, linear programming, image processing, digital signal processing, and Fourier analysis. These topics are woven into the core material of Gaussian elimination and other matrix operations; eigenvalues, eigenvectors, and discrete dynamical systems; and the geometrical aspects of vector spaces. Intended for a one-semester undergraduate course without a strict calculus prerequisite, Applied Linear Algebra and M...

Students’ difficulties in solving linear equation problems

Science.gov (United States)

Wati, S.; Fitriana, L.; Mardiyana

2018-03-01

A linear equation is an algebra material that exists in junior high school to university. It is a very important material for students in order to learn more advanced mathematics topics. Therefore, linear equation material is essential to be mastered. However, the result of 2016 national examination in Indonesia showed that students’ achievement in solving linear equation problem was low. This fact became a background to investigate students’ difficulties in solving linear equation problems. This study used qualitative descriptive method. An individual written test on linear equation tasks was administered, followed by interviews. Twenty-one sample students of grade VIII of SMPIT Insan Kamil Karanganyar did the written test, and 6 of them were interviewed afterward. The result showed that students with high mathematics achievement donot have difficulties, students with medium mathematics achievement have factual difficulties, and students with low mathematics achievement have factual, conceptual, operational, and principle difficulties. Based on the result there is a need of meaningfulness teaching strategy to help students to overcome difficulties in solving linear equation problems.

Preconditioned iterations to calculate extreme eigenvalues

Energy Technology Data Exchange (ETDEWEB)

Brand, C.W.; Petrova, S. [Institut fuer Angewandte Mathematik, Leoben (Austria)

1994-12-31

Common iterative algorithms to calculate a few extreme eigenvalues of a large, sparse matrix are Lanczos methods or power iterations. They converge at a rate proportional to the separation of the extreme eigenvalues from the rest of the spectrum. Appropriate preconditioning improves the separation of the eigenvalues. Davidson`s method and its generalizations exploit this fact. The authors examine a preconditioned iteration that resembles a truncated version of Davidson`s method with a different preconditioning strategy.

A METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS WITH FUZZY PARAMETERS BASED ON MULTIOBJECTIVE LINEAR PROGRAMMING TECHNIQUE

OpenAIRE

M. ZANGIABADI; H. R. MALEKI

2007-01-01

In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, w...

An approach for solving linear fractional programming problems ...

African Journals Online (AJOL)

The paper presents a new approach for solving a fractional linear programming problem in which the objective function is a linear fractional function, while the constraint functions are in the form of linear inequalities. The approach adopted is based mainly upon solving the problem algebraically using the concept of duality ...

Interior Point Method for Solving Fuzzy Number Linear Programming Problems Using Linear Ranking Function

Directory of Open Access Journals (Sweden)

Yi-hua Zhong

2013-01-01

Full Text Available Recently, various methods have been developed for solving linear programming problems with fuzzy number, such as simplex method and dual simplex method. But their computational complexities are exponential, which is not satisfactory for solving large-scale fuzzy linear programming problems, especially in the engineering field. A new method which can solve large-scale fuzzy number linear programming problems is presented in this paper, which is named a revised interior point method. Its idea is similar to that of interior point method used for solving linear programming problems in crisp environment before, but its feasible direction and step size are chosen by using trapezoidal fuzzy numbers, linear ranking function, fuzzy vector, and their operations, and its end condition is involved in linear ranking function. Their correctness and rationality are proved. Moreover, choice of the initial interior point and some factors influencing the results of this method are also discussed and analyzed. The result of algorithm analysis and example study that shows proper safety factor parameter, accuracy parameter, and initial interior point of this method may reduce iterations and they can be selected easily according to the actual needs. Finally, the method proposed in this paper is an alternative method for solving fuzzy number linear programming problems.

On the decision threshold of eigenvalue ratio detector based on moments of joint and marginal distributions of extreme eigenvalues

KAUST Repository

Shakir, Muhammad Zeeshan

2013-03-01

Eigenvalue Ratio (ER) detector based on the two extreme eigenvalues of the received signal covariance matrix is currently one of the most effective solution for spectrum sensing. However, the analytical results of such scheme often depend on asymptotic assumptions since the distribution of the ratio of two extreme eigenvalues is exceptionally complex to compute. In this paper, a non-asymptotic spectrum sensing approach for ER detector is introduced to approximate the marginal and joint distributions of the two extreme eigenvalues. The two extreme eigenvalues are considered as dependent Gaussian random variables such that their joint probability density function (PDF) is approximated by a bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. The PDF approximation approach is based on the moment matching method where we calculate the exact analytical moments of joint and marginal distributions of the two extreme eigenvalues. The decision threshold is calculated by exploiting the statistical mean and the variance of each of the two extreme eigenvalues and the correlation coefficient between them. The performance analysis of our newly proposed approximation approach is compared with the already published asymptotic Tracy-Widom approximation approach. It has been shown that our results are in perfect agreement with the simulation results for any number of secondary users and received samples. © 2002-2012 IEEE.

An algebraic sub-structuring method for large-scale eigenvalue calculation

International Nuclear Information System (INIS)

Yang, C.; Gao, W.; Bai, Z.; Li, X.; Lee, L.; Husbands, P.; Ng, E.

2004-01-01

We examine sub-structuring methods for solving large-scale generalized eigenvalue problems from a purely algebraic point of view. We use the term 'algebraic sub-structuring' to refer to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to provide approximate solutions to the original problem. We are interested in the question of which spectral components one should extract from each sub-structure in order to produce an approximate solution to the original problem with a desired level of accuracy. Error estimate for the approximation to the smallest eigenpair is developed. The estimate leads to a simple heuristic for choosing spectral components (modes) from each sub-structure. The effectiveness of such a heuristic is demonstrated with numerical examples. We show that algebraic sub-structuring can be effectively used to solve a generalized eigenvalue problem arising from the simulation of an accelerator structure. One interesting characteristic of this application is that the stiffness matrix produced by a hierarchical vector finite elements scheme contains a null space of large dimension. We present an efficient scheme to deflate this null space in the algebraic sub-structuring process

Time-optimal feedback control for linear systems

International Nuclear Information System (INIS)

Mirica, S.

1976-01-01

The paper deals with the results of qualitative investigations of the time-optimal feedback control for linear systems with constant coefficients. In the first section, after some definitions and notations, two examples are given and it is shown that even the time-optimal control problem for linear systems with constant coefficients which looked like ''completely solved'' requires a further qualitative investigation of the stability to ''permanent perturbations'' of optimal feedback control. In the second section some basic results of the linear time-optimal control problem are reviewed. The third section deals with the definition of Boltyanskii's ''regular synthesis'' and its connection to Filippov's theory of right-hand side discontinuous differential equations. In the fourth section a theorem is proved concerning the stability to perturbations of time-optimal feedback control for linear systems with scalar control. In the last two sections it is proved that, if the matrix which defines the system has only real eigenvalues or is three-dimensional, the time-optimal feedback control defines a regular synthesis and therefore is stable to perturbations. (author)

A Global Optimization Algorithm for Sum of Linear Ratios Problem

OpenAIRE

Yuelin Gao; Siqiao Jin

2013-01-01

We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the c...

Microlocal analysis of a seismic linearized inverse problem

NARCIS (Netherlands)

Stolk, C.C.

1999-01-01

The seismic inverse problem is to determine the wavespeed c x in the interior of a medium from measurements at the boundary In this paper we analyze the linearized inverse problem in general acoustic media The problem is to nd a left inverse of the linearized forward map F or equivalently to nd the

Correlation between eigenvalues and sorted diagonal matrix elements of a large dimensional matrix

International Nuclear Information System (INIS)

Arima, A.

2008-01-01

Functional dependences of eigenvalues as functions of sorted diagonal elements are given for realistic nuclear shell model (NSM) hamiltonian, the uniform distribution hamiltonian and the GOE hamiltonian. In the NSM case, the dependence is found to be linear. We discuss extrapolation methods for more accurate predictions for low-lying states. (author)

Parametrices and exact paralinearization of semi-linear boundary problems

DEFF Research Database (Denmark)

Johnsen, Jon

2008-01-01

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearization...... of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation....

Linear Scaling Solution of the Time-Dependent Self-Consistent-Field Equations

Directory of Open Access Journals (Sweden)

Matt Challacombe

2014-03-01

Full Text Available A new approach to solving the Time-Dependent Self-Consistent-Field equations is developed based on the double quotient formulation of Tsiper 2001 (J. Phys. B. Dual channel, quasi-independent non-linear optimization of these quotients is found to yield convergence rates approaching those of the best case (single channel Tamm-Dancoff approximation. This formulation is variational with respect to matrix truncation, admitting linear scaling solution of the matrix-eigenvalue problem, which is demonstrated for bulk excitons in the polyphenylene vinylene oligomer and the (4,3 carbon nanotube segment.

A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus

Science.gov (United States)

Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir

2016-07-01

This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.

Generalized eigenvalue based spectrum sensing

KAUST Repository

Shakir, Muhammad

2012-01-01

Spectrum sensing is one of the fundamental components in cognitive radio networks. In this chapter, a generalized spectrum sensing framework which is referred to as Generalized Mean Detector (GMD) has been introduced. In this context, we generalize the detectors based on the eigenvalues of the received signal covariance matrix and transform the eigenvalue based spectrum sensing detectors namely: (i) the Eigenvalue Ratio Detector (ERD) and two newly proposed detectors which are referred to as (ii) the GEometric Mean Detector (GEMD) and (iii) the ARithmetic Mean Detector (ARMD) into an unified framework of generalize spectrum sensing. The foundation of the proposed framework is based on the calculation of exact analytical moments of the random variables of the decision threshold of the respective detectors. The decision threshold has been calculated in a closed form which is based on the approximation of Cumulative Distribution Functions (CDFs) of the respective test statistics. In this context, we exchange the analytical moments of the two random variables of the respective test statistics with the moments of the Gaussian (or Gamma) distribution function. The performance of the eigenvalue based detectors is compared with the several traditional detectors including the energy detector (ED) to validate the importance of the eigenvalue based detectors and the performance of the GEMD and the ARMD particularly in realistic wireless cognitive radio network. Analytical and simulation results show that the newly proposed detectors yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, the presented results based on proposed approximation approaches are in perfect agreement with the empirical results. © 2012 Springer Science+Business Media Dordrecht.

Investigation, development and application of optimal output feedback theory. Vol. 4: Measures of eigenvalue/eigenvector sensitivity to system parameters and unmodeled dynamics

Science.gov (United States)

Halyo, Nesim

1987-01-01

Some measures of eigenvalue and eigenvector sensitivity applicable to both continuous and discrete linear systems are developed and investigated. An infinite series representation is developed for the eigenvalues and eigenvectors of a system. The coefficients of the series are coupled, but can be obtained recursively using a nonlinear coupled vector difference equation. A new sensitivity measure is developed by considering the effects of unmodeled dynamics. It is shown that the sensitivity is high when any unmodeled eigenvalue is near a modeled eigenvalue. Using a simple example where the sensor dynamics have been neglected, it is shown that high feedback gains produce high eigenvalue/eigenvector sensitivity. The smallest singular value of the return difference is shown not to reflect eigenvalue sensitivity since it increases with the feedback gains. Using an upper bound obtained from the infinite series, a procedure to evaluate whether the sensitivity to parameter variations is within given acceptable bounds is developed and demonstrated by an example.

A Global Optimization Algorithm for Sum of Linear Ratios Problem

Directory of Open Access Journals (Sweden)

Yuelin Gao

2013-01-01

Full Text Available We equivalently transform the sum of linear ratios programming problem into bilinear programming problem, then by using the linear characteristics of convex envelope and concave envelope of double variables product function, linear relaxation programming of the bilinear programming problem is given, which can determine the lower bound of the optimal value of original problem. Therefore, a branch and bound algorithm for solving sum of linear ratios programming problem is put forward, and the convergence of the algorithm is proved. Numerical experiments are reported to show the effectiveness of the proposed algorithm.

Modified Bateman solution for identical eigenvalues

International Nuclear Information System (INIS)

Dreher, Raymond

2013-01-01

Highlights: ► Solving indeterminacies due to identical eigenvalues in Bateman’s solution. ► Exact analytical solution of Bateman’s equations for identical eigenvalues. ► Algorithm calculating higher order derivatives appearing in this solution. ► Alternative evaluation of the derivatives through the Taylor polynomial. ► Implementation of an example program demonstrating the developed solution. - Abstract: In this paper we develop a general solution to the Bateman equations taking into account the special case of identical eigenvalues. A characteristic of this new solution is the presence of higher order derivatives. It is shown that the derivatives can be obtained analytically and also computed in an efficient manner

Eigenvalue Decomposition-Based Modified Newton Algorithm

Directory of Open Access Journals (Sweden)

Wen-jun Wang

2013-01-01

Full Text Available When the Hessian matrix is not positive, the Newton direction may not be the descending direction. A new method named eigenvalue decomposition-based modified Newton algorithm is presented, which first takes the eigenvalue decomposition of the Hessian matrix, then replaces the negative eigenvalues with their absolute values, and finally reconstructs the Hessian matrix and modifies the searching direction. The new searching direction is always the descending direction. The convergence of the algorithm is proven and the conclusion on convergence rate is presented qualitatively. Finally, a numerical experiment is given for comparing the convergence domains of the modified algorithm and the classical algorithm.

Eigenvalues calculation algorithms for {lambda}-modes determination. Parallelization approach

Energy Technology Data Exchange (ETDEWEB)

Vidal, V. [Universidad Politecnica de Valencia (Spain). Departamento de Sistemas Informaticos y Computacion; Verdu, G.; Munoz-Cobo, J.L. [Universidad Politecnica de Valencia (Spain). Departamento de Ingenieria Quimica y Nuclear; Ginestart, D. [Universidad Politecnica de Valencia (Spain). Departamento de Matematica Aplicada

1997-03-01

In this paper, we review two methods to obtain the {lambda}-modes of a nuclear reactor, Subspace Iteration method and Arnoldi`s method, which are popular methods to solve the partial eigenvalue problem for a given matrix. In the developed application for the neutron diffusion equation we include improved acceleration techniques for both methods. Also, we propose two parallelization approaches for these methods, a coarse grain parallelization and a fine grain one. We have tested the developed algorithms with two realistic problems, focusing on the efficiency of the methods according to the CPU times. (author).

Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations

NARCIS (Netherlands)

Jarlebring, E.; Hochstenbach, M.E.

2009-01-01

Several recent methods used to analyze asymptotic stability of delay-differential equations (DDEs) involve determining the eigenvalues of a matrix, a matrix pencil or a matrix polynomial constructed by Kronecker products. Despite some similarities between the different types of these so-called

Solution strategies for linear and nonlinear instability phenomena for arbitrarily thin shell structures

International Nuclear Information System (INIS)

Eckstein, U.; Harte, R.; Kraetzig, W.B.; Wittek, U.

1983-01-01

In order to describe nonlinear response and instability behaviour the paper starts with the total potential energy considering the basic kinematic equations of a consistent nonlinear shell theory for large displacements and moderate rotations. The material behaviour is assumed to be hyperelastic and isotropic. The incrementation and discretization of the total potential energy leads to the tangent stiffness relation, which is the central equation of computational algorithms based on combined incremental and iterative techniques. Here a symmetrized form of the RIKS/WEMPNER-algorithm for positive and negative load incrementation represents the basis of the nonlinear solution technique. To detect secondary equilibrium branches at points of neutral equilibrium within nonlinear primary paths a quadratic eigenvalue-problem has to be solved. In order to follow those complicated nonlinear response phenomena the RIKS/WEMPNER incrementation/iteration process is combined with a simultaneous solution of the linearized quadratic eigenvalue-problem. Additionally the essentials of a recently derived family of arbitrarily curved shell elements for linear (LACS) and geometrically nonlinear (NACS) shell problems are presented. The main advantage of these elements is the exact description of all geometric properties as well as the energy-equivalent representation of the applied loads in combination with an efficient algorithm to form the stiffness submatrices. Especially the NACS-elements are designed to improve the accuracy of the solution in the deep postbuckling range including moderate rotations. The derived finite elements and solution strategies are applied to a certain number of typical shell problems to prove the precision of the shell elements and to demonstrate the possibilities of tracing linear and nonlinear bifurcation problems as well as snap-through phenomena with and without secondary bifurcation branches. (orig.)

Evaluation of vectorized Monte Carlo algorithms on GPUs for a neutron Eigenvalue problem

International Nuclear Information System (INIS)

Du, X.; Liu, T.; Ji, W.; Xu, X. G.; Brown, F. B.

2013-01-01

Conventional Monte Carlo (MC) methods for radiation transport computations are 'history-based', which means that one particle history at a time is tracked. Simulations based on such methods suffer from thread divergence on the graphics processing unit (GPU), which severely affects the performance of GPUs. To circumvent this limitation, event-based vectorized MC algorithms can be utilized. A versatile software test-bed, called ARCHER - Accelerated Radiation-transport Computations in Heterogeneous Environments - was used for this study. ARCHER facilitates the development and testing of a MC code based on the vectorized MC algorithm implemented on GPUs by using NVIDIA's Compute Unified Device Architecture (CUDA). The ARCHER GPU code was designed to solve a neutron eigenvalue problem and was tested on a NVIDIA Tesla M2090 Fermi card. We found that although the vectorized MC method significantly reduces the occurrence of divergent branching and enhances the warp execution efficiency, the overall simulation speed is ten times slower than the conventional history-based MC method on GPUs. By analyzing detailed GPU profiling information from ARCHER, we discovered that the main reason was the large amount of global memory transactions, causing severe memory access latency. Several possible solutions to alleviate the memory latency issue are discussed. (authors)

Evaluation of vectorized Monte Carlo algorithms on GPUs for a neutron Eigenvalue problem

Energy Technology Data Exchange (ETDEWEB)

Du, X.; Liu, T.; Ji, W.; Xu, X. G. [Nuclear Engineering Program, Rensselaer Polytechnic Institute, Troy, NY 12180 (United States); Brown, F. B. [Monte Carlo Codes Group, Los Alamos National Laboratory, Los Alamos, NM 87545 (United States)

2013-07-01

Conventional Monte Carlo (MC) methods for radiation transport computations are 'history-based', which means that one particle history at a time is tracked. Simulations based on such methods suffer from thread divergence on the graphics processing unit (GPU), which severely affects the performance of GPUs. To circumvent this limitation, event-based vectorized MC algorithms can be utilized. A versatile software test-bed, called ARCHER - Accelerated Radiation-transport Computations in Heterogeneous Environments - was used for this study. ARCHER facilitates the development and testing of a MC code based on the vectorized MC algorithm implemented on GPUs by using NVIDIA's Compute Unified Device Architecture (CUDA). The ARCHER{sub GPU} code was designed to solve a neutron eigenvalue problem and was tested on a NVIDIA Tesla M2090 Fermi card. We found that although the vectorized MC method significantly reduces the occurrence of divergent branching and enhances the warp execution efficiency, the overall simulation speed is ten times slower than the conventional history-based MC method on GPUs. By analyzing detailed GPU profiling information from ARCHER, we discovered that the main reason was the large amount of global memory transactions, causing severe memory access latency. Several possible solutions to alleviate the memory latency issue are discussed. (authors)

An Optimal Linear Coding for Index Coding Problem

OpenAIRE

Pezeshkpour, Pouya

2015-01-01

An optimal linear coding solution for index coding problem is established. Instead of network coding approach by focus on graph theoric and algebraic methods a linear coding program for solving both unicast and groupcast index coding problem is presented. The coding is proved to be the optimal solution from the linear perspective and can be easily utilize for any number of messages. The importance of this work is lying mostly on the usage of the presented coding in the groupcast index coding ...

Synchronization and Control of Linearly Coupled Singular Systems

Directory of Open Access Journals (Sweden)

Fang Qingxiang

2013-01-01

Full Text Available The synchronization and control problem of linearly coupled singular systems is investigated. The uncoupled dynamical behavior at each node is general and can be chaotic or, otherwise the coupling matrix is not assumed to be symmetrical. Some sufficient conditions for globally exponential synchronization are derived based on Lyapunov stability theory. These criteria, which are in terms of linear matrix inequality (LMI, indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold. The controllers are designed for state feedback control and pinning control, respectively. Finally, a numerical example is provided to illustrate the effectiveness of the proposed conditions.

Construction of accuracy-preserving surrogate for the eigenvalue radiation diffusion and/or transport problem

Energy Technology Data Exchange (ETDEWEB)

Wang, C.; Abdel-Khalik, H. S. [Dept. of Nuclear Engineering, North Caroline State Univ., Raleigh, NC 27695 (United States)

2012-07-01

The construction of surrogate models for high fidelity models is now considered an important objective in support of all engineering activities which require repeated execution of the simulation, such as verification studies, validation exercises, and uncertainty quantification. The surrogate must be computationally inexpensive to allow its repeated execution, and must be computationally accurate in order for its predictions to be credible. This manuscript introduces a new surrogate construction approach that reduces the dimensionality of the state solution via a range-finding algorithm from linear algebra. It then employs a proper orthogonal decomposition-like approach to solve for the reduced state. The algorithm provides an upper bound on the error resulting from the reduction. Different from the state-of-the-art, the new approach allows the user to define the desired accuracy a priori which controls the maximum allowable reduction. We demonstrate the utility of this approach using an eigenvalue radiation diffusion model, where the accuracy is selected to match machine precision. Results indicate that significant reduction is possible for typical reactor assembly models, which are currently considered expensive given the need to employ very fine mesh many group calculations to ensure the highest possible fidelity for the downstream core calculations. Given the potential for significant reduction in the computational cost, we believe it is possible to rethink the manner in which homogenization theory is currently employed in reactor design calculations. (authors)

A numerical method for eigenvalue problems in modeling liquid crystals

Energy Technology Data Exchange (ETDEWEB)

Baglama, J.; Farrell, P.A.; Reichel, L.; Ruttan, A. [Kent State Univ., OH (United States); Calvetti, D. [Stevens Inst. of Technology, Hoboken, NJ (United States)

1996-12-31

Equilibrium configurations of liquid crystals in finite containments are minimizers of the thermodynamic free energy of the system. It is important to be able to track the equilibrium configurations as the temperature of the liquid crystals decreases. The path of the minimal energy configuration at bifurcation points can be computed from the null space of a large sparse symmetric matrix. We describe a new variant of the implicitly restarted Lanczos method that is well suited for the computation of extreme eigenvalues of a large sparse symmetric matrix, and we use this method to determine the desired null space. Our implicitly restarted Lanczos method determines adoptively a polynomial filter by using Leja shifts, and does not require factorization of the matrix. The storage requirement of the method is small, and this makes it attractive to use for the present application.

Sampled-Data Consensus of Linear Multi-agent Systems With Packet Losses.

Science.gov (United States)

Zhang, Wenbing; Tang, Yang; Huang, Tingwen; Kurths, Jurgen

In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function

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.

Problem on eigenfunctions and eigenvalues for effective Hamiltonians in pair channels of four-particle systems

International Nuclear Information System (INIS)

Gurbanovich, N.S.; Zelenskaya, I.N.

1976-01-01

The solution for eigenfunction and eigenvalue for effective Hamiltonians anti Hsub(p) in two-particle channels corresponding to division of four particles into groups (3.1) and (2.2) is very essential in the four-body problem as applied to nuclear reactions. The interaction of anti√sub(p) in each channel may be written in the form of an integral operator which takes account of the structure of a target nucleus or of an incident particle and satisfying the integral equation. While assuming the two-particle potentials to be central, it is possible to expand the effective interactions anti√sub(p) in partial waves and write the radial equation for anti Hsub(p). In the approximation on a mass shell the radial equations for the eigenfunctions of Hsub(p) are reduced to an algebraic equations system. The coefficients of the latter are expressed through the Fourier images for products of wave functions of bound clusters and the two-particle central potential which are localized in a momentum space

A Linearized Relaxing Algorithm for the Specific Nonlinear Optimization Problem

Directory of Open Access Journals (Sweden)

Mio Horai

2016-01-01

Full Text Available We propose a new method for the specific nonlinear and nonconvex global optimization problem by using a linear relaxation technique. To simplify the specific nonlinear and nonconvex optimization problem, we transform the problem to the lower linear relaxation form, and we solve the linear relaxation optimization problem by the Branch and Bound Algorithm. Under some reasonable assumptions, the global convergence of the algorithm is certified for the problem. Numerical results show that this method is more efficient than the previous methods.

Turnpike theory of continuous-time linear optimal control problems

CERN Document Server

Zaslavski, Alexander J

2015-01-01

Individual turnpike results are of great interest due to their numerous applications in engineering and in economic theory; in this book the study is focused on new results of turnpike phenomenon in linear optimal control problems.  The book is intended for engineers as well as for mathematicians interested in the calculus of variations, optimal control, and in applied functional analysis. Two large classes of problems are studied in more depth. The first class studied in Chapter 2 consists of linear control problems with periodic nonsmooth convex integrands. Chapters 3-5 consist of linear control problems with autonomous nonconvex and nonsmooth integrands.  Chapter 6 discusses a turnpike property for dynamic zero-sum games with linear constraints. Chapter 7 examines genericity results. In Chapter 8, the description of structure of variational problems with extended-valued integrands is obtained. Chapter 9 ends the exposition with a study of turnpike phenomenon for dynamic games with extended value integran...

On the numerical solution of coupled eigenvalue differential equations arising in molecular spectroscopy

International Nuclear Information System (INIS)

Friedman, R.S.; Jamieson, M.J.; Preston, S.C.

1990-01-01

A method for solving coupled eigenvalue differential equations is given and its relation to an existing technique is shown. Use of the Gram-Schmidt process to overcome the severe instabilities arising in molecular problems is described in detail. (orig.)

The cosmological constant as an eigenvalue of the Hamiltonian constraint in a varying speed of light theory

Energy Technology Data Exchange (ETDEWEB)

Garattini, Remo [Univ. degli Studi di Bergamo, Dalmine (Italy). Dept. of Engineering and Applied Sciences; I.N.F.N., Sezione di Milano, Milan (Italy); De Laurentis, Mariafelicia [Tomsk State Pedagogical Univ. (Russian Federation). Dept. of Theoretical Physics; INFN, Sezione di Napoli (Italy); Complutense Univ. di Monte S. Angelo, Napoli (Italy)

2017-01-15

In the framework of a Varying Speed of Light theory, we study the eigenvalues associated with the Wheeler-DeWitt equation representing the vacuum expectation values associated with the cosmological constant. We find that the Wheeler-DeWitt equation for the Friedmann-Lemaitre-Robertson-Walker metric is completely equivalent to a Sturm-Liouville problem provided that the related eigenvalue and the cosmological constant be identified. The explicit calculation is performed with the help of a variational procedure with trial wave functionals related to the Bessel function of the second kind K{sub ν}(x). After having verified that in ordinary General Relativity no eigenvalue appears, we find that in a Varying Speed of Light theory this is not the case. Nevertheless, instead of a single eigenvalue, we discover the existence of a family of eigenvalues associated to a negative power of the scale. A brief comment on what happens at the inflationary scale is also included. (copyright 2016 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)

FEAST fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem

Science.gov (United States)

Levin, Alan R.; Zhang, Deyin; Polizzi, Eric

2012-11-01

In a recent article Polizzi (2009) [15], the FEAST algorithm has been presented as a general purpose eigenvalue solver which is ideally suited for addressing the numerical challenges in electronic structure calculations. Here, FEAST is presented beyond the “black-box” solver as a fundamental modeling framework which can naturally address the original numerical complexity of the electronic structure problem as formulated by Slater in 1937 [3]. The non-linear eigenvalue problem arising from the muffin-tin decomposition of the real-space domain is first derived and then reformulated to be solved exactly within the FEAST framework. This new framework is presented as a fundamental and practical solution for performing both accurate and scalable electronic structure calculations, bypassing the various issues of using traditional approaches such as linearization and pseudopotential techniques. A finite element implementation of this FEAST framework along with simulation results for various molecular systems is also presented and discussed.

Special Cases in Optimization Problems for Stationary Linear Closed-Loop Systems

Science.gov (United States)

Aliev, F. A.; Larin, Vladimir Borisovich

2003-03-01

Consideration is given to problems of solving the algebraic Riccati equation (ARE)— J-factorization of matrix polynomials and J-factorization of rational matrices—to which traditional solution algorithms are not applicable. In this connection, solution algorithms for these problems are discussed where the eigenvalues of the Hamiltonian matrix corresponding to the ARE and the zeros of matrix polynomials are located on the imaginary axis. Moreover, a procedure is set forth for asymptotic expansion of a stabilizing solution of the ARE in the neighborhood of a point at which the ARE has no stabilizing solution. It is shown how this expansion can be used for constructing canonical J-factorization of matrix polynomials that is nearly a noncanonical J-factorization. It is pointed out that the algorithms described can be implemented with the help of MATLAB routines

Genetic Algorithm Applied to the Eigenvalue Equalization Filtered-x LMS Algorithm (EE-FXLMS

Directory of Open Access Journals (Sweden)

Stephan P. Lovstedt

2008-01-01

Full Text Available The FXLMS algorithm, used extensively in active noise control (ANC, exhibits frequency-dependent convergence behavior. This leads to degraded performance for time-varying tonal noise and noise with multiple stationary tones. Previous work by the authors proposed the eigenvalue equalization filtered-x least mean squares (EE-FXLMS algorithm. For that algorithm, magnitude coefficients of the secondary path transfer function are modified to decrease variation in the eigenvalues of the filtered-x autocorrelation matrix, while preserving the phase, giving faster convergence and increasing overall attenuation. This paper revisits the EE-FXLMS algorithm, using a genetic algorithm to find magnitude coefficients that give the least variation in eigenvalues. This method overcomes some of the problems with implementing the EE-FXLMS algorithm arising from finite resolution of sampled systems. Experimental control results using the original secondary path model, and a modified secondary path model for both the previous implementation of EE-FXLMS and the genetic algorithm implementation are compared.

A new formulation for the eigenvalue and the eigenfunction in the perturbation theory

International Nuclear Information System (INIS)

Korek, Mahmoud

1999-01-01

Full text.In infrared transitions, the problem of the ro vibrational eigenvalue and eigenfunction of a diatomic molecule is considered. It is shown that, for the transitions vJ↔v'J' the eigenvalues and the eigenfunctions of the two considered states can be expressed respectively in terms of one variable m (transition number), relating these two states, as E vm =Σ i=o e v (i) m i , Ψ vm =Σ i=0 φ v (i) m i and E v'm =Σ i=0 e v' (i) m i , Ψ v'm =Σ i=0 φ v' (i) m i , where m=[J'(J'+1)-J(J+1)]/2, and the coefficients e v (i) , φ v (i) , e v (i) , and φ v (i) , are given by analytical expressions. This m-representation of the eigenvalues and the eigenfunctions is more advantageous for the calculation of many factors in spectroscopy that are given in terms of m as the line intensities, the wave number of a transition, the Herman-Wallis coefficients,...etc. The numerical application to the ground state of the molecule CO shows that the present formulation provides a simple and accurate method for the calculation of the eigenvalues and the eigenfunctions for the two considered states

Extreme eigenvalues of sample covariance and correlation matrices

DEFF Research Database (Denmark)

Heiny, Johannes

This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... matrix of a p-dimensional heavy-tailed time series when p converges to infinity together with the sample size n. We generalize the growth rates of p existing in the literature. Assuming a regular variation condition with tail index ... eigenvalues are essentially determined by the extreme order statistics from an array of iid random variables. The asymptotic behavior of the extreme eigenvalues is then derived routinely from classical extreme value theory. The resulting approximations are strikingly simple considering the high dimension...

Quantization of the Linearized Kepler Problem

OpenAIRE

Guerrero, Julio; Perez, Jose Miguel

2003-01-01

The linearized Kepler problem is considered, as obtained from the Kustaanheimo-Stiefel (K-S)transformation, both for negative and positive energies. The symmetry group for the Kepler problem turns out to be SU(2,2). For negative energies, the Hamiltonian of Kepler problem can be realized as the sum of the energies of four harmonic oscillator with the same frequency, with a certain constrain. For positive energies, it can be realized as the sum of the energies of four repulsive oscillator with...

Hessian eigenvalue distribution in a random Gaussian landscape

Science.gov (United States)

Yamada, Masaki; Vilenkin, Alexander

2018-03-01

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/ N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

Estimates of the eigenvalues of operator arising in swelling pressure model

International Nuclear Information System (INIS)

Kanguzhin, Baltabek; Zhapsarbayeva, Lyailya

2016-01-01

Swelling pressures from materials confined by structures can cause structural deformations and instability. Due to the complexity of interactions between expansive solid and solid-liquid equilibrium, the forces exerting on retaining structures from swelling are highly nonlinear. This work is our initial attempt to study a simplistic spectral problem based on the Euler-elastic beam theory and some simplistic swelling pressure model. In this work estimates of the eigenvalues of some initial/boundary value problem for nonlinear Euler-elastic beam equation are obtained.

Convex variational problems linear, nearly linear and anisotropic growth conditions

CERN Document Server

Bildhauer, Michael

2003-01-01

The author emphasizes a non-uniform ellipticity condition as the main approach to regularity theory for solutions of convex variational problems with different types of non-standard growth conditions. This volume first focuses on elliptic variational problems with linear growth conditions. Here the notion of a "solution" is not obvious and the point of view has to be changed several times in order to get some deeper insight. Then the smoothness properties of solutions to convex anisotropic variational problems with superlinear growth are studied. In spite of the fundamental differences, a non-uniform ellipticity condition serves as the main tool towards a unified view of the regularity theory for both kinds of problems.

Numerical Methods for Solution of the Extended Linear Quadratic Control Problem

DEFF Research Database (Denmark)

Jørgensen, John Bagterp; Frison, Gianluca; Gade-Nielsen, Nicolai Fog

2012-01-01

In this paper we present the extended linear quadratic control problem, its efficient solution, and a discussion of how it arises in the numerical solution of nonlinear model predictive control problems. The extended linear quadratic control problem is the optimal control problem corresponding...... to the Karush-Kuhn-Tucker system that constitute the majority of computational work in constrained nonlinear and linear model predictive control problems solved by efficient MPC-tailored interior-point and active-set algorithms. We state various methods of solving the extended linear quadratic control problem...... and discuss instances in which it arises. The methods discussed in the paper have been implemented in efficient C code for both CPUs and GPUs for a number of test examples....

Computational aspects of linear control

CERN Document Server

2002-01-01

Many devices (we say dynamical systems or simply systems) behave like black boxes: they receive an input, this input is transformed following some laws (usually a differential equation) and an output is observed. The problem is to regulate the input in order to control the output, that is for obtaining a desired output. Such a mechanism, where the input is modified according to the output measured, is called feedback. The study and design of such automatic processes is called control theory. As we will see, the term system embraces any device and control theory has a wide variety of applications in the real world. Control theory is an interdisci­ plinary domain at the junction of differential and difference equations, system theory and statistics. Moreover, the solution of a control problem involves many topics of numerical analysis and leads to many interesting computational problems: linear algebra (QR, SVD, projections, Schur complement, structured matrices, localization of eigenvalues, computation of the...

Generalization of Samuelson's inequality and location of eigenvalues

Indian Academy of Sciences (India)

We prove a generalization of Samuelson's inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex × matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations.

Students' errors in solving linear equation word problems: Case ...

African Journals Online (AJOL)

The study examined errors students make in solving linear equation word problems with a view to expose the nature of these errors and to make suggestions for classroom teaching. A diagnostic test comprising 10 linear equation word problems, was administered to a sample (n=130) of senior high school first year Home ...

Lq-perturbations of leading coefficients of elliptic operators: Asymptotics of eigenvalues

Directory of Open Access Journals (Sweden)

Vladimir Kozlov

2006-01-01

Full Text Available We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L∞-norm.

Inverse problems in linear transport theory

International Nuclear Information System (INIS)

Dressler, K.

1988-01-01

Inverse problems for a class of linear kinetic equations are investigated. The aim is to identify the scattering kernel of a transport equation (corresponding to the structure of a background medium) by observing the 'albedo' part of the solution operator for the corresponding direct initial boundary value problem. This means to get information on some integral operator in an integrodifferential equation through on overdetermined boundary value problem. We first derive a constructive method for solving direct halfspace problems and prove a new factorization theorem for the solutions. Using this result we investigate stationary inverse problems with respect to well posedness (e.g. reduce them to classical ill-posed problems, such as integral equations of first kind). In the time-dependent case we show that a quite general inverse problem is well posed and solve it constructively. (orig.)

The linear ordering problem: an algorithm for the optimal solution ...

African Journals Online (AJOL)

In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such that the sum of the weights which are consistent with this order is as large as possible. It is an NP - Hard combinatorial optimisation ...

Beam dynamics problems for next generation linear colliders

International Nuclear Information System (INIS)

Yokoya, Kaoru

1990-01-01

The most critical issue for the feasibility of high-energy e + e - linear colliders is obviously the development of intense microwave power sources. Remaining problems, however, are not trivial and in fact some of them require several order-of-magnitude improvement from the existing SLC parameters. The present report summarizes the study status of the beam dynamics problems of high energy linear colliders with an exaggeration on the beam-beam phenomenon at the interaction region. There are four laboratories having linear collider plans, SLAC, CERN, Novosibirsk-Protovino, and KEK. The parameters of these projects scatter in some range but seem to converge slowly if one recalls the status five years ago. The beam energy will be below 500GeV. The basic requirements to the damping ring are the short damping time and small equilibrium emittance. All the proposed designs make use of tight focusing optics and strong wiggler magnets to meet these requirements and seem to have no major problems at least compared with other problems in the colliders. One of the major problems in the linac is the transverse beam blow-up due to the wake field created by the head of the bunch and, in the case of multiple bunches per pulse, by the preceeding bunches. (N.K.)

A Direct Algorithm for Pole Placement by State-derivative Feedback for Single-input Linear Systems

Directory of Open Access Journals (Sweden)

Taha H. S. Abdelaziz

2003-01-01

Full Text Available This paper deals with the direct solution of the pole placement problem for single-input linear systems using state-derivative feedback. This pole placement problem is always solvable for any controllable systems if all eigenvalues of the original system are nonzero. Then any arbitrary closed-loop poles can be placed in order to achieve the desired system performance. The solving procedure results in a formula similar to the Ackermann formula. Its derivation is based on the transformation of a linear single-input system into Frobenius canonical form by a special coordinate transformation, then solving the pole placement problem by state derivative feedback. Finally the solution is extended also for single-input time-varying control systems. The simulation results are included to show the effectiveness of the proposed approach.

Spectrum of the linearized operator for the Ginzburg-Landau equation

Directory of Open Access Journals (Sweden)

Tai-Chia Lin

2000-06-01

Full Text Available We study the spectrum of the linearized operator for the Ginzburg-Landau equation about a symmetric vortex solution with degree one. We show that the smallest eigenvalue of the linearized operator has multiplicity two, and then we describe its behavior as a small parameter approaches zero. We also find a positive lower bound for all the other eigenvalues, and find estimates of the first eigenfunction. Then using these results, we give partial results on the dynamics of vortices in the nonlinear heat and Schrodinger equations.

Damped oscillations of linear systems a mathematical introduction

CERN Document Server

Veselić, Krešimir

2011-01-01

The theory of linear damped oscillations was originally developed more than hundred years ago and is still of vital research interest to engineers, mathematicians and physicists alike. This theory plays a central role in explaining the stability of mechanical structures in civil engineering, but it also has applications in other fields such as electrical network systems and quantum mechanics. This volume gives an introduction to linear finite dimensional damped systems as they are viewed by an applied mathematician. After a short overview of the physical principles leading to the linear system model, a largely self-contained mathematical theory for this model is presented. This includes the geometry of the underlying indefinite metric space, spectral theory of J-symmetric matrices and the associated quadratic eigenvalue problem. Particular attention is paid to the sensitivity issues which influence numerical computations. Finally, several recent research developments are included, e.g. Lyapunov stability and ...

Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices

Energy Technology Data Exchange (ETDEWEB)

Khorunzhy, A. [Institute for Low Temperature Physics (Ukraine)], E-mail: khorunjy@ilt.kharkov.ua; Rodgers, G. J. [Brunel University, Uxbridge, Department of Mathematics and Statistics (United Kingdom)], E-mail: g.j.rodgers@brunel.ac.uk

2000-03-15

We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it.

Regularization Techniques for Linear Least-Squares Problems

KAUST Repository

Suliman, Mohamed

2016-04-01

Linear estimation is a fundamental branch of signal processing that deals with estimating the values of parameters from a corrupted measured data. Throughout the years, several optimization criteria have been used to achieve this task. The most astonishing attempt among theses is the linear least-squares. Although this criterion enjoyed a wide popularity in many areas due to its attractive properties, it appeared to suffer from some shortcomings. Alternative optimization criteria, as a result, have been proposed. These new criteria allowed, in one way or another, the incorporation of further prior information to the desired problem. Among theses alternative criteria is the regularized least-squares (RLS). In this thesis, we propose two new algorithms to find the regularization parameter for linear least-squares problems. In the constrained perturbation regularization algorithm (COPRA) for random matrices and COPRA for linear discrete ill-posed problems, an artificial perturbation matrix with a bounded norm is forced into the model matrix. This perturbation is introduced to enhance the singular value structure of the matrix. As a result, the new modified model is expected to provide a better stabilize substantial solution when used to estimate the original signal through minimizing the worst-case residual error function. Unlike many other regularization algorithms that go in search of minimizing the estimated data error, the two new proposed algorithms are developed mainly to select the artifcial perturbation bound and the regularization parameter in a way that approximately minimizes the mean-squared error (MSE) between the original signal and its estimate under various conditions. The first proposed COPRA method is developed mainly to estimate the regularization parameter when the measurement matrix is complex Gaussian, with centered unit variance (standard), and independent and identically distributed (i.i.d.) entries. Furthermore, the second proposed COPRA

Essential linear algebra with applications a problem-solving approach

CERN Document Server

Andreescu, Titu

2014-01-01

This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality. Key features include: • a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory;  • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance; • an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them.   Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course.    ...

Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction-diffusion systems with unilateral terms

Czech Academy of Sciences Publication Activity Database

Kučera, Milan; Navrátil, J.

2018-01-01

Roč. 166, January (2018), s. 154-180 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : global bifurcation * maximal eigenvalue * positively homogeneous operators Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.192, year: 2016 http://www.sciencedirect.com/science/article/pii/S0362546X17302559?via%3Dihub

Eigenvalue distributions of Wilson loops

International Nuclear Information System (INIS)

Lohmayer, Robert

2010-01-01

In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the modulus square of the characteristic polynomial. Furthermore

Separable expansion for realistic multichannel scattering problems

International Nuclear Information System (INIS)

Canton, L.; Cattapan, G.; Pisent, G.

1987-01-01

A new approach to the multichannel scattering problem with realistic local or nonlocal interactions is developed. By employing the negative-energy solutions of uncoupled Sturmian eigenvalue problems referring to simple auxiliary potentials, the coupling interactions appearing to the original multichannel problem are approximated by finite-rank potentials. By resorting to integral-equation tecniques the coupled-channel equations are then reduced to linear algebraic equations which can be straightforwardly solved. Compact algebraic expressions for the relevant scattering matrix elements are thus obtained. The convergence of the method is tasted in the single-channel case with realistic optical potentials. Excellent agreement is obtained with a few terms in the separable expansion for both real and absorptive interactions

The detection of influential subsets in linear regression using an influence matrix

OpenAIRE

Peña, Daniel; Yohai, Víctor J.

1991-01-01

This paper presents a new method to identify influential subsets in linear regression problems. The procedure uses the eigenstructure of an influence matrix which is defined as the matrix of uncentered covariance of the effect on the whole data set of deleting each observation, normalized to include the univariate Cook's statistics in the diagonal. It is shown that points in an influential subset will appear with large weight in at least one of the eigenvector linked to the largest eigenvalue...

Eigenvalues and bifurcation for problems with positively homogeneous operators and reaction-diffusion systems with unilateral terms

Czech Academy of Sciences Publication Activity Database

Kučera, Milan; Navrátil, J.

2018-01-01

Roč. 166, January (2018), s. 154-180 ISSN 0362-546X Institutional support: RVO:67985840 Keywords : global bifurcation * maximal eigenvalue * positively homogeneous operators Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 1.192, year: 2016 http://www. science direct.com/ science /article/pii/S0362546X17302559?via%3Dihub

WKB analysis of PT-symmetric Sturm–Liouville problems

International Nuclear Information System (INIS)

Bender, Carl M; Jones, Hugh F

2012-01-01

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the PT-symmetric Sturm–Liouville problem grow like n 2 for large n. However, the novelty is that a PT eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper)

Linear problems and Baecklund transformations for the Hirota-Ohta system

International Nuclear Information System (INIS)

Adler, V.E.; Postnikov, V.V.

2011-01-01

The auxiliary linear problems are presented for all discretization levels of the Hirota-Ohta system. The structure of these linear problems coincides essentially with the structure of Nonlinear Schroedinger hierarchy. The squared eigenfunction constraints are found which relate Hirota-Ohta and Kulish-Sklyanin vectorial NLS hierarchies.

An introduction to fuzzy linear programming problems theory, methods and applications

CERN Document Server

Kaur, Jagdeep

2016-01-01

The book presents a snapshot of the state of the art in the field of fully fuzzy linear programming. The main focus is on showing current methods for finding the fuzzy optimal solution of fully fuzzy linear programming problems in which all the parameters and decision variables are represented by non-negative fuzzy numbers. It presents new methods developed by the authors, as well as existing methods developed by others, and their application to real-world problems, including fuzzy transportation problems. Moreover, it compares the outcomes of the different methods and discusses their advantages/disadvantages. As the first work to collect at one place the most important methods for solving fuzzy linear programming problems, the book represents a useful reference guide for students and researchers, providing them with the necessary theoretical and practical knowledge to deal with linear programming problems under uncertainty.

Answers to selected problems in multivariable calculus with linear algebra and series

CERN Document Server

Trench, William F

1972-01-01

Answers to Selected Problems in Multivariable Calculus with Linear Algebra and Series contains the answers to selected problems in linear algebra, the calculus of several variables, and series. Topics covered range from vectors and vector spaces to linear matrices and analytic geometry, as well as differential calculus of real-valued functions. Theorems and definitions are included, most of which are followed by worked-out illustrative examples.The problems and corresponding solutions deal with linear equations and matrices, including determinants; vector spaces and linear transformations; eig

Iterative methods for the detection of Hopf bifurcations in finite element discretisation of incompressible flow problems

International Nuclear Information System (INIS)

Cliffe, K.A.; Garratt, T.J.; Spence, A.

1992-03-01

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalised eigenvalue problems arising from mixed finite element discretisations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and can be used in a scheme to determine the stability of steady state solutions and to detect Hopf bifurcations. We introduce a modified Cayley transform of the generalised eigenvalue problem which overcomes a drawback of the usual Cayley transform applied to such problems. Standard iterative methods are then applied to the transformed eigenvalue problem to compute approximations to the eigenvalue of smallest real part. Numerical experiments are performed using a model of double diffusive convection. (author)

Linear Programming and Its Application to Pattern Recognition Problems

Science.gov (United States)

Omalley, M. J.

1973-01-01

Linear programming and linear programming like techniques as applied to pattern recognition problems are discussed. Three relatively recent research articles on such applications are summarized. The main results of each paper are described, indicating the theoretical tools needed to obtain them. A synopsis of the author's comments is presented with regard to the applicability or non-applicability of his methods to particular problems, including computational results wherever given.

An Experiment of Robust Parallel Algorithm for the Eigenvalue problem of a Multigroup Neutron Diffusion based on modified FETI-DP

Energy Technology Data Exchange (ETDEWEB)

Chang, Jonghwa [Korea Atomic Energy Research Institute, Daejeon (Korea, Republic of)

2014-05-15

Parallelization of Monte Carlo simulation is widely adpoted. There are also several parallel algorithms developed for the SN transport theory using the parallel wave sweeping algorithm and for the CPM using parallel ray tracing. For practical purpose of reactor physics application, the thermal feedback and burnup effects on the multigroup cross section should be considered. In this respect, the domain decomposition method(DDM) is suitable for distributing the expensive cross section calculation work. Parallel transport code and diffusion code based on the Raviart-Thomas mixed finite element method was developed. However most of the developed methods rely on the heuristic convergence of flux and current at the domain interfaces. Convergence was not attained in some cases. Mechanical stress computation community has also work on the DDM to solve the stress-strain equation using the finite element methods. The most successful domain decomposition method in terms of robustness is FETI-DP. We have modified the original FETI-DP to solve the eigenvalue problem for the multigroup diffusion problem in this study.

On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.

Science.gov (United States)

Semenov, Yuri S; Bratus, Alexander S; Novozhilov, Artem S

2014-12-01

We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. Copyright © 2014 Elsevier Inc. All rights reserved.

Multisplitting for linear, least squares and nonlinear problems

Energy Technology Data Exchange (ETDEWEB)

Renaut, R.

1996-12-31

In earlier work, presented at the 1994 Iterative Methods meeting, a multisplitting (MS) method of block relaxation type was utilized for the solution of the least squares problem, and nonlinear unconstrained problems. This talk will focus on recent developments of the general approach and represents joint work both with Andreas Frommer, University of Wupertal for the linear problems and with Hans Mittelmann, Arizona State University for the nonlinear problems.

Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels

KAUST Repository

Haidar, Azzam

2011-01-01

This paper introduces a novel implementation in reducing a symmetric dense matrix to tridiagonal form, which is the preprocessing step toward solving symmetric eigenvalue problems. Based on tile algorithms, the reduction follows a two-stage approach, where the tile matrix is first reduced to symmetric band form prior to the final condensed structure. The challenging trade-off between algorithmic performance and task granularity has been tackled through a grouping technique, which consists of aggregating fine-grained and memory-aware computational tasks during both stages, while sustaining the application\\'s overall high performance. A dynamic runtime environment system then schedules the different tasks in an out-of-order fashion. The performance for the tridiagonal reduction reported in this paper is unprecedented. Our implementation results in up to 50-fold and 12-fold improvement (130 Gflop/s) compared to the equivalent routines from LAPACK V3.2 and Intel MKL V10.3, respectively, on an eight socket hexa-core AMD Opteron multicore shared-memory system with a matrix size of 24000×24000. Copyright 2011 ACM.

Solution of linear ill-posed problems using overcomplete dictionaries

OpenAIRE

Pensky, Marianna

2016-01-01

In the present paper we consider application of overcomplete dictionaries to solution of general ill-posed linear inverse problems. Construction of an adaptive optimal solution for such problems usually relies either on a singular value decomposition or representation of the solution via an orthonormal basis. The shortcoming of both approaches lies in the fact that, in many situations, neither the eigenbasis of the linear operator nor a standard orthonormal basis constitutes an appropriate co...

Eigenvalue distributions of Wilson loops

Energy Technology Data Exchange (ETDEWEB)

Lohmayer, Robert

2010-07-01

In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the

Cluster structure in the correlation coefficient matrix can be characterized by abnormal eigenvalues

Science.gov (United States)

Nie, Chun-Xiao

2018-02-01

In a large number of previous studies, the researchers found that some of the eigenvalues of the financial correlation matrix were greater than the predicted values of the random matrix theory (RMT). Here, we call these eigenvalues as abnormal eigenvalues. In order to reveal the hidden meaning of these abnormal eigenvalues, we study the toy model with cluster structure and find that these eigenvalues are related to the cluster structure of the correlation coefficient matrix. In this paper, model-based experiments show that in most cases, the number of abnormal eigenvalues of the correlation matrix is equal to the number of clusters. In addition, empirical studies show that the sum of the abnormal eigenvalues is related to the clarity of the cluster structure and is negatively correlated with the correlation dimension.

Asymmetric modes and complex time eigenvalues of the one-speed neutron transport equation in a homogeneous sphere

International Nuclear Information System (INIS)

Paranjape, S.D.; Kumar, V.; Sahni, D.C.

1993-01-01

The one-speed, time-dependent, isotropically scattering, integral transport equation in a homogeneous sphere has been converted into a criticality-like problem by considering exponential time behaviour of the scalar flux. This criticality problem has been converted into a matrix eigenvalue problem using the Fourier transform technique. The time eigenvalues λ, which are complex in general, have been determined for spherically symmetric as well as asymmetric modes. For the former case, the real decay constants and the real parts of complex decay constants decrease monotonically with increasing system size and form two distinct families of single-valued functions. For the spherically asymmetric modes, certain new features emerge. The real decay constants are found to be multi-valued functions of system size and they do not always decrease monotonically with increasing system size. As the system size increases from zero onwards, the decay constants alternate between complex and real values and the real and complex decay constant curves interlace. (Author)

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Science.gov (United States)

Böttcher, A.; Bogoya, J. M.; Grudsky, S. M.; Maximenko, E. A.

2017-11-01

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of the determinants of Toeplitz matrices are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009-2010. A survey of this new field is presented here. Bibliography: 55 titles.

Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides

Energy Technology Data Exchange (ETDEWEB)

Abdel-Rehim, A M; Stathopoulos, Andreas; Orginos, Kostas

2014-08-01

The technique that was used to build the EigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right-hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems and then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of EigBiCG to compute spectral approximations with accuracy comparable to that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental EigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right-hand sides.

Inverse Modelling Problems in Linear Algebra Undergraduate Courses

Science.gov (United States)

Martinez-Luaces, Victor E.

2013-01-01

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

An efficient method for generalized linear multiplicative programming problem with multiplicative constraints.

Science.gov (United States)

Zhao, Yingfeng; Liu, Sanyang

2016-01-01

We present a practical branch and bound algorithm for globally solving generalized linear multiplicative programming problem with multiplicative constraints. To solve the problem, a relaxation programming problem which is equivalent to a linear programming is proposed by utilizing a new two-phase relaxation technique. In the algorithm, lower and upper bounds are simultaneously obtained by solving some linear relaxation programming problems. Global convergence has been proved and results of some sample examples and a small random experiment show that the proposed algorithm is feasible and efficient.

Stability problems for linear hyperbolic systems

International Nuclear Information System (INIS)

Eckhoff, K.S.

1975-05-01

The stability properties for the trivial solution of a general linear hyperbolic system of partial differential equations of the first order are studied. It is shown that results may be obtained by studying the stability properties of certain systems of ordinary differential equations which can be constructed from the hyperbolic system (the so-called transport equations). In some cases the associated stability problem for the transport equations can in fact be shown to be equivalent to the stability problem for the hyperbolic system, but in general the transport equations will only give the necessary conditions for stability. (Auth.)

The non-linear Perron-Frobenius theorem : Perturbations and aggregation

NARCIS (Netherlands)

Dietzenbacher, E

The dominant eigenvalue and the corresponding eigenvector (or Perron vector) of a non-linear eigensystem are considered. We discuss the effects upon these, of perturbations and of aggregation of the underlying mapping. The results are applied to study the sensivity of the outputs in a non-linear

Eigenvalue for Densely Defined Perturbations of Multivalued Maximal Monotone Operators in Reflexive Banach Spaces

Directory of Open Access Journals (Sweden)

Boubakari Ibrahimou

2013-01-01

maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.

Efficient decomposition and linearization methods for the stochastic transportation problem

International Nuclear Information System (INIS)

Holmberg, K.

1993-01-01

The stochastic transportation problem can be formulated as a convex transportation problem with nonlinear objective function and linear constraints. We compare several different methods based on decomposition techniques and linearization techniques for this problem, trying to find the most efficient method or combination of methods. We discuss and test a separable programming approach, the Frank-Wolfe method with and without modifications, the new technique of mean value cross decomposition and the more well known Lagrangian relaxation with subgradient optimization, as well as combinations of these approaches. Computational tests are presented, indicating that some new combination methods are quite efficient for large scale problems. (authors) (27 refs.)

Periodic Solutions, Eigenvalue Curves, and Degeneracy of the Fractional Mathieu Equation

International Nuclear Information System (INIS)

Parra-Hinojosa, A; Gutiérrez-Vega, J C

2016-01-01

We investigate the eigenvalue curves, the behavior of the periodic solutions, and the orthogonality properties of the Mathieu equation with an additional fractional derivative term using the method of harmonic balance. The addition of the fractional derivative term breaks the hermiticity of the equation in such a way that its eigenvalues need not be real nor its eigenfunctions orthogonal. We show that for a certain choice of parameters the eigenvalue curves reveal the appearance of degenerate eigenvalues. We offer a detailed discussion of the matrix representation of the differential operator corresponding to the fractional Mathieu equation, as well as some numerical examples of its periodic solutions. (paper)

Aspects on increase and decrease within a national economy as eigenvalue problem of linear homogeneous equations

International Nuclear Information System (INIS)

Mueller, E.

2007-01-01

The paper presents an approach which treats topics of macroeconomics by methods familiar in physics and technology, especially in nuclear reactor technology and in quantum mechanics. Such methods are applied to simplified models for the money flows within a national economy, their variation in time and thereby for the annual national growth rate. As usual, money flows stand for economic activities. The money flows between the economic groups are described by a set of difference equations or by a set of approximative differential equations or eventually by a set of linear algebraic equations. Thus this paper especially deals with the time behaviour of model economies which are under the influence of imbalances and of delay processes, thereby dealing also with economic growth and recession rates. These differential equations are solved by a completely numerical Runge-Kutta algorithm. Case studies are presented for cases with 12 groups only and are to show the capability of the methods which have been worked out. (orig.)

Aspects on increase and decrease within a national economy as eigenvalue problem of linear homogeneous equations

Energy Technology Data Exchange (ETDEWEB)

Mueller, E.

2007-12-15

The paper presents an approach which treats topics of macroeconomics by methods familiar in physics and technology, especially in nuclear reactor technology and in quantum mechanics. Such methods are applied to simplified models for the money flows within a national economy, their variation in time and thereby for the annual national growth rate. As usual, money flows stand for economic activities. The money flows between the economic groups are described by a set of difference equations or by a set of approximative differential equations or eventually by a set of linear algebraic equations. Thus this paper especially deals with the time behaviour of model economies which are under the influence of imbalances and of delay processes, thereby dealing also with economic growth and recession rates. These differential equations are solved by a completely numerical Runge-Kutta algorithm. Case studies are presented for cases with 12 groups only and are to show the capability of the methods which have been worked out. (orig.)

The intelligence of dual simplex method to solve linear fractional fuzzy transportation problem.

Science.gov (United States)

Narayanamoorthy, S; Kalyani, S

2015-01-01

An approach is presented to solve a fuzzy transportation problem with linear fractional fuzzy objective function. In this proposed approach the fractional fuzzy transportation problem is decomposed into two linear fuzzy transportation problems. The optimal solution of the two linear fuzzy transportations is solved by dual simplex method and the optimal solution of the fractional fuzzy transportation problem is obtained. The proposed method is explained in detail with an example.

DESIGN OF EDUCATIONAL PROBLEMS ON LINEAR PROGRAMMING USING SYSTEMS OF COMPUTER MATHEMATICS

Directory of Open Access Journals (Sweden)

Volodymyr M. Mykhalevych

2013-11-01

Full Text Available From a perspective of the theory of educational problems a problem of substitution in the conditions of ICT use of one discipline by an educational problem of another discipline is represented. Through the example of mathematical problems of linear programming it is showed that a student’s method of operation in the course of an educational problem solving is determinant in the identification of an educational problem in relation to a specific discipline: linear programming, informatics, mathematical modeling, methods of optimization, automatic control theory, calculus etc. It is substantiated the necessity of linear programming educational problems renovation with the purpose of making students free of bulky similar arithmetic calculations and notes which often becomes a barrier to a deeper understanding of key ideas taken as a basis of algorithms used by them.

Identification problems in linear transformation system

International Nuclear Information System (INIS)

Delforge, Jacques.

1975-01-01

An attempt was made to solve the theoretical and numerical difficulties involved in the identification problem relative to the linear part of P. Delattre's theory of transformation systems. The theoretical difficulties are due to the very important problem of the uniqueness of the solution, which must be demonstrated in order to justify the value of the solution found. Simple criteria have been found when measurements are possible on all the equivalence classes, but the problem remains imperfectly solved when certain evolution curves are unknown. The numerical difficulties are of two kinds: a slow convergence of iterative methods and a strong repercussion of numerical and experimental errors on the solution. In the former case a fast convergence was obtained by transformation of the parametric space, while in the latter it was possible, from sensitivity functions, to estimate the errors, to define and measure the conditioning of the identification problem then to minimize this conditioning as a function of the experimental conditions [fr

Numerical computation of the linear stability of the diffusion model for crystal growth simulation

Energy Technology Data Exchange (ETDEWEB)

Yang, C.; Sorensen, D.C. [Rice Univ., Houston, TX (United States); Meiron, D.I.; Wedeman, B. [California Institute of Technology, Pasadena, CA (United States)

1996-12-31

We consider a computational scheme for determining the linear stability of a diffusion model arising from the simulation of crystal growth. The process of a needle crystal solidifying into some undercooled liquid can be described by the dual diffusion equations with appropriate initial and boundary conditions. Here U{sub t} and U{sub a} denote the temperature of the liquid and solid respectively, and {alpha} represents the thermal diffusivity. At the solid-liquid interface, the motion of the interface denoted by r and the temperature field are related by the conservation relation where n is the unit outward pointing normal to the interface. A basic stationary solution to this free boundary problem can be obtained by writing the equations of motion in a moving frame and transforming the problem to parabolic coordinates. This is known as the Ivantsov parabola solution. Linear stability theory applied to this stationary solution gives rise to an eigenvalue problem of the form.

The Intelligence of Dual Simplex Method to Solve Linear Fractional Fuzzy Transportation Problem

Directory of Open Access Journals (Sweden)

S. Narayanamoorthy

2015-01-01

Full Text Available An approach is presented to solve a fuzzy transportation problem with linear fractional fuzzy objective function. In this proposed approach the fractional fuzzy transportation problem is decomposed into two linear fuzzy transportation problems. The optimal solution of the two linear fuzzy transportations is solved by dual simplex method and the optimal solution of the fractional fuzzy transportation problem is obtained. The proposed method is explained in detail with an example.

Exploring linear algebra labs and projects with Mathematica

CERN Document Server

Arangala, Crista

2014-01-01

Matrix Operations Lab 0: An Introduction to Mathematica Lab 1: Matrix Basics and Operations Lab 2: A Matrix Representation of Linear Systems Lab 3: Powers, Inverses, and Special Matrices Lab 4: Graph Theory and Adjacency Matrices Lab 5: Permutations and Determinants Lab 6: 4 x 4 Determinants and Beyond Project Set 1 Invertibility Lab 7: Singular or Nonsingular? Why Singularity Matters Lab 8: Mod It Out, Matrices with Entries in ZpLab 9: It's a Complex World Lab 10: Declaring Independence: Is It Linear? Project Set 2 Vector Spaces Lab 11: Vector Spaces and SubspacesLab 12: Basing It All on Just a Few Vectors Lab 13: Linear Transformations Lab 14: Eigenvalues and Eigenspaces Lab 15: Markov Chains, An Application of Eigenvalues Project Set 3 Orthogonality Lab 16: Inner Product Spaces Lab 17: The Geometry of Vector and Inner Product SpacesLab 18: Orthogonal Matrices, QR Decomposition, and Least Squares Regression Lab 19: Symmetric Matrices and Quadratic Forms Project Set 4 Matrix Decomposition with Applications L...

Linear algebra a first course with applications

CERN Document Server

Knop, Larry E

2008-01-01

Linear Algebra: A First Course with Applications explores the fundamental ideas of linear algebra, including vector spaces, subspaces, basis, span, linear independence, linear transformation, eigenvalues, and eigenvectors, as well as a variety of applications, from inventories to graphics to Google's PageRank. Unlike other texts on the subject, this classroom-tested book gives students enough time to absorb the material by focusing on vector spaces early on and using computational sections as numerical interludes. It offers introductions to Maple™, MATLAB®, and TI-83 Plus for calculating matri

Linear decomposition approach for a class of nonconvex programming problems.

Science.gov (United States)

Shen, Peiping; Wang, Chunfeng

2017-01-01

This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem can be transformed and decomposed into a polynomial number of equivalent linear programming subproblems. Based on solving a series of liner programming subproblems corresponding to those grid points we can obtain the near-optimal solution of the original problem. Compared to existing results in the literature, the proposed algorithm does not require the assumptions of quasi-concavity and differentiability of the objective function, and it differs significantly giving an interesting approach to solving the problem with a reduced running time.

Experiences with linear solvers for oil reservoir simulation problems

Energy Technology Data Exchange (ETDEWEB)

Joubert, W.; Janardhan, R. [Los Alamos National Lab., NM (United States); Biswas, D.; Carey, G.

1996-12-31

This talk will focus on practical experiences with iterative linear solver algorithms used in conjunction with Amoco Production Company`s Falcon oil reservoir simulation code. The goal of this study is to determine the best linear solver algorithms for these types of problems. The results of numerical experiments will be presented.

Automorphisms of foliations defined by complex linear vectorfields

International Nuclear Information System (INIS)

Shahshahani, S.

1989-06-01

We study biholomorphisms of C n that preserve the foliation associated with a complex linear vectorfield. It is shown that for a large class the only such biholomorphisms are linear conjugacies of the flow. Nonlinear biholomorphisms emerge in the presence of resonance among the eigenvalues of the system. A complete classification is given in dimension two. (author). 8 refs

Positive Eigenvalues of Generalized Words in Two Hermitian Positive Definite Matrices

OpenAIRE

Hillar, Christopher; Johnson, Charles R.

2005-01-01

We define a word in two positive definite (complex Hermitian) matrices $A$ and $B$ as a finite product of real powers of $A$ and $B$. The question of which words have only positive eigenvalues is addressed. This question was raised some time ago in connection with a long-standing problem in theoretical physics, and it was previously approached by the authors for words in two real positive definite matrices with positive integral exponents. A large class of words that do guarantee positive eig...

Advanced Variance Reduction for Global k-Eigenvalue Simulations in MCNP

Energy Technology Data Exchange (ETDEWEB)

Edward W. Larsen

2008-06-01

to the correlations between fission source estimates. In the new FMC method, the eigenvalue problem (expressed in terms of the Boltzmann equation) is integrated over the energy and direction variables. Then these equations are multiplied by J special "tent" functions in space and integrated over the spatial variable. This yields J equations that are exactly satisfied by the eigenvalue k and J space-angle-energy moments of the eigenfunction. Multiplying and dividing by suitable integrals of the eigenfunction, one obtains J algebraic equations for k and the space-angle-energy moments of the eigenfunction, which contain nonlinear functionals that depend weakly on the eigenfunction. In the FMC method, information from the standard Monte Carlo solution for each active cycle is used to estimate the functionals, and at the end of each cycle the J equations for k and the space-angle-energy moments of the eigenfunction are solved. Finally, these results are averaged over N active cycles to obtain estimated means and standard deviations for k and the space-angle-energy moments of the eigenfunction. Our limited testing shows that for large single fissile systems such as a commercial reactor core, (i) the FMC estimate of the eigenvalue is at least one order of magnitude more accurate than estimates obtained from the standard Monte Carlo approach, (ii) the FMC estimate of the eigenfunction converges and is several orders of magnitude more accurate than the standard estimate, and (iii) the FMC estimate of the standard deviation in k is at least one order of magnitude closer to the correct standard deviation than the standard estimate. These advances occur because: (i) the Monte Carlo estimates of the nonlinear functionals are much more accurate than the direct Monte Carlo estimates of the eigenfunction, (ii) the system of discrete equations that determines the FMC estimates of k is robust, and (iii) the functionals are only very weakly correlated between different fission

Spectral inversion of an indefinite Sturm-Liouville problem due to Richardson

International Nuclear Information System (INIS)

Shanley, Paul E

2009-01-01

We study an indefinite Sturm-Liouville problem due to Richardson whose complicated eigenvalue dependence on a parameter has been a puzzle for decades. In atomic physics a process exists that inverts the usual Schroedinger situation of an energy eigenvalue depending on a coupling parameter into the so-called Sturmian problem where the coupling parameter becomes the eigenvalue which then depends on the energy. We observe that the Richardson equation is of the Sturmian type. This means that the Richardson and its related Schroedinger eigenvalue functions are inverses of each other and that the Richardson spectrum is therefore no longer a puzzle

On a non-linear pseudodifferential boundary value problem

International Nuclear Information System (INIS)

Nguyen Minh Chuong.

1989-12-01

A pseudodifferential boundary value problem for operators with symbols taking values in Sobolev spaces and with non-linear right-hand side was studied. Existence and uniqueness theorems were proved. (author). 11 refs

Linear differential equations to solve nonlinear mechanical problems: A novel approach

OpenAIRE

Nair, C. Radhakrishnan

2004-01-01

Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of the linear differe...

Linear algebra meets Lie algebra: the Kostant-Wallach theory

OpenAIRE

Shomron, Noam; Parlett, Beresford N.

2008-01-01

In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.

Sensitivity analysis of linear programming problem through a recurrent neural network

Science.gov (United States)

Das, Raja

2017-11-01

In this paper we study the recurrent neural network for solving linear programming problems. To achieve optimality in accuracy and also in computational effort, an algorithm is presented. We investigate the sensitivity analysis of linear programming problem through the neural network. A detailed example is also presented to demonstrate the performance of the recurrent neural network.

Energy eigenvalues of helium-like atoms in dense plasmas

International Nuclear Information System (INIS)

Hashino, Tasuke; Nakazaki, Shinobu; Kato, Takako; Kashiwabara, Hiromichi.

1987-04-01

Calculations based on a variational method with wave functions including the correlation of electrons are carried out to obtain energy eigenvalues of Schroedinger's equation for helium-like atoms embedded in dense plasmas, taking the Debye-Hueckel approximation. Energy eigenvalues for the 1 1 S, 2 1 S, and 2 3 S states are obtained as a function of Debye screening length. (author)

Eigenvalues of the volume operator in loop quantum gravity

International Nuclear Information System (INIS)

Meissner, Krzysztof A

2006-01-01

We present a simple method to calculate certain sums of the eigenvalues of the volume operator in loop quantum gravity. We derive the asymptotic distribution of the eigenvalues in the classical limit of very large spins, which turns out to be of a very simple form. The results can be useful for example in the statistical approach to quantum gravity

On the completeness of systems of eigenfunctions of the Sturm-Liouville operator with a potential depending on the spectral parameter and a nonlinear problem

International Nuclear Information System (INIS)

Zhidkov, P.E.

1996-01-01

First, the eigenvalue problem on the segment [0,1] for the Sturm-Liouville operator with a potential depending on the spectral parameter with the zero Dirichlet boundary conditions is considered. For this problem, under some hypotheses on the potential, it is proved that the necessary and sufficient condition for an arbitrary system of eigenfunctions, possessing a unique function with n roots in the interval (0,1) for an arbitrary non-negative integer number n, being complete in the space L 2 (0,1) is the linear independence of the functions from this system in the space L 2 (0,1). Then, this result is applied to the investigation of an eigenvalue problem for a nonlinear operator on the Sturm-Liouville type. For this problem, the completeness of the system of its eigenfunctions in the space L 2 (0,1) is proved. (author). 12 refs

Three-dimensional multiple reciprocity boundary element method for one-group neutron diffusion eigenvalue computations

International Nuclear Information System (INIS)

Itagaki, Masafumi; Sahashi, Naoki.

1996-01-01

The multiple reciprocity method (MRM) in conjunction with the boundary element method has been employed to solve one-group eigenvalue problems described by the three-dimensional (3-D) neutron diffusion equation. The domain integral related to the fission source is transformed into a series of boundary-only integrals, with the aid of the higher order fundamental solutions based on the spherical and the modified spherical Bessel functions. Since each degree of the higher order fundamental solutions in the 3-D cases has a singularity of order (1/r), the above series of boundary integrals requires additional terms which do not appear in the 2-D MRM formulation. The critical eigenvalue itself can be also described using only boundary integrals. Test calculations show that Wielandt's spectral shift technique guarantees rapid and stable convergence of 3-D MRM computations. (author)

Analysis of the efficiency of the linearization techniques for solving multi-objective linear fractional programming problems by goal programming

Directory of Open Access Journals (Sweden)

Tunjo Perić

2017-01-01

Full Text Available This paper presents and analyzes the applicability of three linearization techniques used for solving multi-objective linear fractional programming problems using the goal programming method. The three linearization techniques are: (1 Taylor’s polynomial linearization approximation, (2 the method of variable change, and (3 a modification of the method of variable change proposed in [20]. All three linearization techniques are presented and analyzed in two variants: (a using the optimal value of the objective functions as the decision makers’ aspirations, and (b the decision makers’ aspirations are given by the decision makers. As the criteria for the analysis we use the efficiency of the obtained solutions and the difficulties the analyst comes upon in preparing the linearization models. To analyze the applicability of the linearization techniques incorporated in the linear goal programming method we use an example of a financial structure optimization problem.

Wielandt acceleration for MCNP5 Monte Carlo eigenvalue calculations

International Nuclear Information System (INIS)

Brown, F.

2007-01-01

Monte Carlo criticality calculations use the power iteration method to determine the eigenvalue (k eff ) and eigenfunction (fission source distribution) of the fundamental mode. A recently proposed method for accelerating convergence of the Monte Carlo power iteration using Wielandt's method has been implemented in a test version of MCNP5. The method is shown to provide dramatic improvements in convergence rates and to greatly reduce the possibility of false convergence assessment. The method is effective and efficient, improving the Monte Carlo figure-of-merit for many problems. In addition, the method should eliminate most of the underprediction bias in confidence intervals for Monte Carlo criticality calculations. (authors)

Non-linear analytic and coanalytic problems (Lp-theory, Clifford analysis, examples)

International Nuclear Information System (INIS)

Dubinskii, Yu A; Osipenko, A S

2000-01-01

Two kinds of new mathematical model of variational type are put forward: non-linear analytic and coanalytic problems. The formulation of these non-linear boundary-value problems is based on a decomposition of the complete scale of Sobolev spaces into the 'orthogonal' sum of analytic and coanalytic subspaces. A similar decomposition is considered in the framework of Clifford analysis. Explicit examples are presented

An Approximate Proximal Bundle Method to Minimize a Class of Maximum Eigenvalue Functions

Directory of Open Access Journals (Sweden)

Wei Wang

2014-01-01

Full Text Available We present an approximate nonsmooth algorithm to solve a minimization problem, in which the objective function is the sum of a maximum eigenvalue function of matrices and a convex function. The essential idea to solve the optimization problem in this paper is similar to the thought of proximal bundle method, but the difference is that we choose approximate subgradient and function value to construct approximate cutting-plane model to solve the above mentioned problem. An important advantage of the approximate cutting-plane model for objective function is that it is more stable than cutting-plane model. In addition, the approximate proximal bundle method algorithm can be given. Furthermore, the sequences generated by the algorithm converge to the optimal solution of the original problem.

Transport properties of a piecewise linear transformation and deterministic Levy flights

International Nuclear Information System (INIS)

Miyaguchi, Tomoshige

2006-01-01

The transport properties of a 1-dimensional piecewise linear dynamical system are investigated through the spectrum of its Frobenius-Perron operator. For a class of initial densities, eigenvalues and eigenfunctions of the Frobenius-Perron operator are obtained explicitly. It is also found that in the long length wave limit, this system exhibits normal diffusion and super diffusion called Levy flight. The diffusion constant and stable index are derived from the eigenvalues. (author)

Absence of positive eigenvalues for hard-core N-body systems

DEFF Research Database (Denmark)

Ito, K.; Skibsted, Erik

We show absence of positive eigenvalues for generalized 2-body hard-core Schrödinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N-body hard-core Schrödinger operators, N≥ 2, is presented. This scheme inv...

Adomian decomposition method for nonlinear Sturm-Liouville problems

Directory of Open Access Journals (Sweden)

Sennur Somali

2007-09-01

Full Text Available In this paper the Adomian decomposition method is applied to the nonlinear Sturm-Liouville problem-y" + y(tp=λy(t, y(t > 0, t ∈ I = (0, 1, y(0 = y(1 = 0, where p > 1 is a constant and λ > 0 is an eigenvalue parameter. Also, the eigenvalues and the behavior of eigenfuctions of the problem are demonstrated.

On a linear-quadratic problem with Caputo derivative

Directory of Open Access Journals (Sweden)

Dariusz Idczak

2016-01-01

Full Text Available In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration.

Iterative approach for the eigenvalue problems

Indian Academy of Sciences (India)

the Schrödinger equation for the energy levels with a class of confining potentials [3] using Kato–Rellich ... Moreover,. QES problem has its own inner mathematical beauty – it can provide a good starting point for doing ... this technique for calculating the first- and second-order corrections for the ground state as well as the ...

The Cauchy problem for non-linear Klein-Gordon equations

International Nuclear Information System (INIS)

Simon, J.C.H.; Taflin, E.

1993-01-01

We consider in R n+1 , n≥2, the non-linear Klein-Gordon equation. We prove for such an equation that there is neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare-Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincare group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra. (orig.)

Application of linear and higher perturbation theory in reactor physics

International Nuclear Information System (INIS)

Woerner, D.

1978-01-01

For small perturbations in the material composition of a reactor according to the first approximation of perturbation theory the eigenvalue perturbation is proportional to the perturbation of the system. This assumption is true for the neutron flux not influenced by the perturbance. The two-dimensional code LINESTO developed for such problems in this paper on the basis of diffusion theory determines the relative change of the multiplication constant. For perturbations varying the neutron flux in the space of energy and position the eigenvalue perturbation is also influenced by this changed neutron flux. In such cases linear perturbation theory yields larger errors. Starting from the methods of calculus of variations there is additionally developed in this paper a perturbation method of calculation permitting in a quick and simple manner to assess the influence of flux perturbation on the eigenvalue perturbation. While the source of perturbations is evaluated in isotropic approximation of diffusion theory the associated inhomogeneous equation may be used to determine the flux perturbation by means of diffusion or transport theory. Possibilities of application and limitations of this method are studied in further systematic investigations on local perturbations. It is shown that with the integrated code system developed in this paper a number of local perturbations may be checked requiring little computing time. With it flux perturbations in first approximation and perturbations of the multiplication constant in second approximation can be evaluated. (orig./RW) [de

Investigation of cellular detonation structure formation via linear stability theory and 2D and 3D numerical simulations

Science.gov (United States)

Borisov, S. P.; Kudryavtsev, A. N.

2017-10-01

Linear and nonlinear stages of the instability of a plane detonation wave (DW) and the subsequent process of formation of cellular detonation structure are investigated. A simple model with one-step irreversible chemical reaction is used. The linear analysis is employed to predict the DW front structure at the early stages of its formation. An emerging eigenvalue problem is solved with a global method using a Chebyshev pseudospectral method and the LAPACK software library. A local iterative shooting procedure is used for eigenvalue refinement. Numerical simulations of a propagation of a DW in plane and rectangular channels are performed with a shock capturing WENO scheme of 5th order. A special method of a computational domain shift is implemented in order to maintain the DW in the domain. It is shown that the linear analysis gives certain predictions about the DW structure that are in agreement with the numerical simulations of early stages of DW propagation. However, at later stages, a merger of detonation cells occurs so that their number is approximately halved. Computations of DW propagation in a square channel reveal two different types of spatial structure of the DW front, "rectangular" and "diagonal" types. A spontaneous transition from the rectangular to diagonal type of structure is observed during propagation of the DW.

Thick-Restart Lanczos Method for Electronic Structure Calculations

International Nuclear Information System (INIS)

Simon, Horst D.; Wang, L.-W.; Wu, Kesheng

1999-01-01

This paper describes two recent innovations related to the classic Lanczos method for eigenvalue problems, namely the thick-restart technique and dynamic restarting schemes. Combining these two new techniques we are able to implement an efficient eigenvalue problem solver. This paper will demonstrate its effectiveness on one particular class of problems for which this method is well suited: linear eigenvalue problems generated from non-self-consistent electronic structure calculations

Students' errors in solving linear equation word problems: Case ...

African Journals Online (AJOL)

kofi.mereku

Development in most areas of life is based on effective knowledge of science and ... Problem solving, as used in mathematics education literature, refers ... word problems, on the other hand, are those linear equation tasks or ... taught LEWPs in the junior high school, many of them reach the senior high school without a.

An algorithm of α-and γ-mode eigenvalue calculations by Monte Carlo method

International Nuclear Information System (INIS)

Yamamoto, Toshihiro; Miyoshi, Yoshinori

2003-01-01

A new algorithm for Monte Carlo calculation was developed to obtain α- and γ-mode eigenvalues. The α is a prompt neutron time decay constant measured in subcritical experiments, and the γ is a spatial decay constant measured in an exponential method for determining the subcriticality. This algorithm can be implemented into existing Monte Carlo eigenvalue calculation codes with minimum modifications. The algorithm was implemented into MCNP code and the performance of calculating the both mode eigenvalues were verified through comparison of the calculated eigenvalues with the ones obtained by fixed source calculations. (author)

The spectral problem of global microinstabilities in tokamak-like plasmas using a gyrokinetic model

International Nuclear Information System (INIS)

Brunner, S.; Vaclavik, J.; Fivaz, M.; Appert, K.

1996-01-01

Tokamak-like plasmas are modeled by a periodic cylindrical system with magnetic shear and realistic density and temperature profiles. Linear electrostatic microinstabilities in such plasmas are studied by solving the eigenvalue problem starting from gyrokinetic theory. The actual eigenvalue equation is then of integral type. With this approach, finite Larmor radius (FLR) effects to all orders are taken into account. FLR effects provide for the only radial coupling in a cylinder and to lowest order correspond to polarization drift. This effectively one-dimensional problem helped us to gain useful knowledge for solving gyrokinetic equations in a curved system. When searching for the eigenfrequencies of the global modes, two different methods have been tested and compared. Either the true eigenvalue problem is solved by finding the zeros of the characteristic equation, or one considers a system driven by an antenna and looks for resonances in the power response of the plasma. In addition, mode structures were computed as well in direct as in Fourier space. The advantages and disadvantages of these various approaches are discussed. Ion temperature gradient (ITG) instabilities are studied over a wide range of parameters and for wavelengths perpendicular to the magnetic field down to the scale of ion Larmor radii. Flute instabilities driven by magnetic curvature drifts are also considered. Some of these results are compared with a time evolution PIC code. Such comparisons are valuable as the convergence of PIC results is often questioned. Work considering true toroidal geometry is in progress

Solving Eigenvalue response matrix equations with Jacobian-Free Newton-Krylov methods

International Nuclear Information System (INIS)

Roberts, Jeremy A.; Forget, Benoit

2011-01-01

The response matrix method for reactor eigenvalue problems is motivated as a technique for solving coarse mesh transport equations, and the classical approach of power iteration (PI) for solution is described. The method is then reformulated as a nonlinear system of equations, and the associated Jacobian is derived. A Jacobian-Free Newton-Krylov (JFNK) method is employed to solve the system, using an approximate Jacobian coupled with incomplete factorization as a preconditioner. The unpreconditioned JFNK slightly outperforms PI, and preconditioned JFNK outperforms both PI and Steffensen-accelerated PI significantly. (author)

Computation of Double Eigenvalues for Infinite Matrices of a Certain Class

OpenAIRE

宮崎, 佳典; Yoshinori, MIYAZAKI; 静岡産業大学 国際情報学部; Faculty of Communications and Informatics, Shizuoka Sangyo University

2001-01-01

It has been shown that a series of three-term recurrence relations of a certain class is a powerful tool for solving zeros of some special functions and eigenvalue problems (EVPs) of certain differential equations. Such cases include: the zeros of J_v(z); the zeros of zJ′_v(z)+HJ_v(z); the EVP of the Mathieu differential equation; and the EVP of the spheroidal wave equation. Previously by the author, it was demonstrated that the three-term recurrence relations of the class may be reformulated...

Eigenvalue problem and nonlinear evolution of kink modes in a toroidal plasma

International Nuclear Information System (INIS)

Ogino, T.; Takeda, S.; Sanuki, H.; Kamimura, T.

1979-04-01

The internal kink modes of a cylindrical plasma are investigated by a linear eigen value problem and their nonlinear evolution is studied by 3-dimensional MHD simulation based on the rectangular column model under the fixed boundary condition. The growth rates in two cases, namely uniform and diffused current profiles are analyzed in detail, which agree with the analytical estimation by Shafranov. The time evolution of the m = 1 mode showed that q > 1 is satisfied at the relaxation time (q safety factor), a stable configuration like a horse shoe (a new equilibrium) was formed. Also, the time evolution of the pressure p for the m = 2 mode showed that a stable configuration like a pair of anchors was formed. (author)

A property of assignment type mixed integer linear programming problems

NARCIS (Netherlands)

Benders, J.F.; van Nunen, J.A.E.E.

1982-01-01

In this paper we will proof that rather tight upper bounds can be given for the number of non-unique assignments that are achieved after solving the linear programming relaxation of some types of mixed integer linear assignment problems. Since in these cases the number of splitted assignments is

Health physics problems encountered in the Saclay linear accelerator

International Nuclear Information System (INIS)

Delsaut, R.

1979-01-01

The safety and health physics problems specific to the Saclay linear accelerator are presented: activation (of gases, dust, water, structural materials, targets); individual dosimetry; the safety engineering [fr

Invariant imbedding equations for linear scattering problems

International Nuclear Information System (INIS)

Apresyan, L.

1988-01-01

A general form of the invariant imbedding equations is investigated for the linear problem of scattering by a bounded scattering volume. The conditions for the derivability of such equations are described. It is noted that the possibility of the explicit representation of these equations for a sphere and for a layer involves the separation of variables in the unperturbed wave equation

Linear operator pencils on Lie algebras and Laurent biorthogonal polynomials

International Nuclear Information System (INIS)

Gruenbaum, F A; Vinet, Luc; Zhedanov, Alexei

2004-01-01

We study operator pencils on generators of the Lie algebras sl 2 and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss 2 F 1 and degenerate 1 F 1 hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen-van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi-Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered

Recurrence quantity analysis based on matrix eigenvalues

Science.gov (United States)

Yang, Pengbo; Shang, Pengjian

2018-06-01

Recurrence plots is a powerful tool for visualization and analysis of dynamical systems. Recurrence quantification analysis (RQA), based on point density and diagonal and vertical line structures in the recurrence plots, is considered to be alternative measures to quantify the complexity of dynamical systems. In this paper, we present a new measure based on recurrence matrix to quantify the dynamical properties of a given system. Matrix eigenvalues can reflect the basic characteristics of the complex systems, so we show the properties of the system by exploring the eigenvalues of the recurrence matrix. Considering that Shannon entropy has been defined as a complexity measure, we propose the definition of entropy of matrix eigenvalues (EOME) as a new RQA measure. We confirm that EOME can be used as a metric to quantify the behavior changes of the system. As a given dynamical system changes from a non-chaotic to a chaotic regime, the EOME will increase as well. The bigger EOME values imply higher complexity and lower predictability. We also study the effect of some factors on EOME,including data length, recurrence threshold, the embedding dimension, and additional noise. Finally, we demonstrate an application in physiology. The advantage of this measure lies in a high sensitivity and simple computation.

Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

OpenAIRE

Nakamura, Gen; Vashisth, Manmohan

2017-01-01

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\\sqrt{\\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotr...

Method for solving fully fuzzy linear programming problems using deviation degree measure

Institute of Scientific and Technical Information of China (English)

Haifang Cheng; Weilai Huang; Jianhu Cai

2013-01-01

A new ful y fuzzy linear programming (FFLP) prob-lem with fuzzy equality constraints is discussed. Using deviation degree measures, the FFLP problem is transformed into a crispδ-parametric linear programming (LP) problem. Giving the value of deviation degree in each constraint, the δ-fuzzy optimal so-lution of the FFLP problem can be obtained by solving this LP problem. An algorithm is also proposed to find a balance-fuzzy optimal solution between two goals in conflict: to improve the va-lues of the objective function and to decrease the values of the deviation degrees. A numerical example is solved to il ustrate the proposed method.

Fundamental solution of the problem of linear programming and method of its determination

Science.gov (United States)

Petrunin, S. V.

1978-01-01

The idea of a fundamental solution to a problem in linear programming is introduced. A method of determining the fundamental solution and of applying this method to the solution of a problem in linear programming is proposed. Numerical examples are cited.

An Entropic Estimator for Linear Inverse Problems

Directory of Open Access Journals (Sweden)

Amos Golan

2012-05-01

Full Text Available In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknown parameters (when such information exists, and constraints on the parameters can be incorporated in the statement of the problem. The method builds on the basics of the maximum entropy principle and consists of transforming the original problem into an estimation of a probability density on an appropriate space naturally associated with the statement of the problem. This estimation method is generic in the sense that it provides a framework for analyzing non-normal models, it is easy to implement and is suitable for all types of inverse problems such as small and or ill-conditioned, noisy data. First order approximation, large sample properties and convergence in distribution are developed as well. Analytical examples, statistics for model comparisons and evaluations, that are inherent to this method, are discussed and complemented with explicit examples.

On the automorphisms of foliations defined by complex linear vectorfields

International Nuclear Information System (INIS)

Shahshahani, S.

1989-04-01

We study biholomorphisms of C n that preserve the foliation associated to a complex linear vector fields. It is shown that for hyperbolic Poincare vector fields the only such biholomorphisms are linear. Nonlinear biholomorphisms emerge in the presence of resonance among the eigenvalues of the system. A complete classification is given in dimension 2. (author). 8 refs, 1 fig

The Guderley problem revisited

International Nuclear Information System (INIS)

Ramsey, Scott D.; Kamm, James R.; Bolstad, John H.

2009-01-01

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of numerous investigations since its publication. In this paper, we review the simplifications and group invariance properties that lead to a self-similar formulation of this problem from the compressible flow equations for a polytropic gas. The complete solution to the self-similar problem reduces to two coupled nonlinear eigenvalue problems: the eigenvalue of the first is the so-called similarity exponent for the converging flow, and that of the second is a trajectory multiplier for the diverging regime. We provide a clear exposition concerning the reflected shock configuration. Additionally, we introduce a new approximation for the similarity exponent, which we compare with other estimates and numerically computed values. Lastly, we use the Guderley problem as the basis of a quantitative verification analysis of a cell-centered, finite volume, Eulerian compressible flow algorithm.

Formulated linear programming problems from game theory and its ...

African Journals Online (AJOL)

Formulated linear programming problems from game theory and its computer implementation using Tora package. ... Game theory, a branch of operations research examines the various concepts of decision ... AJOL African Journals Online.

A Fast Condensing Method for Solution of Linear-Quadratic Control Problems

DEFF Research Database (Denmark)

Frison, Gianluca; Jørgensen, John Bagterp

2013-01-01

consider a condensing (or state elimination) method to solve an extended version of the LQ control problem, and we show how to exploit the structure of this problem to both factorize the dense Hessian matrix and solve the system. Furthermore, we present two efficient implementations. The first......In both Active-Set (AS) and Interior-Point (IP) algorithms for Model Predictive Control (MPC), sub-problems in the form of linear-quadratic (LQ) control problems need to be solved at each iteration. The solution of these sub-problems is usually the main computational effort. In this paper we...... implementation is formally identical to the Riccati recursion based solver and has a computational complexity that is linear in the control horizon length and cubic in the number of states. The second implementation has a computational complexity that is quadratic in the control horizon length as well...

Using deflation in the pole assignment problem with output feedback

Science.gov (United States)

Miminis, George

1989-01-01

A direct algorithm is suggested for the computation of a linear output feedback for a multi input, multi output system such that the resultant closed-loop matrix has eigenvalues that include a specified set of eigenvalues. The algorithm uses deflation based on unitary similarity transformations. Thus researchers hope the algorithm is numerically stable; however, this has not been proven as yet.

Non-linear analytic and coanalytic problems ( L_p-theory, Clifford analysis, examples)

Science.gov (United States)

Dubinskii, Yu A.; Osipenko, A. S.

2000-02-01

Two kinds of new mathematical model of variational type are put forward: non-linear analytic and coanalytic problems. The formulation of these non-linear boundary-value problems is based on a decomposition of the complete scale of Sobolev spaces into the "orthogonal" sum of analytic and coanalytic subspaces. A similar decomposition is considered in the framework of Clifford analysis. Explicit examples are presented.

Eigenstructures of MIMO Fading Channel Correlation Matrices and Optimum Linear Precoding Designs for Maximum Ergodic Capacity

Directory of Open Access Journals (Sweden)

Hamid Reza Bahrami

2007-01-01

Full Text Available The ergodic capacity of MIMO frequency-flat and -selective channels depends greatly on the eigenvalue distribution of spatial correlation matrices. Knowing the eigenstructure of correlation matrices at the transmitter is very important to enhance the capacity of the system. This fact becomes of great importance in MIMO wireless systems where because of the fast changing nature of the underlying channel, full channel knowledge is difficult to obtain at the transmitter. In this paper, we first investigate the effect of eigenvalues distribution of spatial correlation matrices on the capacity of frequency-flat and -selective channels. Next, we introduce a practical scheme known as linear precoding that can enhance the ergodic capacity of the channel by changing the eigenstructure of the channel by applying a linear transformation. We derive the structures of precoders using eigenvalue decomposition and linear algebra techniques in both cases and show their similarities from an algebraic point of view. Simulations show the ability of this technique to change the eigenstructure of the channel, and hence enhance the ergodic capacity considerably.

Estimates of the first Dirichlet eigenvalue from exit time moment spectra

DEFF Research Database (Denmark)

Hurtado, Ana; Markvorsen, Steen; Palmer, Vicente

2013-01-01

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball...

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

International Nuclear Information System (INIS)

Egozcue, J.; Meziat, R.; Pedregal, P.

2002-01-01

We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature

Particle swarm optimization - Genetic algorithm (PSOGA) on linear transportation problem

Science.gov (United States)

Rahmalia, Dinita

2017-08-01

Linear Transportation Problem (LTP) is the case of constrained optimization where we want to minimize cost subject to the balance of the number of supply and the number of demand. The exact method such as northwest corner, vogel, russel, minimal cost have been applied at approaching optimal solution. In this paper, we use heurisitic like Particle Swarm Optimization (PSO) for solving linear transportation problem at any size of decision variable. In addition, we combine mutation operator of Genetic Algorithm (GA) at PSO to improve optimal solution. This method is called Particle Swarm Optimization - Genetic Algorithm (PSOGA). The simulations show that PSOGA can improve optimal solution resulted by PSO.

An Optimal Lower Eigenvalue System

Directory of Open Access Journals (Sweden)

Yingfan Liu

2011-01-01

Full Text Available An optimal lower eigenvalue system is studied, and main theorems including a series of necessary and suffcient conditions concerning existence and a Lipschitz continuity result concerning stability are obtained. As applications, solvability results to some von-Neumann-type input-output inequalities, growth, and optimal growth factors, as well as Leontief-type balanced and optimal balanced growth paths, are also gotten.

On two-spectra inverse problems

OpenAIRE

Guliyev, Namig J.

2018-01-01

We consider a two-spectra inverse problem for the one-dimensional Schr\\"{o}dinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this problem.

Paradox in a non-linear capacitated transportation problem

Directory of Open Access Journals (Sweden)

Dahiya Kalpana

2006-01-01

Full Text Available This paper discusses a paradox in fixed charge capacitated transportation problem where the objective function is the sum of two linear fractional functions consisting of variables costs and fixed charges respectively. A paradox arises when the transportation problem admits of an objective function value which is lower than the optimal objective function value, by transporting larger quantities of goods over the same route. A sufficient condition for the existence of a paradox is established. Paradoxical range of flow is obtained for any given flow in which the corresponding objective function value is less than the optimum value of the given transportation problem. Numerical illustration is included in support of theory.

Numerical algorithms for contact problems in linear elastostatics

International Nuclear Information System (INIS)

Barbosa, H.J.C.; Feijoo, R.A.

1984-01-01

In this work contact problems in linear elasticity are analysed by means of Finite Elements and Mathematical Programming Techniques. The principle of virtual work leads in this case to a variational inequality which in turn is equivalent, for Hookean materials and infinitesimal strains, to the minimization of the total potential energy over the set of all admissible virtual displacements. The use of Gauss-Seidel algorithm with relaxation and projection and also Lemke's algorithm and Uzawa's algorithm for solving the minimization problem is discussed. Finally numerical examples are presented. (Author) [pt

Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold.

Science.gov (United States)

Palacios, Jonathan; Yeh, Harry; Wang, Wenping; Zhang, Yue; Laramee, Robert S; Sharma, Ritesh; Schultz, Thomas; Zhang, Eugene

2016-03-01

Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.

Analytical vs. Simulation Solution Techniques for Pulse Problems in Non-linear Stochastic Dynamics

DEFF Research Database (Denmark)

Iwankiewicz, R.; Nielsen, Søren R. K.

Advantages and disadvantages of available analytical and simulation techniques for pulse problems in non-linear stochastic dynamics are discussed. First, random pulse problems, both those which do and do not lead to Markov theory, are presented. Next, the analytical and analytically-numerical tec......Advantages and disadvantages of available analytical and simulation techniques for pulse problems in non-linear stochastic dynamics are discussed. First, random pulse problems, both those which do and do not lead to Markov theory, are presented. Next, the analytical and analytically...

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

Energy Technology Data Exchange (ETDEWEB)

Slattery, S. R.; Wilson, P. P. H. [Engineering Physics Department, University of Wisconsin - Madison, 1500 Engineering Dr., Madison, WI 53706 (United States); Evans, T. M. [Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN 37830 (United States)

2013-07-01

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results. (authors)

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

International Nuclear Information System (INIS)

Slattery, S. R.; Wilson, P. P. H.; Evans, T. M.

2013-01-01

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear operator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approximation and the mean chord approximation are applied to estimate the leakage fraction of stochastic histories from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem to test the models for symmetric operators. In general, the derived approximations show good agreement with measured computational results. (authors)

On a Volume Constrained for the First Eigenvalue of the P-Laplacian Operator

International Nuclear Information System (INIS)

Ly, Idrissa

2009-10-01

In this paper, we are interested in a shape optimization problem which consists in minimizing the functional that associates to an open set the first eigenvalue for p-Laplacian operator with homogeneous boundary condition. The minimum is taken among all open subsets with prescribed measure of a given bounded domain. We study an existence result for the associate variational problem. Our technique consists in enlarging the class of admissible functions to the whole space W 0 1,p (D), penalizing those functions whose level sets have a measure which is less than those required. In fact, we study the minimizers of a family of penalized functionals J λ , λ > 0 showing they are Hoelder continuous. And we prove that such functions minimize the initial problem provided the penalization parameter λ is large enough. (author)

A goal programming procedure for solving fuzzy multiobjective fractional linear programming problems

Directory of Open Access Journals (Sweden)

Tunjo Perić

2014-12-01

Full Text Available This paper presents a modification of Pal, Moitra and Maulik's goal programming procedure for fuzzy multiobjective linear fractional programming problem solving. The proposed modification of the method allows simpler solving of economic multiple objective fractional linear programming (MOFLP problems, enabling the obtained solutions to express the preferences of the decision maker defined by the objective function weights. The proposed method is tested on the production planning example.

LinvPy : a Python package for linear inverse problems

OpenAIRE

Beaud, Guillaume François Paul

2016-01-01

The goal of this project is to make a Python package including the tau-estimator algorithm to solve linear inverse problems. The package must be distributed, well documented, easy to use and easy to extend for future developers.

Perfect observables for the hierarchical non-linear O(N)-invariant σ-model

International Nuclear Information System (INIS)

Wieczerkowski, C.; Xylander, Y.

1995-05-01

We compute moving eigenvalues and the eigenvectors of the linear renormalization group transformation for observables along the renormalized trajectory of the hierarchical non-linear O(N)-invariant σ-model by means of perturbation theory in the running coupling constant. Moving eigenvectors are defined as solutions to a Callan-Symanzik type equation. (orig.)

Eigenvalues and expansion of bipartite graphs

DEFF Research Database (Denmark)

Høholdt, Tom; Janwa, Heeralal

2012-01-01

We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by ...

Arbitrary Lagrangian-Eulerian method for non-linear problems of geomechanics

International Nuclear Information System (INIS)

Nazem, M; Carter, J P; Airey, D W

2010-01-01

In many geotechnical problems it is vital to consider the geometrical non-linearity caused by large deformation in order to capture a more realistic model of the true behaviour. The solutions so obtained should then be more accurate and reliable, which should ultimately lead to cheaper and safer design. The Arbitrary Lagrangian-Eulerian (ALE) method originated from fluid mechanics, but has now been well established for solving large deformation problems in geomechanics. This paper provides an overview of the ALE method and its challenges in tackling problems involving non-linearities due to material behaviour, large deformation, changing boundary conditions and time-dependency, including material rate effects and inertia effects in dynamic loading applications. Important aspects of ALE implementation into a finite element framework will also be discussed. This method is then employed to solve some interesting and challenging geotechnical problems such as the dynamic bearing capacity of footings on soft soils, consolidation of a soil layer under a footing, and the modelling of dynamic penetration of objects into soil layers.

A Compensatory Approach to Multiobjective Linear Transportation Problem with Fuzzy Cost Coefficients

Directory of Open Access Journals (Sweden)

Hale Gonce Kocken

2011-01-01

Full Text Available This paper deals with the Multiobjective Linear Transportation Problem that has fuzzy cost coefficients. In the solution procedure, many objectives may conflict with each other; therefore decision-making process becomes complicated. And also due to the fuzziness in the costs, this problem has a nonlinear structure. In this paper, fuzziness in the objective functions is handled with a fuzzy programming technique in the sense of multiobjective approach. And then we present a compensatory approach to solve Multiobjective Linear Transportation Problem with fuzzy cost coefficients by using Werner's and operator. Our approach generates compromise solutions which are both compensatory and Pareto optimal. A numerical example has been provided to illustrate the problem.

Eigenvalue Problems.

Science.gov (United States)

1987-06-01

11.55), we get (11.56) 1 bu(u - ZhU)dx < bull - hUi{ !s Ch2V(a), where C = C(aOa 1lbo,blV 0 (a),X). Next we consider I(S-Sh)uIL and Ij(T-Th)uaL2 It is...Math. Ann. 97, 711-736. Courant, R. [1943]: Variational methods for the solution of prob- .-lems of equilibrium and vibrations, Bull . Amer. Math. Soc...fournir des bornes superieures ou inferieures, C.R. Acad. Sci., Paris 235, 995-997. .V Prodi, G. (1962]: Theoremi di tipo locale per il sistema de Navier

A Complete Parametric Solutions of Eigenstructure Assignment by State-Derivative Feedback for Linear Control Systems

Directory of Open Access Journals (Sweden)

T. H. S. Abdelaziz

2005-01-01

Full Text Available In this paper we introduce a complete parametric approach for solving the problem of eigenstructure assignment via state-derivative feedback for linear systems. This problem is always solvable for any controllable systems iff the open-loop system matrix is nonsingular. In this work, two parametric solutions to the feedback gain matrix are introduced that describe the available degrees of freedom offered by the state-derivative feedback in selecting the associated eigenvectors from an admissible class. These freedoms can be utilized to improve robustness of the closed-loop system. Accordingly, the sensitivity of the assigned eigenvalues to perturbations in the system and gain matrix is minimized. Numerical examples are included to show the effectiveness of the proposed approach. 

Existence of solutions to fractional boundary-value problems with a parameter

Directory of Open Access Journals (Sweden)

Ya-Ning Li

2013-06-01

Full Text Available This article concerns the existence of solutions to the fractional boundary-value problem $$displaylines{ -frac{d}{dt} ig(frac{1}{2} {}_0D_t^{-eta}+ frac{1}{2}{}_tD_{T}^{-eta}igu'(t=lambda u(t+abla F(t,u(t,quad hbox{a.e. } tin[0,T], cr u(0=0,quad u(T=0. }$$ First for the eigenvalue problem associated with it, we prove that there is a sequence of positive and increasing real eigenvalues; a characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity F(t,u, we show the existence of weak solutions of the problem when $lambda$ lies in various intervals. Our main tools are variational methods and critical point theorems.

Accurate Valence Ionization Energies from Kohn-Sham Eigenvalues with the Help of Potential Adjustors.

Science.gov (United States)

Thierbach, Adrian; Neiss, Christian; Gallandi, Lukas; Marom, Noa; Körzdörfer, Thomas; Görling, Andreas

2017-10-10

An accurate yet computationally very efficient and formally well justified approach to calculate molecular ionization potentials is presented and tested. The first as well as higher ionization potentials are obtained as the negatives of the Kohn-Sham eigenvalues of the neutral molecule after adjusting the eigenvalues by a recently [ Görling Phys. Rev. B 2015 , 91 , 245120 ] introduced potential adjustor for exchange-correlation potentials. Technically the method is very simple. Besides a Kohn-Sham calculation of the neutral molecule, only a second Kohn-Sham calculation of the cation is required. The eigenvalue spectrum of the neutral molecule is shifted such that the negative of the eigenvalue of the highest occupied molecular orbital equals the energy difference of the total electronic energies of the cation minus the neutral molecule. For the first ionization potential this simply amounts to a ΔSCF calculation. Then, the higher ionization potentials are obtained as the negatives of the correspondingly shifted Kohn-Sham eigenvalues. Importantly, this shift of the Kohn-Sham eigenvalue spectrum is not just ad hoc. In fact, it is formally necessary for the physically correct energetic adjustment of the eigenvalue spectrum as it results from ensemble density-functional theory. An analogous approach for electron affinities is equally well obtained and justified. To illustrate the practical benefits of the approach, we calculate the valence ionization energies of test sets of small- and medium-sized molecules and photoelectron spectra of medium-sized electron acceptor molecules using a typical semilocal (PBE) and two typical global hybrid functionals (B3LYP and PBE0). The potential adjusted B3LYP and PBE0 eigenvalues yield valence ionization potentials that are in very good agreement with experimental values, reaching an accuracy that is as good as the best G 0 W 0 methods, however, at much lower computational costs. The potential adjusted PBE eigenvalues result in

Обратные задачи для уравнения Штурма–Лиувилля со спектральным параметром в краевом условии

OpenAIRE

Guliyev , Namig

2004-01-01

International audience; Inverse problems of recovering the coefficients of Sturm–Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Multiscale finite element methods for high-contrast problems using local spectral basis functions

KAUST Repository

Efendiev, Yalchin

2011-02-01

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ*)1/2, where Λ* is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings. © 2010.

The total Hartree-Fock energy-eigenvalue sum relationship in atoms

International Nuclear Information System (INIS)

Sen, K.D.

1979-01-01

Using the well known relationships for the isoelectronic changes in the total Hartree-Fock energy, nucleus-electron attraction energy and electron-electron repulsion energy in atoms a simple polynomial expansion in Z is obtained for the sum of the eigenvalues which can be used to calculate the total Hartree-Fock energy. Numerical results are presented for 2-10 electron series to show that the present relationship is a better approximation than the other available energy-eigenvalue relationships. (author)

Problems of linear electron (polaron) transport theory in semiconductors

CERN Document Server

Klinger, M I

1979-01-01

Problems of Linear Electron (Polaron) Transport Theory in Semiconductors summarizes and discusses the development of areas in electron transport theory in semiconductors, with emphasis on the fundamental aspects of the theory and the essential physical nature of the transport processes. The book is organized into three parts. Part I focuses on some general topics in the theory of transport phenomena: the general dynamical theory of linear transport in dissipative systems (Kubo formulae) and the phenomenological theory. Part II deals with the theory of polaron transport in a crystalline semicon

A primal-dual exterior point algorithm for linear programming problems

Directory of Open Access Journals (Sweden)

Samaras Nikolaos

2009-01-01

Full Text Available The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal - Dual method is a Simplex - type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal - Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.

Averaging and Linear Programming in Some Singularly Perturbed Problems of Optimal Control

Energy Technology Data Exchange (ETDEWEB)

Gaitsgory, Vladimir, E-mail: vladimir.gaitsgory@mq.edu.au [Macquarie University, Department of Mathematics (Australia); Rossomakhine, Sergey, E-mail: serguei.rossomakhine@flinders.edu.au [Flinders University, Flinders Mathematical Sciences Laboratory, School of Computer Science, Engineering and Mathematics (Australia)

2015-04-15

The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples.

Computations of zeros of special functions and eigenvalues of differential equations by matrix method

OpenAIRE

Miyazaki, Yoshinori

2000-01-01

This paper is strongly based on two powerful general theorems proved by Ikebe, et. al in 1993[15] and 1996[13], which will be referred to as Theorem A and Theorem B in this paper. They were recently published and justify the approximate computations of simple eigenvalues of infinite matrices of certain types by truncation, giving an extremely accurate error estimates. So far, they have applied to some important problems in engineering, such as computing the zeros of some special functions, an...

Modification of the MORSE code for Monte Carlo eigenvalue problems by coarse-mesh rebalance acceleration

International Nuclear Information System (INIS)

Nishida, Takahiko; Horikami, Kunihiko; Suzuki, Tadakazu; Nakahara, Yasuaki; Taji, Yukichi

1975-09-01

The coarse-mesh rebalancing technique is introduced into the general-purpose neutron and gamma-ray Monte Carlo transport code MORSE, to accelerate the convergence rate of the iteration process for eigenvalue calculation in a nuclear reactor system. Two subroutines are thus attached to the code. One is bookkeeping routine 'COARSE' for obtaining the quantities related with the neutron balance in each coarse mesh cell, such as the number of neutrons absorbed in the cell, from random walks of neutrons in a batch. The other is rebalance factor calculation routine 'REBAL' for obtaining the scaling factor whereby the neutron flux in the cell is multiplied to attain the neutron balance. The two subroutines and algorithm of the coarse mesh rebalancing acceleration in a Monte Carlo game are described. (auth.)

An eigenvalue approach to quantum plasmonics based on a self-consistent hydrodynamics method.

Science.gov (United States)

Ding, Kun; Chan, C T

2018-02-28

Plasmonics has attracted much attention not only because it has useful properties such as strong field enhancement, but also because it reveals the quantum nature of matter. To handle quantum plasmonics effects, ab initio packages or empirical Feibelman d-parameters have been used to explore the quantum correction of plasmonic resonances. However, most of these methods are formulated within the quasi-static framework. The self-consistent hydrodynamics model offers a reliable approach to study quantum plasmonics because it can incorporate the quantum effect of the electron gas into classical electrodynamics in a consistent manner. Instead of the standard scattering method, we formulate the self-consistent hydrodynamics method as an eigenvalue problem to study quantum plasmonics with electrons and photons treated on the same footing. We find that the eigenvalue approach must involve a global operator, which originates from the energy functional of the electron gas. This manifests the intrinsic nonlocality of the response of quantum plasmonic resonances. Our model gives the analytical forms of quantum corrections to plasmonic modes, incorporating quantum electron spill-out effects and electrodynamical retardation. We apply our method to study the quantum surface plasmon polariton for a single flat interface.

High-order modulation on a single discrete eigenvalue for optical communications based on nonlinear Fourier transform.

Science.gov (United States)

Gui, Tao; Lu, Chao; Lau, Alan Pak Tao; Wai, P K A

2017-08-21

In this paper, we experimentally investigate high-order modulation over a single discrete eigenvalue under the nonlinear Fourier transform (NFT) framework and exploit all degrees of freedom for encoding information. For a fixed eigenvalue, we compare different 4 bit/symbol modulation formats on the spectral amplitude and show that a 2-ring 16-APSK constellation achieves optimal performance. We then study joint spectral phase, spectral magnitude and eigenvalue modulation and found that while modulation on the real part of the eigenvalue induces pulse timing drift and leads to neighboring pulse interactions and nonlinear inter-symbol interference (ISI), it is more bandwidth efficient than modulation on the imaginary part of the eigenvalue in practical settings. We propose a spectral amplitude scaling method to mitigate such nonlinear ISI and demonstrate a record 4 GBaud 16-APSK on the spectral amplitude plus 2-bit eigenvalue modulation (total 6 bit/symbol at 24 Gb/s) transmission over 1000 km.

Two linearization methods for atmospheric remote sensing

International Nuclear Information System (INIS)

Doicu, A.; Trautmann, T.

2009-01-01

We present two linearization methods for a pseudo-spherical atmosphere and general viewing geometries. The first approach is based on an analytical linearization of the discrete ordinate method with matrix exponential and incorporates two models for matrix exponential calculation: the matrix eigenvalue method and the Pade approximation. The second method referred to as the forward-adjoint approach is based on the adjoint radiative transfer for a pseudo-spherical atmosphere. We provide a compact description of the proposed methods as well as a numerical analysis of their accuracy and efficiency.

To the confinement problem

International Nuclear Information System (INIS)

Savvidi, G.K.

1985-01-01

Such a viewpoint is proposed for separation of the physical quantities into observable and unobservable ones, when the latters are connected with the Hermitian operator for which the eigenvalue problem is unsolvable

Applied linear algebra

CERN Document Server

Olver, Peter J

2018-01-01

This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the un...

Leak detection of complex pipelines based on the filter diagonalization method: robust technique for eigenvalue assessment

International Nuclear Information System (INIS)

Lay-Ekuakille, Aimé; Pariset, Carlo; Trotta, Amerigo

2010-01-01

The FDM (filter diagonalization method), an interesting technique used in nuclear magnetic resonance data processing for tackling FFT (fast Fourier transform) limitations, can be used by considering pipelines, especially complex configurations, as a vascular apparatus with arteries, veins, capillaries, etc. Thrombosis, which might occur in humans, can be considered as a leakage for the complex pipeline, the human vascular apparatus. The choice of eigenvalues in FDM or in spectra-based techniques is a key issue in recovering the solution of the main equation (for FDM) or frequency domain transformation (for FFT) in order to determine the accuracy in detecting leaks in pipelines. This paper deals with the possibility of improving the leak detection accuracy of the FDM technique thanks to a robust algorithm by assessing the problem of eigenvalues, making it less experimental and more analytical using Tikhonov-based regularization techniques. The paper starts from the results of previous experimental procedures carried out by the authors

Solution of unidimensional problems from monoenergetics neutrons diffusion through finite differences

International Nuclear Information System (INIS)

Filio Lopez, Carlos.

1979-01-01

A calculation program (URA 6.F4) was elaborated on FORTRAN IV language, that through finite differences solves the unidimensional scalar Helmholtz equation, assuming only one energy group, in spherical cylindrical or plane geometry. The purpose is the determination of the flow distribution in a reactor of spherical cylindrical or plane geometry and the critical dimensions. Feeding as entrance datas to the program the geometry, diffusion coefficients and macroscopic transversals cross sections of absorption and fission for each region. The differential diffusion equation is converted with its boundary conditions, to one system of homogeneous algebraic linear equations using the box integration technique. The investigation on criticality is converted then in a succession of eigenvalue problems for the critical eigenvalue. In general, only is necessary to solve the first eigenvalue and its corresponding eigenvector, employing the power method. The obtained results by the program for the critical dimensions of the clean reactors are admissible, the existing error as respect to the analytic is less of 0.5%; by the analysed reactors of three regions, the relative error with respect to the semianalytic result is less of 0.2%. With this program is possible to obtain one quantitative description of one reactor if the transversal sections that appears in the monoenergetic model are adequatedly averaged by the energy group used. (author)

Relativistic energy eigenvalues for the Dirac equation in the presence of vector and scalar potentials via the simple similarity transformation

International Nuclear Information System (INIS)

Barakat, T

2012-01-01

Based on the simple similarity transformation, we were able to transform the Dirac equation whose potential contains vector V (r) = -A/r + B 1 r and scalar S(r) = B 2 r types into a form nearly identical to the Schrödinger equation. The transformed equation is so simple that one can solve it by means of the asymptotic iteration method. Moreover, within the same framework we were able to obtain the relativistic energy eigenvalues for the Dirac equation with vector Coulomb plus scalar linear, and with pure scalar linear potentials; V (r) = -A/r, S(r) = B 2 r, and V (r) = 0, S(r) = B 2 r, respectively.

Recent developments in semiclassical mechanics: eigenvalues and reaction rate constants

International Nuclear Information System (INIS)

Miller, W.H.

1976-04-01

A semiclassical treatment of eigenvalues for a multidimensional non-separable potential function and of the rate constant for a chemical reaction with an activation barrier is presented. Both phenomena are seen to be described by essentially the same semiclassical formalism, which is based on a construction of the total Hamiltonian in terms of the complete set of ''good'' action variables (or adiabatic invariants) associated with the minimum in the potential energy surface for the eigenvalue case, or the saddle point in the potential energy surface for the case of chemical reaction

Reduced-Size Integer Linear Programming Models for String Selection Problems: Application to the Farthest String Problem.

Science.gov (United States)

Zörnig, Peter

2015-08-01

We present integer programming models for some variants of the farthest string problem. The number of variables and constraints is substantially less than that of the integer linear programming models known in the literature. Moreover, the solution of the linear programming-relaxation contains only a small proportion of noninteger values, which considerably simplifies the rounding process. Numerical tests have shown excellent results, especially when a small set of long sequences is given.

Eigenvalue pinching on spinc manifolds

Science.gov (United States)

Roos, Saskia

2017-02-01

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Solutions to estimation problems for scalar hamilton-jacobi equations using linear programming

KAUST Repository

Claudel, Christian G.; Chamoin, Timothee; Bayen, Alexandre M.

2014-01-01

This brief presents new convex formulations for solving estimation problems in systems modeled by scalar Hamilton-Jacobi (HJ) equations. Using a semi-analytic formula, we show that the constraints resulting from a HJ equation are convex, and can be written as a set of linear inequalities. We use this fact to pose various (and seemingly unrelated) estimation problems related to traffic flow-engineering as a set of linear programs. In particular, we solve data assimilation and data reconciliation problems for estimating the state of a system when the model and measurement constraints are incompatible. We also solve traffic estimation problems, such as travel time estimation or density estimation. For all these problems, a numerical implementation is performed using experimental data from the Mobile Century experiment. In the context of reproducible research, the code and data used to compute the results presented in this brief have been posted online and are accessible to regenerate the results. © 2013 IEEE.

The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

Directory of Open Access Journals (Sweden)

Hongchun Sun

2012-01-01

Full Text Available For the extended mixed linear complementarity problem (EML CP, we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

Analysis of eigenvalue correction applied to biometrics

NARCIS (Netherlands)

Hendrikse, A.J.; Veldhuis, Raymond N.J.; Spreeuwers, Lieuwe Jan; Bazen, A.M.

Eigenvalue estimation plays an important role in biometrics. However, if the number of samples is limited, estimates are significantly biased. In this article we analyse the influence of this bias on the error rates of PCA/LDA based verification systems, using both synthetic data with realistic

First-order optical systems with unimodular eigenvalues

NARCIS (Netherlands)

Bastiaans, M.J.; Alieva, T.

2006-01-01

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues, is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the

Single-machine common/slack due window assignment problems with linear decreasing processing times

Science.gov (United States)

Zhang, Xingong; Lin, Win-Chin; Wu, Wen-Hsiang; Wu, Chin-Chia

2017-08-01

This paper studies linear non-increasing processing times and the common/slack due window assignment problems on a single machine, where the actual processing time of a job is a linear non-increasing function of its starting time. The aim is to minimize the sum of the earliness cost, tardiness cost, due window location and due window size. Some optimality results are discussed for the common/slack due window assignment problems and two O(n log n) time algorithms are presented to solve the two problems. Finally, two examples are provided to illustrate the correctness of the corresponding algorithms.

Rigorous Asymptotics for the Lamé and Mathieu Functions and their Respective Eigenvalues with a Large Parameter

Science.gov (United States)

Ogilvie, Karen; Olde Daalhuis, Adri B.

2015-11-01

By application of the theory for second-order linear differential equations with two turning points developed in [Olver F.W.J., Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 137-174], uniform asymptotic approximations are obtained in the first part of this paper for the Lamé and Mathieu functions with a large real parameter. These approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid in their respective real open intervals. In all cases explicit bounds are supplied for the error terms associated with the approximations. Approximations are also obtained for the large order behaviour for the respective eigenvalues. We restrict ourselves to a two term uniform approximation. Theoretically more terms in these approximations could be computed, but the coefficients would be very complicated. In the second part of this paper we use a simplified method to obtain uniform asymptotic expansions for these functions. The coefficients are just polynomials and satisfy simple recurrence relations. The price to pay is that these asymptotic expansions hold only in a shrinking interval as their respective parameters become large; this interval however encapsulates all the interesting oscillatory behaviour of the functions. This simplified method also gives many terms in asymptotic expansions for these eigenvalues, derived simultaneously with the coefficients in the function expansions. We provide rigorous realistic error bounds for the function expansions when truncated and order estimates for the error when the eigenvalue expansions are truncated. With this paper we confirm that many of the formal results in the literature are correct.

Multivariate analysis of eigenvalues and eigenvectors in tensor based morphometry

Science.gov (United States)

Rajagopalan, Vidya; Schwartzman, Armin; Hua, Xue; Leow, Alex; Thompson, Paul; Lepore, Natasha

2015-01-01

We develop a new algorithm to compute voxel-wise shape differences in tensor-based morphometry (TBM). As in standard TBM, we non-linearly register brain T1-weighed MRI data from a patient and control group to a template, and compute the Jacobian of the deformation fields. In standard TBM, the determinants of the Jacobian matrix at each voxel are statistically compared between the two groups. More recently, a multivariate extension of the statistical analysis involving the deformation tensors derived from the Jacobian matrices has been shown to improve statistical detection power.7 However, multivariate methods comprising large numbers of variables are computationally intensive and may be subject to noise. In addition, the anatomical interpretation of results is sometimes difficult. Here instead, we analyze the eigenvalues and the eigenvectors of the Jacobian matrices. Our method is validated on brain MRI data from Alzheimer's patients and healthy elderly controls from the Alzheimer's Disease Neuro Imaging Database.

The application of the fall-vector method in decomposition schemes for the solution of integer linear programming problems

International Nuclear Information System (INIS)

Sergienko, I.V.; Golodnikov, A.N.

1984-01-01

This article applies the methods of decompositions, which are used to solve continuous linear problems, to integer and partially integer problems. The fall-vector method is used to solve the obtained coordinate problems. An algorithm of the fall-vector is described. The Kornai-Liptak decomposition principle is used to reduce the integer linear programming problem to integer linear programming problems of a smaller dimension and to a discrete coordinate problem with simple constraints

Large deviations of the maximum eigenvalue in Wishart random matrices

International Nuclear Information System (INIS)

Vivo, Pierpaolo; Majumdar, Satya N; Bohigas, Oriol

2007-01-01

We analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W = X T X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value (λ) = N/c decreases for large N as ∼exp[-β/2 N 2 Φ - (2√c + 1: c)], where β = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M ≤ 1 and Φ - (x; c) is a rate (sometimes also called large deviation) function that we compute explicitly. The result for the anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions

Large deviations of the maximum eigenvalue in Wishart random matrices

Energy Technology Data Exchange (ETDEWEB)

Vivo, Pierpaolo [School of Information Systems, Computing and Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH (United Kingdom) ; Majumdar, Satya N [Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France); Bohigas, Oriol [Laboratoire de Physique Theorique et Modeles Statistiques (UMR 8626 du CNRS), Universite Paris-Sud, Batiment 100, 91405 Orsay Cedex (France)

2007-04-20

We analytically compute the probability of large fluctuations to the left of the mean of the largest eigenvalue in the Wishart (Laguerre) ensemble of positive definite random matrices. We show that the probability that all the eigenvalues of a (N x N) Wishart matrix W = X{sup T}X (where X is a rectangular M x N matrix with independent Gaussian entries) are smaller than the mean value ({lambda}) = N/c decreases for large N as {approx}exp[-{beta}/2 N{sup 2}{phi}{sub -} (2{radical}c + 1: c)], where {beta} = 1, 2 corresponds respectively to real and complex Wishart matrices, c = N/M {<=} 1 and {phi}{sub -}(x; c) is a rate (sometimes also called large deviation) function that we compute explicitly. The result for the anti-Wishart case (M < N) simply follows by exchanging M and N. We also analytically determine the average spectral density of an ensemble of Wishart matrices whose eigenvalues are constrained to be smaller than a fixed barrier. Numerical simulations are in excellent agreement with the analytical predictions.

Matrix with Prescribed Eigenvectors

Science.gov (United States)

Ahmad, Faiz

2011-01-01

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

Solving fault diagnosis problems linear synthesis techniques

CERN Document Server

Varga, Andreas

2017-01-01

This book addresses fault detection and isolation topics from a computational perspective. Unlike most existing literature, it bridges the gap between the existing well-developed theoretical results and the realm of reliable computational synthesis procedures. The model-based approach to fault detection and diagnosis has been the subject of ongoing research for the past few decades. While the theoretical aspects of fault diagnosis on the basis of linear models are well understood, most of the computational methods proposed for the synthesis of fault detection and isolation filters are not satisfactory from a numerical standpoint. Several features make this book unique in the fault detection literature: Solution of standard synthesis problems in the most general setting, for both continuous- and discrete-time systems, regardless of whether they are proper or not; consequently, the proposed synthesis procedures can solve a specific problem whenever a solution exists Emphasis on the best numerical algorithms to ...

Numerical stability in problems of linear algebra.

Science.gov (United States)

Babuska, I.

1972-01-01

Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.

Genetic programming over context-free languages with linear constraints for the knapsack problem: first results.

Science.gov (United States)

Bruhn, Peter; Geyer-Schulz, Andreas

2002-01-01

In this paper, we introduce genetic programming over context-free languages with linear constraints for combinatorial optimization, apply this method to several variants of the multidimensional knapsack problem, and discuss its performance relative to Michalewicz's genetic algorithm with penalty functions. With respect to Michalewicz's approach, we demonstrate that genetic programming over context-free languages with linear constraints improves convergence. A final result is that genetic programming over context-free languages with linear constraints is ideally suited to modeling complementarities between items in a knapsack problem: The more complementarities in the problem, the stronger the performance in comparison to its competitors.

Existence, stability, and dynamics of harmonically trapped one-dimensional multi-component solitary waves: The near-linear limit

Science.gov (United States)

Xu, H.; Kevrekidis, P. G.; Kapitula, T.

2017-06-01

In the present work, we consider a variety of two-component, one-dimensional states in nonlinear Schrödinger equations in the presence of a parabolic trap, inspired by the atomic physics context of Bose-Einstein condensates. The use of Lyapunov-Schmidt reduction methods allows us to identify persistence criteria for the different families of solutions which we classify as (m, n), in accordance with the number of zeros in each component. Upon developing the existence theory, we turn to a stability analysis of the different configurations, using the Krein signature and the Hamiltonian-Krein index as topological tools identifying the number of potentially unstable eigendirections for each branch. A perturbation expansion for the eigenvalue problems associated with nonlinear states found near the linear limit permits us to obtain explicit asymptotic expressions for the eigenvalues. Finally, when the states are found to be unstable, typically by virtue of Hamiltonian Hopf bifurcations, their dynamics is studied in order to identify the nature of the respective instability. The dynamics is generally found to lead to a vibrational evolution over long time scales.

Analysis of 2D reactor core using linear perturbation theory and nodal finite element methods

International Nuclear Information System (INIS)

Adrian Mugica; Edmundo del Valle

2005-01-01

In this work the multigroup steady state neutron diffusion equations are solved using the nodal finite element method (NFEM) and the Linear Perturbation Theory (LPT) for XY geometry. The NFEM used corresponds to the Raviart-Thomas schemes RT0 and RT1, interpolating 5 and 12 parameters respectively in each node of the space discretization. The accuracy of these methods is related with the dimension of the space approximation and the mesh size. Therefore, using fine meshes and the RT0 or RT1 nodal methods leads to a large an interesting eigenvalue problem. The finite element method used to discretize the weak formulation of the diffusion equations is the Galerkin one. The algebraic structure of the discrete eigenvalue problem is obtained and solved using the Wielandt technique and the BGSTAB iterative method using the SPARSKIT package developed by Yousef Saad. The results obtained with LPT show good agreement with the results obtained directly for the perturbed problem. In fact, the cpu time to solve a single problem, the unperturbed and the perturbed one, is practically the same but when one is focused in shuffling many times two different assemblies in the core then the LPT technique becomes quite useful to get good approximations in a short time. This particular problem was solved for one quarter-core with NFEM. Thus, the computer program based on LPT can be used to perform like an analysis tool in the fuel reload optimization or combinatory analysis to get reload patterns in nuclear power plants once that it had been incorporated with the thermohydraulic aspects needed to simulate accurately a real problem. The maximum differences between the NFEM and LPT for the three LWR reactor cores are about 250 pcm. This quantity is considered an acceptable value for this kind of analysis. (authors)

Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs

Science.gov (United States)

Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.

2017-10-01

This paper focuses on the optimal control design for a system of coupled parabolic-hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear-quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

Higher-order relationship between eigen-value separation and static flux tilts

International Nuclear Information System (INIS)

Beckner, W.D.

1975-01-01

Spatial kinetics phenomena in nuclear reactors, such as xenon-induced spatial flux oscillations, are currently being analyzed using the higher harmonic solutions to the static reactor balance equation. An important parameter in such an analysis is a global quantity called eigenvalue separation. It is desirable to be able to experimentally measure this parameter in power reactors in order to confirm design calculations. Since spatial distortions in the flux shape depend on the eigenvalue separation of the reactor, an attempt has been made previously to use this fact as a means of measuring the parameter. It was postulated that an induced flux distortion or ''static flux tilt'' could be measured and theoretically related to eigenvalue separation. Unfortunately, the behavior of experimental data did not exactly agree with theoretical predictions, and values of the parameter found using the original static flux tilt technique were consistently low. The theory has been re-evaluated here and the previously observed discrepancy eliminated. Techniques have been also developed to allow for more accurate interpretation of experimental data. In order to make the method applicable to real systems, the theory has been extended to two spatial dimensions; extension to three dimensions follows directly. Possible trouble areas have been investigated, and experimental procedures for use of the technique to measure the eigenvalue separation in power reactors are presented

The Core Problem within a Linear Approximation Problem $AX/approx B$ with Multiple Right-Hand Sides

Czech Academy of Sciences Publication Activity Database

Hnětynková, Iveta; Plešinger, Martin; Strakoš, Z.

2013-01-01

Roč. 34, č. 3 (2013), s. 917-931 ISSN 0895-4798 R&D Projects: GA ČR GA13-06684S Grant - others:GA ČR(CZ) GA201/09/0917; GA MŠk(CZ) EE2.3.09.0155; GA MŠk(CZ) EE2.3.30.0065 Program:GA Institutional support: RVO:67985807 Keywords : total least squares problem * multiple right-hand sides * core problem * linear approximation problem * error-in-variables modeling * orthogonal regression * singular value decomposition Subject RIV: BA - General Mathematics Impact factor: 1.806, year: 2013

Polymorphic Uncertain Linear Programming for Generalized Production Planning Problems

Directory of Open Access Journals (Sweden)

Xinbo Zhang

2014-01-01

Full Text Available A polymorphic uncertain linear programming (PULP model is constructed to formulate a class of generalized production planning problems. In accordance with the practical environment, some factors such as the consumption of raw material, the limitation of resource and the demand of product are incorporated into the model as parameters of interval and fuzzy subsets, respectively. Based on the theory of fuzzy interval program and the modified possibility degree for the order of interval numbers, a deterministic equivalent formulation for this model is derived such that a robust solution for the uncertain optimization problem is obtained. Case study indicates that the constructed model and the proposed solution are useful to search for an optimal production plan for the polymorphic uncertain generalized production planning problems.

Exact results for the many-body problem in one dimension with repulsive delta-function interaction

International Nuclear Information System (INIS)

Yang, C.N.

1983-01-01

The repulsive δ interaction problem in one dimension for N particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations R of the permutation group S/sub N/. For some R's this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-1/2 fermions is reduced to a generalized Fredholm equation

Computational methods in calculating superconducting current problems

Science.gov (United States)

Brown, David John, II

Various computational problems in treating superconducting currents are examined. First, field inversion in spatial Fourier transform space is reviewed to obtain both one-dimensional transport currents flowing down a long thin tape, and a localized two-dimensional current. The problems associated with spatial high-frequency noise, created by finite resolution and experimental equipment, are presented, and resolved with a smooth Gaussian cutoff in spatial frequency space. Convergence of the Green's functions for the one-dimensional transport current densities is discussed, and particular attention is devoted to the negative effects of performing discrete Fourier transforms alone on fields asymptotically dropping like 1/r. Results of imaging simulated current densities are favorably compared to the original distributions after the resulting magnetic fields undergo the imaging procedure. The behavior of high-frequency spatial noise, and the behavior of the fields with a 1/r asymptote in the imaging procedure in our simulations is analyzed, and compared to the treatment of these phenomena in the published literature. Next, we examine calculation of Mathieu and spheroidal wave functions, solutions to the wave equation in elliptical cylindrical and oblate and prolate spheroidal coordinates, respectively. These functions are also solutions to Schrodinger's equations with certain potential wells, and are useful in solving time-varying superconducting problems. The Mathieu functions are Fourier expanded, and the spheroidal functions expanded in associated Legendre polynomials to convert the defining differential equations to recursion relations. The infinite number of linear recursion equations is converted to an infinite matrix, multiplied by a vector of expansion coefficients, thus becoming an eigenvalue problem. The eigenvalue problem is solved with root solvers, and the eigenvector problem is solved using a Jacobi-type iteration method, after preconditioning the

Complex eigenvalue analysis of railway wheel/rail squeal

African Journals Online (AJOL)

DR OKE

Squeal noise from wheel/rail and brake disc/pad frictional contact is typical in railways. ... squeal noise by multibody simulation of a rail car running on rigid rails. ... system, traditional complex eigenvalue analysis by finite element was used.

A teaching proposal for the study of Eigenvectors and Eigenvalues

Directory of Open Access Journals (Sweden)

María José Beltrán Meneu

2017-03-01

Full Text Available In this work, we present a teaching proposal which emphasizes on visualization and physical applications in the study of eigenvectors and eigenvalues. These concepts are introduced using the notion of the moment of inertia of a rigid body and the GeoGebra software. The proposal was motivated after observing students’ difficulties when treating eigenvectors and eigenvalues from a geometric point of view. It was designed following a particular sequence of activities with the schema: exploration, introduction of concepts, structuring of knowledge and application, and considering the three worlds of mathematical thinking provided by Tall: embodied, symbolic and formal.

Multiobjective fuzzy stochastic linear programming problems with inexact probability distribution

Energy Technology Data Exchange (ETDEWEB)

Hamadameen, Abdulqader Othman [Optimization, Department of Mathematical Sciences, Faculty of Science, UTM (Malaysia); Zainuddin, Zaitul Marlizawati [Department of Mathematical Sciences, Faculty of Science, UTM (Malaysia)

2014-06-19

This study deals with multiobjective fuzzy stochastic linear programming problems with uncertainty probability distribution which are defined as fuzzy assertions by ambiguous experts. The problem formulation has been presented and the two solutions strategies are; the fuzzy transformation via ranking function and the stochastic transformation when α{sup –}. cut technique and linguistic hedges are used in the uncertainty probability distribution. The development of Sen’s method is employed to find a compromise solution, supported by illustrative numerical example.

Inference from correlated patterns: a unified theory for perceptron learning and linear vector channels

Energy Technology Data Exchange (ETDEWEB)

Kabashima, Y [Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama 226-8502 (Japan)], E-mail: kaba@dis.titech.ac.jp

2008-01-15

A framework to analyze inference performance in densely connected single-layer feed-forward networks is developed for situations where a given data set is composed of correlated patterns. The framework is based on the assumption that the left and right singular value bases of the given pattern matrix are generated independently and uniformly from Haar measures. This assumption makes it possible to characterize the objective system by a single function of two variables which is determined by the eigenvalue spectrum of the cross-correlation matrix of the pattern matrix. Links to existing methods for analysis of perceptron learning and Gaussian linear vector channels and an application to a simple but nontrivial problem are also shown.

Inference from correlated patterns: a unified theory for perceptron learning and linear vector channels

International Nuclear Information System (INIS)

Kabashima, Y

2008-01-01

A framework to analyze inference performance in densely connected single-layer feed-forward networks is developed for situations where a given data set is composed of correlated patterns. The framework is based on the assumption that the left and right singular value bases of the given pattern matrix are generated independently and uniformly from Haar measures. This assumption makes it possible to characterize the objective system by a single function of two variables which is determined by the eigenvalue spectrum of the cross-correlation matrix of the pattern matrix. Links to existing methods for analysis of perceptron learning and Gaussian linear vector channels and an application to a simple but nontrivial problem are also shown

Inference from correlated patterns: a unified theory for perceptron learning and linear vector channels

Science.gov (United States)

Kabashima, Y.

2008-01-01

A framework to analyze inference performance in densely connected single-layer feed-forward networks is developed for situations where a given data set is composed of correlated patterns. The framework is based on the assumption that the left and right singular value bases of the given pattern matrix are generated independently and uniformly from Haar measures. This assumption makes it possible to characterize the objective system by a single function of two variables which is determined by the eigenvalue spectrum of the cross-correlation matrix of the pattern matrix. Links to existing methods for analysis of perceptron learning and Gaussian linear vector channels and an application to a simple but nontrivial problem are also shown.

EZLP: An Interactive Computer Program for Solving Linear Programming Problems. Final Report.

Science.gov (United States)

Jarvis, John J.; And Others

Designed for student use in solving linear programming problems, the interactive computer program described (EZLP) permits the student to input the linear programming model in exactly the same manner in which it would be written on paper. This report includes a brief review of the development of EZLP; narrative descriptions of program features,…

Classification of the linear canonical transformation and its associated real symplectic matrix

NARCIS (Netherlands)

Bastiaans, M.J.; Alieva, T.

2007-01-01

Based on the eigenvalues of the real symplectic ABCD-matrix that characterizes the linear canonical integral transformation, a classification of this transformation and the associated ABCD-system is proposed and some nuclei (i.e. elementary members) in each class are described. In the