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)

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.

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

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

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

Inequalities among eigenvalues of Sturm–Liouville problems

Directory of Open Access Journals (Sweden)

Kong Q

1999-01-01

Full Text Available There are well-known inequalities among the eigenvalues of Sturm–Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary conditions. In this paper, for an arbitrary coupled self-adjoint boundary condition, we identify two separated boundary conditions corresponding to the Dirichlet and Neumann conditions in the classical case, and establish analogous inequalities. It is also well-known that the lowest periodic eigenvalue is simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm–Liouville problems with coupled boundary conditions are the same. An important step in our approach is to obtain a representation of the fundamental solutions for sufficiently negative values of the spectral parameter. Our approach yields the existence and boundedness from below of the eigenvalues of arbitrary self-adjoint regular Sturm–Liouville problems without using operator theory.

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.

Nonlinear acceleration of transport criticality problems

International Nuclear Information System (INIS)

Park, H.; Knoll, D.A.; Newman, C.K.

2011-01-01

We present a nonlinear acceleration algorithm for the transport criticality problem. The algorithm combines the well-known nonlinear diffusion acceleration (NDA) with a recently developed, Newton-based, nonlinear criticality acceleration (NCA) algorithm. The algorithm first employs the NDA to reduce the system to scalar flux, then the NCA is applied to the resulting drift-diffusion system. We apply a nonlinear elimination technique to eliminate the eigenvalue from the Jacobian matrix. Numerical results show that the algorithm reduces the CPU time a factor of 400 in a very diffusive system, and a factor of 5 in a non-diffusive system. (author)

Transmission eigenvalues

Science.gov (United States)

Cakoni, Fioralba; Haddar, Houssem

2013-10-01

In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission

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...

MAIA, Eigenvalues for MHD Equation of Tokamak Plasma Stability Problems

International Nuclear Information System (INIS)

Tanaka, Y.; Azumi, M.; Kurita, G.; Tsunematsu, T.; Takeda, T.

1986-01-01

1 - Description of program or function: This program solves an eigenvalue problem zBx=Ax where A and B are real block tri-diagonal matrices. This eigenvalue problem is derived from a reduced set of linear resistive MHD equations which is often employed to study tokamak plasma stability problem. 2 - Method of solution: Both the determinant and inverse iteration methods are employed. 3 - Restrictions on the complexity of the problem: The eigenvalue z must be real

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.

Modern algorithms for large sparse eigenvalue problems

International Nuclear Information System (INIS)

Meyer, A.

1987-01-01

The volume is written for mathematicians interested in (numerical) linear algebra and in the solution of large sparse eigenvalue problems, as well as for specialists in engineering, who use the considered algorithms in the investigation of eigenoscillations of structures, in reactor physics, etc. Some variants of the algorithms based on the idea of a gradient-type direction of movement are presented and their convergence properties are discussed. From this, a general strategy for the direct use of preconditionings for the eigenvalue problem is derived. In this new approach the necessity of the solution of large linear systems is entirely avoided. Hence, these methods represent a new alternative to some other modern eigenvalue algorithms, as they show a slightly slower convergence on the one hand but essentially lower numerical and data processing problems on the other hand. A brief description and comparison of some well-known methods (i.e. simultaneous iteration, Lanczos algorithm) completes this volume. (author)

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)

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

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)

TWO-DIMENSIONAL APPROXIMATION OF EIGENVALUE PROBLEMS IN SHELL THEORY: FLEXURAL SHELLS

Institute of Scientific and Technical Information of China (English)

无

2000-01-01

The eigenvalue problem for a thin linearly elastic shell, of thickness 2e, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as ε→0,the eigenvalue problem for the two-dimensional"flexural shell"model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.

EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.

Science.gov (United States)

Tal-Ezer, Hillel

2016-05-19

Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler.

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

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.

Bounds and estimates for the linearly perturbed eigenvalue problem

International Nuclear Information System (INIS)

Raddatz, W.D.

1983-01-01

This thesis considers the problem of bounding and estimating the discrete portion of the spectrum of a linearly perturbed self-adjoint operator, M(x). It is supposed that one knows an incomplete set of data consisting in the first few coefficients of the Taylor series expansions of one or more of the eigenvalues of M(x) about x = 0. The foundations of the variational study of eigen-values are first presented. These are then used to construct the best possible upper bounds and estimates using various sets of given information. Lower bounds are obtained by estimating the error in the upper bounds. The extension of these bounds and estimates to the eigenvalues of the doubly-perturbed operator M(x,y) is discussed. The results presented have numerous practical application in the physical sciences, including problems in atomic physics and the theory of vibrations of acoustical and mechanical systems

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.

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

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

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.

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...

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)

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.

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.

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

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

On the quantum inverse problem for a new type of nonlinear Schroedinger equation for Alfven waves in plasma

International Nuclear Information System (INIS)

Sen, S.; Roy Chowdhury, A.

1989-06-01

The nonlinear Alfven waves are governed by the Vector Derivative nonlinear Schroedinger (VDNLS) equation, which for parallel or quasi parallel propagation reduces to the Derivative Nonlinear Schroedinger (DNLS) equation for the circularly polarized waves. We have formulated the Quantum Inverse problem for a new type of Nonlinear Schroedinger Equation which has many properties similar to the usual NLS problem but the structure of classical and quantum R matrix are distinctly different. The commutation rules of the scattering data are obtained and the Algebraic Bethe Ansatz is formulated to derive the eigenvalue equation for the energy of the excited states. 10 refs

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.

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.

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 large-scale sparse eigenvalue problems and linear systems of equations for accelerator modeling

International Nuclear Information System (INIS)

Gene Golub; Kwok Ko

2009-01-01

The solutions of sparse eigenvalue problems and linear systems constitute one of the key computational kernels in the discretization of partial differential equations for the modeling of linear accelerators. The computational challenges faced by existing techniques for solving those sparse eigenvalue problems and linear systems call for continuing research to improve on the algorithms so that ever increasing problem size as required by the physics application can be tackled. Under the support of this award, the filter algorithm for solving large sparse eigenvalue problems was developed at Stanford to address the computational difficulties in the previous methods with the goal to enable accelerator simulations on then the world largest unclassified supercomputer at NERSC for this class of problems. Specifically, a new method, the Hemitian skew-Hemitian splitting method, was proposed and researched as an improved method for solving linear systems with non-Hermitian positive definite and semidefinite matrices.

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

Directory of Open Access Journals (Sweden)

Ben C. Yee

2017-09-01

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

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)

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 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

Nonlinear analysis

CERN Document Server

Gasinski, Leszek

2005-01-01

Hausdorff Measures and Capacity. Lebesgue-Bochner and Sobolev Spaces. Nonlinear Operators and Young Measures. Smooth and Nonsmooth Analysis and Variational Principles. Critical Point Theory. Eigenvalue Problems and Maximum Principles. Fixed Point Theory.

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.

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.

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

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)

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.

On the solution of two-point linear differential eigenvalue problems. [numerical technique with application to Orr-Sommerfeld equation

Science.gov (United States)

Antar, B. N.

1976-01-01

A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.

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)

Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions

OpenAIRE

Guliyev, Namig J.

2008-01-01

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.

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)

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)

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

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

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).

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.

Nonlinear acceleration of SN transport calculations

Energy Technology Data Exchange (ETDEWEB)

Fichtl, Erin D [Los Alamos National Laboratory; Warsa, James S [Los Alamos National Laboratory; Calef, Matthew T [Los Alamos National Laboratory

2010-12-20

The use of nonlinear iterative methods, Jacobian-Free Newton-Krylov (JFNK) in particular, for solving eigenvalue problems in transport applications has recently become an active subject of research. While JFNK has been shown to be effective for k-eigenvalue problems, there are a number of input parameters that impact computational efficiency, making it difficult to implement efficiently in a production code using a single set of default parameters. We show that different selections for the forcing parameter in particular can lead to large variations in the amount of computational work for a given problem. In contrast, we present a nonlinear subspace method that sits outside and effectively accelerates nonlinear iterations of a given form and requires only a single input parameter, the subspace size. It is shown to consistently and significantly reduce the amount of computational work when applied to fixed-point iteration, and this combination of methods is shown to be more efficient than JFNK for our application.

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.

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.

Convergence diagnostics for Eigenvalue problems with linear regression model

International Nuclear Information System (INIS)

Shi, Bo; Petrovic, Bojan

2011-01-01

Although the Monte Carlo method has been extensively used for criticality/Eigenvalue problems, a reliable, robust, and efficient convergence diagnostics method is still desired. Most methods are based on integral parameters (multiplication factor, entropy) and either condense the local distribution information into a single value (e.g., entropy) or even disregard it. We propose to employ the detailed cycle-by-cycle local flux evolution obtained by using mesh tally mechanism to assess the source and flux convergence. By applying a linear regression model to each individual mesh in a mesh tally for convergence diagnostics, a global convergence criterion can be obtained. We exemplify this method on two problems and obtain promising diagnostics results. (author)

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.

Nonlinear acceleration of S_n transport calculations

International Nuclear Information System (INIS)

Fichtl, Erin D.; Warsa, James S.; Calef, Matthew T.

2011-01-01

The use of nonlinear iterative methods, Jacobian-Free Newton-Krylov (JFNK) in particular, for solving eigenvalue problems in transport applications has recently become an active subject of research. While JFNK has been shown to be effective for k-eigenvalue problems, there are a number of input parameters that impact computational efficiency, making it difficult to implement efficiently in a production code using a single set of default parameters. We show that different selections for the forcing parameter in particular can lead to large variations in the amount of computational work for a given problem. In contrast, we employ a nonlinear subspace method that sits outside and effectively accelerates nonlinear iterations of a given form and requires only a single input parameter, the subspace size. It is shown to consistently and significantly reduce the amount of computational work when applied to fixed-point iteration, and this combination of methods is shown to be more efficient than JFNK for our application. (author)

Stokes phenomena and monodromy deformation problem for nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Chowdury, A.R.; Naskar, M.

1986-01-01

Following Flaschka and Newell, the inverse problem for Painleve IV is formulated with the help of similarity variables. The Painleve IV arises as the eliminant of the two second-order ordinary differential equations originating from the nonlinear Schrodinger equation. Asymptotic expansions are obtained near the singularities at zero and infinity of the complex eigenvalue plane. The corresponding analysis then displays the Stokes phenomena. The monodromy matrices connecting the solution Y /sub j/ in the sector S /sub j/ to that in S /sub j+1/ are fixed in structure by the imposition of certain conditions. It is then shown that a deformation keeping the monodromy data fixed leads to the nonlinear Schrodinger equation. While Flaschka and Newell did not make any absolute determination of the Stokes parameters, the present approach yields the values of the Stokes parameters in an explicit way, which in turn can determine the matrix connecting the solutions near zero and infinity. Finally, it is shown that the integral equation originating from the analyticity and asymptotic nature of the problem leads to the similarity solution previously determined by Boiti and Pampinelli

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....

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} (|\

A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems

Directory of Open Access Journals (Sweden)

Fatemeh Mohammad

2014-05-01

Full Text Available In this paper‎, ‎we represent an inexact inverse‎ ‎subspace iteration method for computing a few eigenpairs of the‎ ‎generalized eigenvalue problem $Ax = \\lambda Bx$[Q.~Ye and P.~Zhang‎, ‎Inexact inverse subspace iteration for generalized eigenvalue‎ ‎problems‎, ‎Linear Algebra and its Application‎, ‎434 (2011 1697-1715‎‎]‎. ‎In particular‎, ‎the linear convergence property of the inverse‎ ‎subspace iteration is preserved‎.

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)

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).

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

Unbounded Perturbations of Nonlinear Second-Order Difference Equations at Resonance

Directory of Open Access Journals (Sweden)

Ma Ruyun

2007-01-01

Full Text Available We study the existence of solutions of nonlinear discrete boundary value problems , , , where is the first eigenvalue of the linear problem , , , satisfies some “asymptotic nonuniform” resonance conditions, and for .

Nonlinear dynamics of intense EM pulses in plasma

International Nuclear Information System (INIS)

Mahajan, Ranju; Gill, Tarsem Singh; Kaur, Ravinder

2010-01-01

The evolution of laser beam in underdense/overdense plasma medium which is key to understanding of several nonlinear processes and underlying physics is governed by nonlinear parabolic equation. The nonlinearity considered here is of relativistic as well as of ponderomotive type. We have set Lagrangian for the problem and reduced Lagrangian problem is solved using appropriate trial function. Equation for the beam width and phase are derived. Further, these equations are used to solve eigenvalue problem for the stability of laser beam evolution and Hurwitz condition is satisfied.

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.

Mathematical and numerical study of nonlinear boundary problems related to plasma physics

International Nuclear Information System (INIS)

Sermange, M.

1982-06-01

After the study of some equations based on the Hodgkin-Huxley model, the work presented here is concerned with nonlinear boundary problems in MHD. They are gathered in two subjects: equilibrium equations and stability equations. The axisymmetric MHD equilibrium equations with free boundary have been studied by different authors, particularly the existence, regularity, unicity and non-unicity. Here, bifurcation, convergence of calculation methods existence of solutions in a discontinuous frame are studied. MHD stability can be determined by the principle of Bernstein et al; the mathematical work concerned here bears on the equivalence, in the case of two-dimensional or axisymmetric stability, between this model and a scalar eigenvalue problem which is introduced. At last, modules for computing MHD equilibrium for the simulation of plasma confinement in a tokamak are described [fr

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.

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...

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)

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

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 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

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.

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)

Model reduction tools for nonlinear structural dynamics

NARCIS (Netherlands)

Slaats, P.M.A.; Jongh, de J.; Sauren, A.A.H.J.

1995-01-01

Three mode types are proposed for reducing nonlinear dynamical system equations, resulting from finite element discretizations: tangent modes, modal derivatives, and newly added static modes. Tangent modes are obtained from an eigenvalue problem with a momentary tangent stiffness matrix. Their

Problems in nonlinear resistive MHD

International Nuclear Information System (INIS)

Turnbull, A.D.; Strait, E.J.; La Haye, R.J.; Chu, M.S.; Miller, R.L.

1998-01-01

Two experimentally relevant problems can relatively easily be tackled by nonlinear MHD codes. Both problems require plasma rotation in addition to the nonlinear mode coupling and full geometry already incorporated into the codes, but no additional physics seems to be crucial. These problems discussed here are: (1) nonlinear coupling and interaction of multiple MHD modes near the B limit and (2) nonlinear coupling of the m/n = 1/1 sawtooth mode with higher n gongs and development of seed islands outside q = 1

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)

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.

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

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 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

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

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.)

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.

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)

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 $widetilde{A}widetilde{X}= widetilde{lambda} widetilde{X}.$

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.

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

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.

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

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

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

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.)

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.

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.

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

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)

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)

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.

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…

Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos.

Science.gov (United States)

Basko, D M

2014-02-01

We study the discrete nonlinear Schröinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved but, on the other, strong enough for the dynamics of the normal mode amplitudes to be chaotic for almost all modes. We show that in this regime and in the limit of high temperature, the macroscopic density ρ satisfies the nonlinear diffusion equation with a density-dependent diffusion coefficient, D(ρ) = D(0)ρ(2). An explicit expression for D(0) is obtained in terms of the eigenfunctions and eigenvalues of the linear problem, which is then evaluated numerically. The role of the second conserved quantity (energy) in the transport is also quantitatively discussed.

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

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

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

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)

Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions

Directory of Open Access Journals (Sweden)

Ciprian G. Gal

2017-01-01

Full Text Available Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞, we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α1, α2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L∞-setting.

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

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

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

A nonlinear oscillatory problem

International Nuclear Information System (INIS)

Zhou Qingqing.

1991-10-01

We have studied the nonlinear oscillatory problem of orthotropic cylindrical shell, we have analyzed the character of the oscillatory system. The stable condition of the oscillatory system has been given. (author). 6 refs

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.)

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 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.

On Euler's problem

International Nuclear Information System (INIS)

Egorov, Yurii V

2013-01-01

We consider the classical problem on the tallest column which was posed by Euler in 1757. Bernoulli-Euler theory serves today as the basis for the design of high buildings. This problem is reduced to the problem of finding the potential for the Sturm-Liouville equation corresponding to the maximum of the first eigenvalue. The problem has been studied by many mathematicians but we give the first rigorous proof of the existence and uniqueness of the optimal column and we give new formulae which let us find it. Our method is based on a new approach consisting in the study of critical points of a related nonlinear functional. Bibliography: 6 titles.

Overlapping Schwarz for Nonlinear Problems. An Element Agglomeration Nonlinear Additive Schwarz Preconditioned Newton Method for Unstructured Finite Element Problems

Energy Technology Data Exchange (ETDEWEB)

Cai, X C; Marcinkowski, L; Vassilevski, P S

2005-02-10

This paper extends previous results on nonlinear Schwarz preconditioning ([4]) to unstructured finite element elliptic problems exploiting now nonlocal (but small) subspaces. The non-local finite element subspaces are associated with subdomains obtained from a non-overlapping element partitioning of the original set of elements and are coarse outside the prescribed element subdomain. The coarsening is based on a modification of the agglomeration based AMGe method proposed in [8]. Then, the algebraic construction from [9] of the corresponding non-linear finite element subproblems is applied to generate the subspace based nonlinear preconditioner. The overall nonlinearly preconditioned problem is solved by an inexact Newton method. Numerical illustration is also provided.

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)

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.

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

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

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.

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.)

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

Multigrid Reduction in Time for Nonlinear Parabolic Problems

Energy Technology Data Exchange (ETDEWEB)

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

2016-01-04

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

Recent advances in computational-analytical integral transforms for convection-diffusion problems

Science.gov (United States)

Cotta, R. M.; Naveira-Cotta, C. P.; Knupp, D. C.; Zotin, J. L. Z.; Pontes, P. C.; Almeida, A. P.

2017-10-01

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numerical-analytical methodology for partial differential equations. Special emphasis is given to novel algorithm implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

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)

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.)

A Multiscale Enrichment Procedure for Nonlinear Monotone Operators

KAUST Repository

Efendiev, Yalchin R.

2014-03-11

In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners for high-contrast anisotropic flows in multiscale media. Submitted.; Y. Efendiev, J. Galvis and X. Wu, J. Comput. Phys. 230 (2011) 937–955; J. Galvis and Y. Efendiev, SIAM Multiscale Model. Simul. 8 (2010) 1461–1483.] is extended to treat a class of nonlinear elliptic operators. The proposed method requires the solutions of (small dimension and local) nonlinear eigenvalue problems in order to systematically enrich the coarse solution space. Convergence of the method is shown to relate to the dimension of the coarse space (due to the enrichment procedure) as well as the coarse mesh size. In addition, it is shown that the coarse mesh spaces can be effectively used in two-level domain decomposition preconditioners. A number of numerical results are presented to complement the analysis.

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.

Spectral properties of a confined nonlinear quantum oscillator in one and three dimensions

International Nuclear Information System (INIS)

Schulze-Halberg, Axel; Gordon, Christopher R.

2013-01-01

We analyze the spectral behaviour of a nonlinear quantum oscillator model under confinement. The underlying potential is given by a harmonic oscillator interaction plus a nonlinear term that can be weakened or strengthened through a parameter. Numerical eigenvalues of the model in one and three dimensions are presented. The asymptotic behaviour of the eigenvalues for confinement relaxation and for vanishing nonlinear term in the potential is investigated. Our findings are compared with existing results.

On Euler's problem

Energy Technology Data Exchange (ETDEWEB)

Egorov, Yurii V [Institute de Mathematique de Toulouse, Toulouse (France)

2013-04-30

We consider the classical problem on the tallest column which was posed by Euler in 1757. Bernoulli-Euler theory serves today as the basis for the design of high buildings. This problem is reduced to the problem of finding the potential for the Sturm-Liouville equation corresponding to the maximum of the first eigenvalue. The problem has been studied by many mathematicians but we give the first rigorous proof of the existence and uniqueness of the optimal column and we give new formulae which let us find it. Our method is based on a new approach consisting in the study of critical points of a related nonlinear functional. Bibliography: 6 titles.

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.

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.

Combined algorithms in nonlinear problems of magnetostatics

International Nuclear Information System (INIS)

Gregus, M.; Khoromskij, B.N.; Mazurkevich, G.E.; Zhidkov, E.P.

1988-01-01

To solve boundary problems of magnetostatics in unbounded two- and three-dimensional regions, we construct combined algorithms based on a combination of the method of boundary integral equations with the grid methods. We study the question of substantiation of the combined method of nonlinear magnetostatic problem without the preliminary discretization of equations and give some results on the convergence of iterative processes that arise in non-linear cases. We also discuss economical iterative processes and algorithms that solve boundary integral equations on certain surfaces. Finally, examples of numerical solutions of magnetostatic problems that arose when modelling the fields of electrophysical installations are given too. 14 refs.; 2 figs.; 1 tab

Higher-order techniques for some problems of nonlinear control

Directory of Open Access Journals (Sweden)

Sarychev Andrey V.

2002-01-01

Full Text Available A natural first step when dealing with a nonlinear problem is an application of some version of linearization principle. This includes the well known linearization principles for controllability, observability and stability and also first-order optimality conditions such as Lagrange multipliers rule or Pontryagin's maximum principle. In many interesting and important problems of nonlinear control the linearization principle fails to provide a solution. In the present paper we provide some examples of how higher-order methods of differential geometric control theory can be used for the study nonlinear control systems in such cases. The presentation includes: nonlinear systems with impulsive and distribution-like inputs; second-order optimality conditions for bang–bang extremals of optimal control problems; methods of high-order averaging for studying stability and stabilization of time-variant control systems.

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.

Advanced Research Workshop on Nonlinear Hyperbolic Problems

CERN Document Server

Serre, Denis; Raviart, Pierre-Arnaud

1987-01-01

The field of nonlinear hyperbolic problems has been expanding very fast over the past few years, and has applications - actual and potential - in aerodynamics, multifluid flows, combustion, detonics amongst other. The difficulties that arise in application are of theoretical as well as numerical nature. In fact, the papers in this volume of proceedings deal to a greater extent with theoretical problems emerging in the resolution of nonlinear hyperbolic systems than with numerical methods. The volume provides an excellent up-to-date review of the current research trends in this area.

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.

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 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.

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.

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.

Nonlinear singular perturbation problems of arbitrary real orders

International Nuclear Information System (INIS)

Bijura, Angelina M.

2003-10-01

Higher order asymptotic solutions of singularly perturbed nonlinear fractional integral and derivatives of order 1/2 are investigated. It is particularly shown that whilst certain asymptotic expansions are applied successfully to linear equations and particular nonlinear problems, the standard formal asymptotic expansion is appropriate for the general class of nonlinear equations. This theory is then generalised to the general equation (of order β, 0 < β < 1). (author)

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

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

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)

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)

Dual plane problems for creeping flow of power-law incompressible medium

Directory of Open Access Journals (Sweden)

Dmitriy S. Petukhov

2016-09-01

Full Text Available In this paper, we consider the class of solutions for a creeping plane flow of incompressible medium with power-law rheology, which are written in the form of the product of arbitrary power of the radial coordinate by arbitrary function of the angular coordinate of the polar coordinate system covering the plane. This class of solutions represents the asymptotics of fields in the vicinity of singular points in the domain occupied by the examined medium. We have ascertained the duality of two problems for a plane with wedge-shaped notch, at which boundaries in one of the problems the vector components of the surface force vanish, while in the other—the vanishing components are the vector components of velocity, We have investigated the asymptotics and eigensolutions of the dual nonlinear eigenvalue problems in relation to the rheological exponent and opening angle of the notch for the branch associated with the eigenvalue of the Hutchinson–Rice–Rosengren problem learned from the problem of stress distribution over a notched plane for a power law medium. In the context of the dual problem we have determined the velocity distribution in the flow of power-law medium at the vertex of a rigid wedge, We have also found another two eigenvalues, one of which was determined by V. V. Sokolovsky for the problem of power-law fluid flow in a convergent channel.

The spectral transform as a tool for solving nonlinear discrete evolution equations

International Nuclear Information System (INIS)

Levi, D.

1979-01-01

In this contribution we study nonlinear differential difference equations which became important to the description of an increasing number of problems in natural science. Difference equations arise for instance in the study of electrical networks, in statistical problems, in queueing problems, in ecological problems, as computer models for differential equations and as models for wave excitation in plasma or vibrations of particles in an anharmonic lattice. We shall first review the passages necessary to solve linear discrete evolution equations by the discrete Fourier transfrom, then, starting from the Zakharov-Shabat discretized eigenvalue, problem, we shall introduce the spectral transform. In the following part we obtain the correlation between the evolution of the potentials and scattering data through the Wronskian technique, giving at the same time many other properties as, for example, the Baecklund transformations. Finally we recover some of the important equations belonging to this class of nonlinear discrete evolution equations and extend the method to equations with n-dependent coefficients. (HJ)

Analytical Solutions to Non-linear Mechanical Oscillation Problems

DEFF Research Database (Denmark)

Kaliji, H. D.; Ghadimi, M.; Barari, Amin

2011-01-01

In this paper, the Max-Min Method is utilized for solving the nonlinear oscillation problems. The proposed approach is applied to three systems with complex nonlinear terms in their motion equations. By means of this method, the dynamic behavior of oscillation systems can be easily approximated u...

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.

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

Science.gov (United States)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

A solution to nonlinearity problems

International Nuclear Information System (INIS)

Neuffer, D.V.

1989-01-01

New methods of correcting dynamic nonlinearities resulting from the multipole content of a synchrotron or transport line are presented. In a simplest form, correction elements are places at the center (C) of the accelerator half-cells as well as near the focusing (F) and defocusing (D) quadrupoles. In a first approximation, the corrector strengths follow Simpson's Rule, forming an accurate quasi-local canceling approximation to the nonlinearity. The F, C, and D correctors may also be used to obtain precise control of the horizontal, coupled, and vertical motion. Correction by three or more orders of magnitude can be obtained, and simple solutions to a fundamental problem in beam transport have been obtained. 13 refs., 1 fig., 1 tab

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.

Cross-constrained problems for nonlinear Schrodinger equation with harmonic potential

Directory of Open Access Journals (Sweden)

Runzhang Xu

2012-11-01

Full Text Available This article studies a nonlinear Schodinger equation with harmonic potential by constructing different cross-constrained problems. By comparing the different cross-constrained problems, we derive different sharp criterion and different invariant manifolds that separate the global solutions and blowup solutions. Moreover, we conclude that some manifolds are empty due to the essence of the cross-constrained problems. Besides, we compare the three cross-constrained problems and the three depths of the potential wells. In this way, we explain the gaps in [J. Shu and J. Zhang, Nonlinear Shrodinger equation with harmonic potential, Journal of Mathematical Physics, 47, 063503 (2006], which was pointed out in [R. Xu and Y. Liu, Remarks on nonlinear Schrodinger equation with harmonic potential, Journal of Mathematical Physics, 49, 043512 (2008].

Universality in an information-theoretic motivated nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Parwani, R; Tabia, G

2007-01-01

Using perturbative methods, we analyse a nonlinear generalization of Schrodinger's equation that had previously been obtained through information-theoretic arguments. We obtain analytical expressions for the leading correction, in terms of the nonlinearity scale, to the energy eigenvalues of the linear Schrodinger equation in the presence of an external potential and observe some generic features. In one space dimension these are (i) for nodeless ground states, the energy shifts are subleading in the nonlinearity parameter compared to the shifts for the excited states; (ii) the shifts for the excited states are due predominantly to contribution from the nodes of the unperturbed wavefunctions, and (iii) the energy shifts for excited states are positive for small values of a regulating parameter and negative at large values, vanishing at a universal critical value that is not manifest in the equation. Some of these features hold true for higher dimensional problems. We also study two exactly solved nonlinear Schrodinger equations so as to contrast our observations. Finally, we comment on the possible significance of our results if the nonlinearity is physically realized

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.

Nonlinear elliptic equations and nonassociative algebras

CERN Document Server

Nadirashvili, Nikolai; Vlăduţ, Serge

2014-01-01

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions...

Renormgroup symmetries in problems of nonlinear geometrical optics

International Nuclear Information System (INIS)

Kovalev, V.F.

1996-01-01

Utilization and further development of the previously announced approach [1,2] enables one to construct renormgroup symmetries for a boundary value problem for the system of equations which describes propagation of a powerful radiation in a nonlinear medium in geometrical optics approximation. With the help of renormgroup symmetries new rigorous and approximate analytical solutions of nonlinear geometrical optics equations are obtained. Explicit analytical expressions are presented that characterize spatial evolution of laser beam which has an arbitrary intensity dependence at the boundary of the nonlinear medium. (author)

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)

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-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 \

Renormalization-group approach to nonlinear radiation-transport problems

International Nuclear Information System (INIS)

Chapline, G.F.

1980-01-01

A Monte Carlo method is derived for solving nonlinear radiation-transport problems that allows one to average over the effects of many photon absorptions and emissions at frequencies where the opacity is large. This method should allow one to treat radiation-transport problems with large optical depths, e.g., line-transport problems, with little increase in computational effort over that which is required for optically thin problems

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.

A study of discrete nonlinear systems

International Nuclear Information System (INIS)

Dhillon, H.S.

2001-04-01

An investigation of various spatially discrete time-independent nonlinear models was undertaken. These models are generically applicable to many different physical systems including electron-phonon interactions in solids, magnetic multilayers, layered superconductors and classical lattice systems. To characterise the possible magnetic structures created on magnetic multilayers a model has been formulated and studied. The Euler-Lagrange equation for this model is a discrete version of the Sine-Gordon equation. Solutions of this equation are generated by applying the methods of Chaotic Dynamics - treating the space variable associated with the layer number as a discrete time variable. The states found indicate periodic, quasiperiodic and chaotic structures. Analytic solutions to the discrete nonlinear Schroedinger Equation (DNSE) with cubic nonlinearity are presented in the strong coupling limit. Using these as a starting point, a procedure is developed to determine the wave function and the energy eigenvalue for moderate coupling. The energy eigenvalues of the different structures of the wave function are found to be in excellent agreement with the exact strong coupling result. The solutions to the DNSE indicate commensurate and incommensurate spatial structures associated with different localisation patterns of the wave function. The states which arise may be fractal, periodic, quasiperiodic or chaotic. This work is then extended to solve a first order discrete nonlinear equation. The exact solutions for both the first and second order discrete nonlinear equations with cubic nonlinearity suggests that this method of studying discrete nonlinear equations may be applied to solve discrete equations with any order difference and cubic nonlinearity. (author)

Numerical methods for solution of some nonlinear problems of mathematical physics

International Nuclear Information System (INIS)

Zhidkov, E.P.

1981-01-01

The continuous analog of the Newton method and its application to some nonlinear problems of mathematical physics using a computer is considered. It is shown that the application of this method in JINR to the wide range of nonlinear problems has shown its universality and high efficiency [ru

Existence of Positive Solutions for Nonlinear Eigenvalue Problems

Directory of Open Access Journals (Sweden)

Wong Fu-Hsiang

2010-01-01

Full Text Available We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, , , with some suitable boundary conditions, where is a parameter.

Existence of Positive Solutions for Nonlinear Eigenvalue Problems

Directory of Open Access Journals (Sweden)

Fan-Kai Kung

2010-01-01

Full Text Available We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, u″+λf(t,u=0, 0

Morozov-type discrepancy principle for nonlinear ill-posed problems ...

Indian Academy of Sciences (India)

For proving the existence of a regularization parameter under a Morozov-type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freéchet derivative and requirement of the Lipschitz ...

Morozov-type discrepancy principle for nonlinear ill-posed problems ...

Indian Academy of Sciences (India)

2016-08-26

Aug 26, 2016 ... For proving the existence of a regularization parameter under a Morozov-type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freéchet derivative and requirement ...

Mixed-mode loading of the structural elements with defect

Directory of Open Access Journals (Sweden)

Larisa V. Stepanova

2015-06-01

Full Text Available In the article the problem of determining the stress-strain state near the mixed-mode crack tip in a power-law material under plane stress conditions is considered. The eigenfunction method is used for the mixed-mode crack tip problem. It is shown that the eigenfunction expansion method results in the nonlinear eigenvalue problem. The numeric solution of the nonlinear eigenvalue problem formulated is obtained. The power of the distance from the crack tip is the eigenvalue of the nonlinear eigenvalue problem considered whereas the angular distributions of the stress components are the eigenfunctions. The new eigenvalues different from the eigenvalues of the Hutchinson–Rice–Rosengren are found. It is shown that the new asymptotic solution can be interpreted as the self-similar intermediate asymptotics of the stress field in the vicinity of the crack tip at distances which are very small compared to the crack length or the size of the specimen and at distances which are large compared to the length of the completely damaged zone. The developed method allows us to construct the geometry of the completely damaged zone in vicinity of the crack tip.

Global Optimization of Nonlinear Blend-Scheduling Problems

Directory of Open Access Journals (Sweden)

Pedro A. Castillo Castillo

2017-04-01

Full Text Available The scheduling of gasoline-blending operations is an important problem in the oil refining industry. This problem not only exhibits the combinatorial nature that is intrinsic to scheduling problems, but also non-convex nonlinear behavior, due to the blending of various materials with different quality properties. In this work, a global optimization algorithm is proposed to solve a previously published continuous-time mixed-integer nonlinear scheduling model for gasoline blending. The model includes blend recipe optimization, the distribution problem, and several important operational features and constraints. The algorithm employs piecewise McCormick relaxation (PMCR and normalized multiparametric disaggregation technique (NMDT to compute estimates of the global optimum. These techniques partition the domain of one of the variables in a bilinear term and generate convex relaxations for each partition. By increasing the number of partitions and reducing the domain of the variables, the algorithm is able to refine the estimates of the global solution. The algorithm is compared to two commercial global solvers and two heuristic methods by solving four examples from the literature. Results show that the proposed global optimization algorithm performs on par with commercial solvers but is not as fast as heuristic approaches.

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.

Bonus algorithm for large scale stochastic nonlinear programming problems

CERN Document Server

Diwekar, Urmila

2015-01-01

This book presents the details of the BONUS algorithm and its real world applications in areas like sensor placement in large scale drinking water networks, sensor placement in advanced power systems, water management in power systems, and capacity expansion of energy systems. A generalized method for stochastic nonlinear programming based on a sampling based approach for uncertainty analysis and statistical reweighting to obtain probability information is demonstrated in this book. Stochastic optimization problems are difficult to solve since they involve dealing with optimization and uncertainty loops. There are two fundamental approaches used to solve such problems. The first being the decomposition techniques and the second method identifies problem specific structures and transforms the problem into a deterministic nonlinear programming problem. These techniques have significant limitations on either the objective function type or the underlying distributions for the uncertain variables. Moreover, these ...

Initial boundary value problems of nonlinear wave equations in an exterior domain

International Nuclear Information System (INIS)

Chen Yunmei.

1987-06-01

In this paper, we investigate the existence and uniqueness of the global solutions to the initial boundary value problems of nonlinear wave equations in an exterior domain. When the space dimension n >= 3, the unique global solution of the above problem is obtained for small initial data, even if the nonlinear term is fully nonlinear and contains the unknown function itself. (author). 10 refs

Some New Results in Astrophysical Problems of Nonlinear Theory of Radiative Transfer

Science.gov (United States)

Pikichyan, H. V.

2017-07-01

In the interpretation of the observed astrophysical spectra, a decisive role is related to nonlinear problems of radiative transfer, because the processes of multiple interactions of matter of cosmic medium with the exciting intense radiation ubiquitously occur in astrophysical objects, and in their vicinities. Whereas, the intensity of the exciting radiation changes the physical properties of the original medium, and itself was modified, simultaneously, in a self-consistent manner under its influence. In the present report, we show that the consistent application of the principle of invariance in the nonlinear problem of bilateral external illumination of a scattering/absorbing one-dimensional anisotropic medium of finite geometrical thickness allows for simplifications that were previously considered as a prerogative only of linear problems. The nonlinear problem is analyzed through the three methods of the principle of invariance: (i) an adding of layers, (ii) its limiting form, described by differential equations of invariant imbedding, and (iii) a transition to the, so-called, functional equations of the "Ambartsumyan's complete invariance". Thereby, as an alternative to the Boltzmann equation, a new type of equations, so-called "kinetic equations of equivalence", are obtained. By the introduction of new functions - the so-called "linear images" of solution of nonlinear problem of radiative transfer, the linear structure of the solution of the nonlinear problem under study is further revealed. Linear images allow to convert naturally the statistical characteristics of random walk of a "single quantum" or their "beam of unit intensity", as well as widely known "probabilistic interpretation of phenomena of transfer", to the field of nonlinear problems. The structure of the equations obtained for determination of linear images is typical of linear problems.

Methods of intermediate problems for eigenvalues

CERN Document Server

Weinstein, Alexander

1972-01-01

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat

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.

SEACAS Theory Manuals: Part 1. Problem Formulation in Nonlinear Solid Mechancis

Energy Technology Data Exchange (ETDEWEB)

Attaway, S.W.; Laursen, T.A.; Zadoks, R.I.

1998-08-01

This report gives an introduction to the basic concepts and principles involved in the formulation of nonlinear problems in solid mechanics. By way of motivation, the discussion begins with a survey of some of the important sources of nonlinearity in solid mechanics applications, using wherever possible simple one dimensional idealizations to demonstrate the physical concepts. This discussion is then generalized by presenting generic statements of initial/boundary value problems in solid mechanics, using linear elasticity as a template and encompassing such ideas as strong and weak forms of boundary value problems, boundary and initial conditions, and dynamic and quasistatic idealizations. The notational framework used for the linearized problem is then extended to account for finite deformation of possibly inelastic solids, providing the context for the descriptions of nonlinear continuum mechanics, constitutive modeling, and finite element technology given in three companion reports.

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

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...

Implicit solvers for large-scale nonlinear problems

International Nuclear Information System (INIS)

Keyes, David E; Reynolds, Daniel R; Woodward, Carol S

2006-01-01

Computational scientists are grappling with increasingly complex, multi-rate applications that couple such physical phenomena as fluid dynamics, electromagnetics, radiation transport, chemical and nuclear reactions, and wave and material propagation in inhomogeneous media. Parallel computers with large storage capacities are paving the way for high-resolution simulations of coupled problems; however, hardware improvements alone will not prove enough to enable simulations based on brute-force algorithmic approaches. To accurately capture nonlinear couplings between dynamically relevant phenomena, often while stepping over rapid adjustments to quasi-equilibria, simulation scientists are increasingly turning to implicit formulations that require a discrete nonlinear system to be solved for each time step or steady state solution. Recent advances in iterative methods have made fully implicit formulations a viable option for solution of these large-scale problems. In this paper, we overview one of the most effective iterative methods, Newton-Krylov, for nonlinear systems and point to software packages with its implementation. We illustrate the method with an example from magnetically confined plasma fusion and briefly survey other areas in which implicit methods have bestowed important advantages, such as allowing high-order temporal integration and providing a pathway to sensitivity analyses and optimization. Lastly, we overview algorithm extensions under development motivated by current SciDAC applications

Response of Non-Linear Shock Absorbers-Boundary Value Problem Analysis

Science.gov (United States)

Rahman, M. A.; Ahmed, U.; Uddin, M. S.

2013-08-01

A nonlinear boundary value problem of two degrees-of-freedom (DOF) untuned vibration damper systems using nonlinear springs and dampers has been numerically studied. As far as untuned damper is concerned, sixteen different combinations of linear and nonlinear springs and dampers have been comprehensively analyzed taking into account transient terms. For different cases, a comparative study is made for response versus time for different spring and damper types at three important frequency ratios: one at r = 1, one at r > 1 and one at r <1. The response of the system is changed because of the spring and damper nonlinearities; the change is different for different cases. Accordingly, an initially stable absorber may become unstable with time and vice versa. The analysis also shows that higher nonlinearity terms make the system more unstable. Numerical simulation includes transient vibrations. Although problems are much more complicated compared to those for a tuned absorber, a comparison of the results generated by the present numerical scheme with the exact one shows quite a reasonable agreement

Modal representation of geometrically nonlinear behavior by the finite element method

International Nuclear Information System (INIS)

Nagy, D.A.

1977-01-01

A method is presented for representing mild geometrically nonlinear static behavior of thin-type structures, within the finite element method, in terms of linear elastic and linear (bifurcation) buckling analysis results for structural loading or geometry situations which violate the idealized restrictive (perfect) interpretation of linear behavior up to bifurcation. Formulation of the finite element displacement method for material linearity but retaining the full, nonlinear strain-displacement relations (geometric nonlinearity) leads to highly nonlinear equations relating the unknown nodal generalized displacements r to the applied loading R. Restriction to small strains alone does not linearize these equations for thin-type structural configurations; only explicitly requiring that all products of displacement gadients be much smaller than the gadients themselves reduces the equations to the familiar linear form Ksub(e)r=R, where Ksub(e) is the elastic stiffness. Assuming then that the solutions r of the linear equations also satisfies the full nonlinear equations (i.e., that the above explicit requirement is satisfied), a second solution to the full equations can be sought for a one-parameter loading path lambdaR, leading to the well-known linear (bifurcation) buckling eigenvalue problem Ksub(e)X=-Ksub(g)XΛ where Ksub(g) is the geometric stiffness, X the matrix whose columns are the eigenvectors (so-called buckling mode shapes) and Λ is a diagonal matrix of eigenvalues lambda(i) (so-called load scale factors). From the viewpoint of the practising structural analyst using finite element software, the method presented here gives broader and deeper significance to an existing linear (bifurcation) buckling analysis capability, in that the additional computations are minimal beyond those already required for a linear static and buckling analysis, and should be easily performable within any well-designed general purpose finite element system

Nonlinear travelling waves in rotating Hagen–Poiseuille flow

Science.gov (United States)

Pier, Benoît; Govindarajan, Rama

2018-03-01

The dynamics of viscous flow through a rotating pipe is considered. Small-amplitude stability characteristics are obtained by linearizing the Navier–Stokes equations around the base flow and solving the resulting eigenvalue problems. For linearly unstable configurations, the dynamics leads to fully developed finite-amplitude perturbations that are computed by direct numerical simulations of the complete Navier–Stokes equations. By systematically investigating all linearly unstable combinations of streamwise wave number k and azimuthal mode number m, for streamwise Reynolds numbers {{Re}}z ≤slant 500 and rotational Reynolds numbers {{Re}}{{Ω }} ≤slant 500, the complete range of nonlinear travelling waves is obtained and the associated flow fields are characterized.

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

APPLICATION OF PARAMETER CONTINUATION METHOD FOR INVESTIGATION OF VIBROIMPACT SYSTEMS DYNAMIC BEHAVIOUR. PROBLEM STATE. SHORT SURVEY OF WORLD SCIENTIFIC LITERATURE

Directory of Open Access Journals (Sweden)

V.A. Bazhenov

2014-12-01

Full Text Available Authors in their works study vibroimpact system dynamic behaviour by numerical parametric continuation technique combined with shooting and Newton-Raphson’s methods. The technique is adapted to two-mass two-degree-of-freedom vibroimpact system under periodic excitation. Impact is simulated by nonlinear contact interaction force based on Hertz’s contact theory. Stability or instability of obtained periodic solutions is determined by monodromy matrix eigenvalues (multipliers based on Floquet’s theory. In the present paper we describe the state of problem of parameter continuation method using for nonlinear tasks solution. Also we give the short survey of numerous contemporary literature in English and Russian about parameter continuation method application for nonlinear problems. This method is applied for vibroimpact problem solving more rarely because of the difficulties connected with repeated impacts.

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.

Morozov-type discrepancy principle for nonlinear ill-posed problems ...

Indian Academy of Sciences (India)

[3] Engl H W, Kunisch K and Neubauer A, Convergence rates for Tikhonov regularization of nonliner problems, Inverse Problems 5 (1989) 523–540. [4] Hanke M, Neubauer A and Scherzer O, A convergence analysis of Landweber iteration for nonlinear ill-posed problems, Numer. Math. 72 (1995) 21–37. [5] Hofmann B and ...

Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

Directory of Open Access Journals (Sweden)

Tieshan He

2011-01-01

Full Text Available This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.

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.

Nonlinear problems in theoretical physics

International Nuclear Information System (INIS)

Ranada, A.F.

1979-01-01

This volume contains the lecture notes and review talks delivered at the 9th GIFT international seminar on theoretical physics on the general subject 'Nonlinear Problems in Theoretical Physics'. Mist contributions deal with recent developments in the theory of the spectral transformation and solitons, but there are also articles from the field of transport theory and plasma physics and an unconventional view of classical and quantum electrodynamics. All contributions to this volume will appear under their corresponding subject categories. (HJ)

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

Nonlinear problems in fluid dynamics and inverse scattering: Nonlinear waves and inverse scattering

Science.gov (United States)

Ablowitz, Mark J.

1994-12-01

Research investigations involving the fundamental understanding and applications of nonlinear wave motion and related studies of inverse scattering and numerical computation have been carried out and a number of significant results have been obtained. A class of nonlinear wave equations which can be solved by the inverse scattering transform (IST) have been studied, including the Kadaomtsev-Petviashvili (KP) equation, the Davey-Stewartson equation, and the 2+1 Toda system. The solutions obtained by IST correspond to the Cauchy initial value problem with decaying initial data. We have also solved two important systems via the IST method: a 'Volterra' system in 2+1 dimensions and a new one dimensional nonlinear equation which we refer to as the Toda differential-delay equation. Research in computational chaos in moderate to long time numerical simulations continues.

Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory

Directory of Open Access Journals (Sweden)

Iman Eshraghi

2016-09-01

Full Text Available Imperfection sensitivity of large amplitude vibration of curved single-walled carbon nanotubes (SWCNTs is considered in this study. The SWCNT is modeled as a Timoshenko nano-beam and its curved shape is included as an initial geometric imperfection term in the displacement field. Geometric nonlinearities of von Kármán type and nonlocal elasticity theory of Eringen are employed to derive governing equations of motion. Spatial discretization of governing equations and associated boundary conditions is performed using differential quadrature (DQ method and the corresponding nonlinear eigenvalue problem is iteratively solved. Effects of amplitude and location of the geometric imperfection, and the nonlocal small-scale parameter on the nonlinear frequency for various boundary conditions are investigated. The results show that the geometric imperfection and non-locality play a significant role in the nonlinear vibration characteristics of curved SWCNTs.

Selected Problems in Nonlinear Dynamics and Sociophysics

Science.gov (United States)

Westley, Alexandra Renee

This Ph.D. dissertation focuses on a collection of problems on the dynamical behavior of nonlinear many-body systems, drawn from two substantially different areas. First, the dynamical behavior seen in strongly nonlinear lattices such as in the Fermi-Pasta-Ulam-Tsingou (FPUT) system (part I) and second, time evolution behavior of interacting living objects which can be broadly considered as sociophysics systems (part II). The studies on FPUT-like systems will comprise of five chapters, dedicated to the properties of solitary and anti-solitary waves in the system, how localized nonlinear excitations decay and spread throughout these lattices, how two colliding solitary waves can precipitate highly localized and stable excitations, a possible alternative way to view these localized excitations through Duffing oscillators, and finally an exploration of parametric resonance in an FPUT-like lattice. Part II consists of two problems in the context of sociophysics. I use molecular dynamics inspired simulations to study the size and the stability of social groups of chimpanzees (such as those seen in central Africa) and compare the results with existing observations on the stability of chimpanzee societies. Secondly, I use an agent-based model to simulate land battles between an intelligent army and an insurgency when both have access to equally powerful weaponry. The study considers genetic algorithm based adaptive strategies to infer the strategies needed for the intelligent army to win the battles.

Analytic approximations to nonlinear boundary value problems modeling beam-type nano-electromechanical systems

Energy Technology Data Exchange (ETDEWEB)

Zou, Li [Dalian Univ. of Technology, Dalian City (China). State Key Lab. of Structural Analysis for Industrial Equipment; Liang, Songxin; Li, Yawei [Dalian Univ. of Technology, Dalian City (China). School of Mathematical Sciences; Jeffrey, David J. [Univ. of Western Ontario, London (Canada). Dept. of Applied Mathematics

2017-06-01

Nonlinear boundary value problems arise frequently in physical and mechanical sciences. An effective analytic approach with two parameters is first proposed for solving nonlinear boundary value problems. It is demonstrated that solutions given by the two-parameter method are more accurate than solutions given by the Adomian decomposition method (ADM). It is further demonstrated that solutions given by the ADM can also be recovered from the solutions given by the two-parameter method. The effectiveness of this method is demonstrated by solving some nonlinear boundary value problems modeling beam-type nano-electromechanical systems.

A new integrability theory for certain nonlinear physical problems

International Nuclear Information System (INIS)

Berger, M.S.

1993-01-01

A new mathematically sound integrability theory for certain nonlinear problems defined by ordinary or partial differential equations is defined. The new theory works in an arbitrary finite number of space dimensions. Moreover, if a system is integrable in the new sense described here, it has a remarkable stability property that distinguishes if from any previously known integrability ideas. The new theory proceeds by establishing a ''global normal form'' for the problem at hand. This normal form holds subject to canonical coordinate transformations, extending such classical ideas by using new nonlinear methods of infinite dimensional functional analysis. The global normal form in question is related to the mathematical theory of singularities of mappings of H. Whitney and R. Thom extended globally and form finite to infinite dimensions. Thus bifurcation phenomena are naturally included in the new integrability theory. Typical examples include the classically nonintegrable Riccati equation, certain non-Euclidean mean field theories, certain parabolic reaction diffusion equations and the hyperbolic nonlinear telegrapher's equation. (Author)

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

A Note on the Solutions of the Van der Pol and Duffing Equations Using a Linearisation Method

Directory of Open Access Journals (Sweden)

Sandile S. Motsa

2012-01-01

Full Text Available We present a novel application of the successive linearisation method to the classical Van der Pol and Duffing oscillator equations. By recasting the governing equations as nonlinear eigenvalue problems we obtain accurate values of the frequency and amplitude. We demonstrate that the proposed method can be used to obtain the limit cycle and bifurcation diagrams of the governing equations. Comparison with exact and other results in the literature shows that the method is accurate and effective in finding solutions of nonlinear equations with oscillatory solutions, nonlinear eigenvalue problems, and other nonlinear problems with bifurcations.

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)

Probabilistic finite elements for transient analysis in nonlinear continua

Science.gov (United States)

Liu, W. K.; Belytschko, T.; Mani, A.

1985-01-01

The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

Some nonlinear problems in the manipulation of beams

International Nuclear Information System (INIS)

Sessler, A.M.

1990-01-01

An overview is given of nonlinear problems that arise in the manipulation of beams. Beams can be made of material particles or photons, can be intense or dilute, can be energetic or not, and they can be propagating in vacuum or in a medium. The nonlinear aspects of the motion are different in each case, and this diversity of behavior is categorized. Many examples are given, which serves to illustrate the categorization and, furthermore, display the richness of behavior encountered in the physics of beams. 25 refs., 5 figs

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.

Solution of Contact Problems for Nonlinear Gao Beam and Obstacle

Directory of Open Access Journals (Sweden)

J. Machalová

2015-01-01

Full Text Available Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.

A fast nonlinear conjugate gradient based method for 3D frictional contact problems

NARCIS (Netherlands)

Zhao, J.; Vollebregt, E.A.H.; Oosterlee, C.W.

2014-01-01

This paper presents a fast numerical solver for a nonlinear constrained optimization problem, arising from a 3D frictional contact problem. It incorporates an active set strategy with a nonlinear conjugate gradient method. One novelty is to consider the tractions of each slip element in a polar

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.)

Linear instability and nonlinear motion of rotating plasma

International Nuclear Information System (INIS)

Liu, J.

1985-01-01

Two coupled nonlinear equations describing the flute dynamics of the magnetically confined low-β collisionless rotating plasma are derived. The linear instability and nonlinear dynamics of the rotating column are analyzed theoretically. In the linear stability analysis, a new sufficient condition of stability is obtained. From the exact solution of eigenvalue equation for Gaussian density profile and uniform rotation of the plasma, the stability of the system strongly depends on the direction of plasma rotation, FLR effect and the location of the conducting wall. An analytic expression showing the finite wall effect on different normal modes is obtained and it explains the different behavior of (1,0) normal mode from other modes. The sheared rotation driven instability is investigated by using three model equilibrium profiles, and the analytic expressions of eigenvalues which includes the wall effect are obtained. The analogy between shear rotation driven instability and the instability driven by sheared plane parallel flow in the inviscid fluid is analyzed. Applying the linear analysis to the central cell of tandem mirror system, the trapped particle instability with only passing electronics is analyzed. For uniform rotation and Gaussian density profile, an analytic expression that determines the stability boundary is found. The nonlinear analysis shows that the nonlinear equations have a solitary vortex solution which is very similar to the vortex solution of nonlinear Rossby wave equation

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.

Intrinsic nonlinearity and method of disturbed observations in inverse problems of celestial mechanics

Science.gov (United States)

Avdyushev, Victor A.

2017-12-01

Orbit determination from a small sample of observations over a very short observed orbital arc is a strongly nonlinear inverse problem. In such problems an evaluation of orbital uncertainty due to random observation errors is greatly complicated, since linear estimations conventionally used are no longer acceptable for describing the uncertainty even as a rough approximation. Nevertheless, if an inverse problem is weakly intrinsically nonlinear, then one can resort to the so-called method of disturbed observations (aka observational Monte Carlo). Previously, we showed that the weaker the intrinsic nonlinearity, the more efficient the method, i.e. the more accurate it enables one to simulate stochastically the orbital uncertainty, while it is strictly exact only when the problem is intrinsically linear. However, as we ascertained experimentally, its efficiency was found to be higher than that of other stochastic methods widely applied in practice. In the present paper we investigate the intrinsic nonlinearity in complicated inverse problems of Celestial Mechanics when orbits are determined from little informative samples of observations, which typically occurs for recently discovered asteroids. To inquire into the question, we introduce an index of intrinsic nonlinearity. In asteroid problems it evinces that the intrinsic nonlinearity can be strong enough to affect appreciably probabilistic estimates, especially at the very short observed orbital arcs that the asteroids travel on for about a hundredth of their orbital periods and less. As it is known from regression analysis, the source of intrinsic nonlinearity is the nonflatness of the estimation subspace specified by a dynamical model in the observation space. Our numerical results indicate that when determining asteroid orbits it is actually very slight. However, in the parametric space the effect of intrinsic nonlinearity is exaggerated mainly by the ill-conditioning of the inverse problem. Even so, as for the

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

On a Highly Nonlinear Self-Obstacle Optimal Control Problem

Energy Technology Data Exchange (ETDEWEB)

Di Donato, Daniela, E-mail: daniela.didonato@unitn.it [University of Trento, Department of Mathematics (Italy); Mugnai, Dimitri, E-mail: dimitri.mugnai@unipg.it [Università di Perugia, Dipartimento di Matematica e Informatica (Italy)

2015-10-15

We consider a non-quadratic optimal control problem associated to a nonlinear elliptic variational inequality, where the obstacle is the control itself. We show that, fixed a desired profile, there exists an optimal solution which is not far from it. Detailed characterizations of the optimal solution are given, also in terms of approximating problems.

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.)

Experimental analysis of nonlinear problems in solid mechanics

International Nuclear Information System (INIS)

1982-01-01

The booklet presents abstracts of papers from the Euromech Colloqium No. 152 held from Sept. 20th to 24th, 1982 in Wuppertal, Federal Republic of Germany. All the papers are dealing with Experimental Analysis of Nonlinear Problems in Solid Mechanics. (RW)

Lectures on nonlinear evolution equations initial value problems

CERN Document Server

Racke, Reinhard

2015-01-01

This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value p...

A nonsmooth nonlinear conjugate gradient method for interactive contact force problems

DEFF Research Database (Denmark)

Silcowitz, Morten; Abel, Sarah Maria Niebe; Erleben, Kenny

2010-01-01

of a nonlinear complementarity problem (NCP), which can be solved using an iterative splitting method, such as the projected Gauss–Seidel (PGS) method. We present a novel method for solving the NCP problem by applying a Fletcher–Reeves type nonlinear nonsmooth conjugate gradient (NNCG) type method. We analyze...... and present experimental convergence behavior and properties of the new method. Our results show that the NNCG method has at least the same convergence rate as PGS, and in many cases better....

New Exact Penalty Functions for Nonlinear Constrained Optimization Problems

Directory of Open Access Journals (Sweden)

Bingzhuang Liu

2014-01-01

Full Text Available For two kinds of nonlinear constrained optimization problems, we propose two simple penalty functions, respectively, by augmenting the dimension of the primal problem with a variable that controls the weight of the penalty terms. Both of the penalty functions enjoy improved smoothness. Under mild conditions, it can be proved that our penalty functions are both exact in the sense that local minimizers of the associated penalty problem are precisely the local minimizers of the original constrained problem.

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)

A Lax integrable hierarchy, bi-Hamiltonian structure and finite-dimensional Liouville integrable involutive systems

International Nuclear Information System (INIS)

Xia Tiecheng; Chen Xiaohong; Chen Dengyuan

2004-01-01

An eigenvalue problem and the associated new Lax integrable hierarchy of nonlinear evolution equations are presented in this paper. As two reductions, the generalized nonlinear Schroedinger equations and the generalized mKdV equations are obtained. Zero curvature representation and bi-Hamiltonian structure are established for the whole hierarchy based on a pair of Hamiltonian operators (Lenard's operators), and it is shown that the hierarchy of nonlinear evolution equations is integrable in Liouville's sense. Thus the hierarchy of nonlinear evolution equations has infinitely many commuting symmetries and conservation laws. Moreover the eigenvalue problem is nonlinearized as a finite-dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenvalue functions. Finally finite-dimensional Liouville integrable system are found, and the involutive solutions of the hierarchy of equations are given. In particular, the involutive solutions are developed for the system of generalized nonlinear Schroedinger equations

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

Directory of Open Access Journals (Sweden)

Mickael D. Chekroun

2017-07-01

Full Text Available Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated here in terms of optimal control of energy balance climate models posed on the sphere $\\mathbb{S}^2$.

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.

NOLB: Nonlinear Rigid Block Normal Mode Analysis Method

OpenAIRE

Hoffmann , Alexandre; Grudinin , Sergei

2017-01-01

International audience; We present a new conceptually simple and computationally efficient method for nonlinear normal mode analysis called NOLB. It relies on the rotations-translations of blocks (RTB) theoretical basis developed by Y.-H. Sanejouand and colleagues. We demonstrate how to physically interpret the eigenvalues computed in the RTB basis in terms of angular and linear velocities applied to the rigid blocks and how to construct a nonlinear extrapolation of motion out of these veloci...

Center manifold for nonintegrable nonlinear Schroedinger equations on the line

International Nuclear Information System (INIS)

Weder, R.

2000-01-01

In this paper we study the following nonlinear Schroedinger equation on the line, where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x → ± ∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of H:= -d 2 /dx 2 +V consists of one simple negative eigenvalue and absolutely-continuous spectrum filling (0,∞). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all small solutions approach a particular periodic orbit in the center manifold as t→ ± ∞. In general, the periodic orbits are different for t→ ± ∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic as t→ ± ∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→ ± ∞. (orig.)

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.

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

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...

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...

Radiation from nonlinear coupling of plasma waves

International Nuclear Information System (INIS)

Fung, S.F.

1986-01-01

The author examines the generation of electromagnetic radiation by nonlinear resonant interactions of plasma waves in a cold, uniformly magnetized plasma. In particular, he considers the up-conversion of two electrostatic wave packets colliding to produce high frequency electromagnetic radiation. Efficient conversion of electrostatic to electromagnetic wave energy occurs when the pump amplitudes approach and exceed the pump depletion threshold. Results from the inverse scattering transform analysis of the three-wave interaction equations are applied. When the wave packets are initially separated, the fully nonlinear set of coupling equations, which describe the evolution of the wave packets, can be reduced to three separate eigenvalue problems; each can be considered as a scattering problem, analogous to eh Schroedinger equation. In the scattering space, the wave packet profiles act as the scattering potentials. When the wavepacket areas approach (or exceed) π/2, the wave functions are localized (bound states) and the scattering potentials are said to contain solitons. Exchange of solitons occurs during the interaction. The transfer of solitons from the pump waves to the electromagnetic wave leads to pump depletion and the production of strong radiation. The emission of radio waves is considered by the coupling of two upper-hybrid branch wave packets, and an upper-hybrid and a lower hybrid branch wave packet

A quadratic approximation-based algorithm for the solution of multiparametric mixed-integer nonlinear programming problems

KAUST Repository

Domí nguez, Luis F.; Pistikopoulos, Efstratios N.

2012-01-01

An algorithm for the solution of convex multiparametric mixed-integer nonlinear programming problems arising in process engineering problems under uncertainty is introduced. The proposed algorithm iterates between a multiparametric nonlinear

A fast nonlinear conjugate gradient based method for 3D concentrated frictional contact problems

NARCIS (Netherlands)

J. Zhao (Jing); E.A.H. Vollebregt (Edwin); C.W. Oosterlee (Cornelis)

2015-01-01

htmlabstractThis paper presents a fast numerical solver for a nonlinear constrained optimization problem, arising from 3D concentrated frictional shift and rolling contact problems with dry Coulomb friction. The solver combines an active set strategy with a nonlinear conjugate gradient method. One

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.

On the solvability of initial boundary value problems for nonlinear ...

African Journals Online (AJOL)

In this paper, we study the initial boundary value problems for a non-linear time dependent Schrödinger equation with Dirichlet and Neumann boundary conditions, respectively. We prove the existence and uniqueness of solutions of the initial boundary value problems by using Galerkin's method. Keywords: Initial boundary ...

First international conference on nonlinear problems in aviation and aerospace

International Nuclear Information System (INIS)

Sivasundaram, S.

1994-01-01

The International Conference on Nonlinear Problems in Aviation and Aerospace was held at Embry-Riddle Aeronautical University, Daytona Beach, Florida on May 9-11, 1996. This conference was sponsored by the International Federation of Nonlinear Analysts, International Federation of Information Processing, and Embry-Riddle Aeronautical University. Over one hundred engineers, scientists, and mathematicians from seventeen countries attended. These proceedings include keynote addresses, invited lectures, and contributed papers presented during the conference

Some problems on nonlinear hyperbolic equations and applications

CERN Document Server

Peng, YueJun

2010-01-01

This volume is composed of two parts: Mathematical and Numerical Analysis for Strongly Nonlinear Plasma Models and Exact Controllability and Observability for Quasilinear Hyperbolic Systems and Applications. It presents recent progress and results obtained in the domains related to both subjects without attaching much importance to the details of proofs but rather to difficulties encountered, to open problems and possible ways to be exploited. It will be very useful for promoting further study on some important problems in the future.

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.

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

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.

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

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.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

Science.gov (United States)

Polyanin, A. D.; Sorokin, V. G.

2017-12-01

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

On Newton-Raphson formulation and algorithm for displacement based structural dynamics problem with quadratic damping nonlinearity

Directory of Open Access Journals (Sweden)

Koh Kim Jie

2017-01-01

Full Text Available Quadratic damping nonlinearity is challenging for displacement based structural dynamics problem as the problem is nonlinear in time derivative of the primitive variable. For such nonlinearity, the formulation of tangent stiffness matrix is not lucid in the literature. Consequently, ambiguity related to kinematics update arises when implementing the time integration-iterative algorithm. In present work, an Euler-Bernoulli beam vibration problem with quadratic damping nonlinearity is addressed as the main source of quadratic damping nonlinearity arises from drag force estimation, which is generally valid only for slender structures. Employing Newton-Raphson formulation, tangent stiffness components associated with quadratic damping nonlinearity requires velocity input for evaluation purpose. For this reason, two mathematically equivalent algorithm structures with different kinematics arrangement are tested. Both algorithm structures result in the same accuracy and convergence characteristic of solution.

Free vibration of geometrically nonlinear micro-switches under electrostatic and Casimir forces

International Nuclear Information System (INIS)

Jia, X L; Kitipornchai, S; Lim, C W; Yang, J

2010-01-01

This paper investigates the free vibration characteristics of micro-switches under combined electrostatic, intermolecular forces and axial residual stress, with an emphasis on the effect of geometric nonlinear deformation due to mid-plane stretching and the influence of Casimir force. The micro-switch considered in this study is made of either homogeneous material or non-homogeneous functionally graded material with two material phases. The Euler–Bernoulli beam theory with von Karman type nonlinear kinematics is applied in the theoretical formulation. The principle of virtual work is used to derive the nonlinear governing differential equation. The eigenvalue problem which describes free vibration of the micro-beam at its statically deflected state is then solved using the differential quadrature method. The natural frequencies and mode shapes of micro-switches for four different boundary conditions (i.e. clamped–clamped, clamped–simply supported, simply supported and clamped–free) are obtained. The solutions are validated through direct comparisons with experimental and other existing results reported in previous studies. A parametric study is conducted to show the significant effects of geometric nonlinearity, Casimir force, axial residual stress and material composition for the natural frequencies

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.

Consensus problem in directed networks of multi-agents via nonlinear protocols

International Nuclear Information System (INIS)

Liu Xiwei; Chen Tianping; Lu Wenlian

2009-01-01

In this Letter, the consensus problem via distributed nonlinear protocols for directed networks is investigated. Its dynamical behaviors are described by ordinary differential equations (ODEs). Based on graph theory, matrix theory and the Lyapunov direct method, some sufficient conditions of nonlinear protocols guaranteeing asymptotical or exponential consensus are presented and rigorously proved. The main contribution of this work is that for nonlinearly coupled networks, we generalize the results for undirected networks to directed networks. Consensus under pinning control technique is also developed here. Simulations are also given to show the validity of the theories.

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...

Thermoelectrically induced nonlinear free vibration analysis of piezo laminated composite conical shell panel with random fiber orientation

Directory of Open Access Journals (Sweden)

Lal Achchhe

2017-09-01

Full Text Available This paper presents the free vibration response of piezo laminated composite geometrically nonlinear conical shell panel subjected to a thermo-electrical loading. The temperature field is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E2 only. The material properties are assumed to be independent of the temperature and the electric field. The basic formulation is based on higher order shear deformation plate theory (HSDT with von-Karman nonlinearity. A C0 nonlinear finite element method based on direct iterative approach is outlined and applied to solve nonlinear generalized eigenvalue problem. Parametric studies are carried out to examine the effect of amplitude ratios, stacking sequences, cone angles, piezoelectric layers, applied voltages, circumferential length to thickness ratios, change in temperatures and support boundary conditions on the nonlinear natural frequency of laminated conical shell panels. The present outlined approach has been validated with those available results in the literature.

Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation

Energy Technology Data Exchange (ETDEWEB)

Kaikina, Elena I., E-mail: ekaikina@matmor.unam.mx [Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 58089, Michoacán (Mexico)

2013-11-15

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.

Asymptotics for inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation

International Nuclear Information System (INIS)

Kaikina, Elena I.

2013-01-01

We consider the inhomogeneous Dirichlet initial-boundary value problem for the nonlinear Schrödinger equation, formulated on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time

Application of HPEM to investigate the response and stability of nonlinear problems in vibration

DEFF Research Database (Denmark)

Mohammadi, M.H.; Mohammadi, A.; Kimiaeifar, A.

2010-01-01

In this work, a powerful analytical method, called He's Parameter Expanding Methods (HPEM) is used to obtain the exact solution of nonlinear problems in nonlinear vibration. In this work, the governing equation is obtained by using Lagrange method, then the nonlinear governing equation is solved...

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.

On discrete maximum principles for nonlinear elliptic problems

Czech Academy of Sciences Publication Activity Database

Karátson, J.; Korotov, S.; Křížek, Michal

2007-01-01

Roč. 76, č. 1 (2007), s. 99-108 ISSN 0378-4754 R&D Projects: GA MŠk 1P05ME749; GA AV ČR IAA1019201 Institutional research plan: CEZ:AV0Z10190503 Keywords : nonlinear elliptic problem * mixed boundary conditions * finite element method Subject RIV: BA - General Mathematics Impact factor: 0.738, year: 2007

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

Assessment of Two Analytical Methods in Solving the Linear and Nonlinear Elastic Beam Deformation Problems

DEFF Research Database (Denmark)

Barari, Amin; Ganjavi, B.; Jeloudar, M. Ghanbari

2010-01-01

and fluid mechanics. Design/methodology/approach – Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics. Findings – Analytical solutions often fit under classical perturbation methods. However......, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all......Purpose – In the last two decades with the rapid development of nonlinear science, there has appeared ever-increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. This paper considers linear and nonlinear systems that are not only regarded as general...

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

On some nonlinear problems arising in the physics of ionized gases

International Nuclear Information System (INIS)

Hilhorst-Goldman, D.

1981-01-01

The author reports results obtained by rigorous analysis of a nonlinear differential equation for the electron density nsub(e) in a specific type of electrical discharge. The problem is essentially two-dimensional. She discusses in particular the escape of electrons to infinity above a critical temperature and the boundary layer exhibited by nsub(e) near zero temperature. A singular boundary value problem arising in a pre-breakdown gas discharge is discussed. A Coulomb gas is considered in a special experimental situation: the pre-breakdown gas discharge between two electrodes. The equation for the negative charge density can be formulated as a nonlinear parabolic equation degenerate at the origin. The existence and uniqueness of the solution are proved as well as the asymptotic stability of its unique steady state. Some results are also given about the rate of convergence. The variational characterisation of the limit solution of a singular perturbation problem and variational analysis of a perturbed free boundary problem are considered. (Auth./C.F.)

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)

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.

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)

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.

Inverse periodic problem for the discrete approximation of the Schroedinger nonlinear equation

International Nuclear Information System (INIS)

Bogolyubov, N.N.; Prikarpatskij, A.K.; AN Ukrainskoj SSR, Lvov. Inst. Prikladnykh Problem Mekhaniki i Matematiki)

1982-01-01

The problem of numerical solution of the Schroedinger nonlinear equation (1) iPSIsub(t) = PSIsub(xx)+-2(PSI)sup(2)PSI. The numerical solution of nonlinear differential equation supposes its discrete approximation is required for the realization of the computer calculation process. Tor the equation (1) there exists the following discrete approximation by variable x(2) iPSIsub(n, t) = (PSIsub(n+1)-2PSIsub(n)+PSIsub(n-1))/(Δx)sup(2)+-(PSIsub(n))sup(2)(PSIsub(n+1)+PSIsub(n-1)), n=0, +-1, +-2... where PSIsub(n)(+) is the corresponding value of PSI(x, t) function in the node and divisions with the equilibrium step Δx. The main problem is obtaining analytically exact solutions of the equations (2). The analysis of the equation system (2) is performed on the base of the discrete analogue of the periodic variant of the inverse scattering problem method developed with the aid of nonlinear equations of the Korteweg-de Vries type. Obtained in explicit form are analytical solutions of the equations system (2). The solutions are expressed through the Riemann THETA-function [ru

Application of nonlinear Krylov acceleration to radiative transfer problems

International Nuclear Information System (INIS)

Till, A. T.; Adams, M. L.; Morel, J. E.

2013-01-01

The iterative solution technique used for radiative transfer is normally nested, with outer thermal iterations and inner transport iterations. We implement a nonlinear Krylov acceleration (NKA) method in the PDT code for radiative transfer problems that breaks nesting, resulting in more thermal iterations but significantly fewer total inner transport iterations. Using the metric of total inner transport iterations, we investigate a crooked-pipe-like problem and a pseudo-shock-tube problem. Using only sweep preconditioning, we compare NKA against a typical inner / outer method employing GMRES / Newton and find NKA to be comparable or superior. Finally, we demonstrate the efficacy of applying diffusion-based preconditioning to grey problems in conjunction with NKA. (authors)

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.

Analytical Solution of Nonlinear Problems in Classical Dynamics by Means of Lagrange-Ham

DEFF Research Database (Denmark)

Kimiaeifar, Amin; Mahdavi, S. H; Rabbani, A.

2011-01-01

In this work, a powerful analytical method, called Homotopy Analysis Methods (HAM) is coupled with Lagrange method to obtain the exact solution for nonlinear problems in classic dynamics. In this work, the governing equations are obtained by using Lagrange method, and then the nonlinear governing...

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)

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.

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

The solution of a coupled system of nonlinear physical problems using the homotopy analysis method

International Nuclear Information System (INIS)

El-Wakil, S A; Abdou, M A

2010-01-01

In this article, the homotopy analysis method (HAM) has been applied to solve coupled nonlinear evolution equations in physics. The validity of this method has been successfully demonstrated by applying it to two nonlinear evolution equations, namely coupled nonlinear diffusion reaction equations and the (2+1)-dimensional Nizhnik-Novikov Veselov system. The results obtained by this method show good agreement with the ones obtained by other methods. The proposed method is a powerful and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The basic ideas of this approach can be widely employed to solve other strongly nonlinear problems.

COYOTE: a finite element computer program for nonlinear heat conduction problems

International Nuclear Information System (INIS)

Gartling, D.K.

1978-06-01

COYOTE is a finite element computer program designed for the solution of two-dimensional, nonlinear heat conduction problems. The theoretical and mathematical basis used to develop the code is described. Program capabilities and complete user instructions are presented. Several example problems are described in detail to demonstrate the use of the program

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.

Nonlinear triple-point problems on time scales

Directory of Open Access Journals (Sweden)

Douglas R. Anderson

2004-04-01

Full Text Available We establish the existence of multiple positive solutions to the nonlinear second-order triple-point boundary-value problem on time scales, $$displaylines{ u^{Delta abla}(t+h(tf(t,u(t=0, cr u(a=alpha u(b+delta u^Delta(a,quad eta u(c+gamma u^Delta(c=0 }$$ for $tin[a,c]subsetmathbb{T}$, where $mathbb{T}$ is a time scale, $eta, gamma, deltage 0$ with $Beta+gamma>0$, $0

Least Squares Problems with Absolute Quadratic Constraints

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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.

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)

TENSOLVE: A software package for solving systems of nonlinear equations and nonlinear least squares problems using tensor methods

Energy Technology Data Exchange (ETDEWEB)

Bouaricha, A. [Argonne National Lab., IL (United States). Mathematics and Computer Science Div.; Schnabel, R.B. [Colorado Univ., Boulder, CO (United States). Dept. of Computer Science

1996-12-31

This paper describes a modular software package for solving systems of nonlinear equations and nonlinear least squares problems, using a new class of methods called tensor methods. It is intended for small to medium-sized problems, say with up to 100 equations and unknowns, in cases where it is reasonable to calculate the Jacobian matrix or approximate it by finite differences at each iteration. The software allows the user to select between a tensor method and a standard method based upon a linear model. The tensor method models F({ital x}) by a quadratic model, where the second-order term is chosen so that the model is hardly more expensive to form, store, or solve than the standard linear model. Moreover, the software provides two different global strategies, a line search and a two- dimensional trust region approach. Test results indicate that, in general, tensor methods are significantly more efficient and robust than standard methods on small and medium-sized problems in iterations and function evaluations.

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.

Multiple solutions for inhomogeneous nonlinear elliptic problems arising in astrophyiscs

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Marco Calahorrano

2004-04-01

Full Text Available Using variational methods we prove the existence and multiplicity of solutions for some nonlinear inhomogeneous elliptic problems on a bounded domain in $mathbb{R}^n$, with $ngeq 2$ and a smooth boundary, and when the domain is $mathbb{R}_+^n$

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.

Lavrentiev regularization method for nonlinear ill-posed problems

International Nuclear Information System (INIS)

Kinh, Nguyen Van

2002-10-01

In this paper we shall be concerned with Lavientiev regularization method to reconstruct solutions x 0 of non ill-posed problems F(x)=y o , where instead of y 0 noisy data y δ is an element of X with absolut(y δ -y 0 ) ≤ δ are given and F:X→X is an accretive nonlinear operator from a real reflexive Banach space X into itself. In this regularization method solutions x α δ are obtained by solving the singularly perturbed nonlinear operator equation F(x)+α(x-x*)=y δ with some initial guess x*. Assuming certain conditions concerning the operator F and the smoothness of the element x*-x 0 we derive stability estimates which show that the accuracy of the regularized solutions is order optimal provided that the regularization parameter α has been chosen properly. (author)

Iterative solution of a nonlinear system arising in phase change problems

International Nuclear Information System (INIS)

Williams, M.A.

1987-01-01

We consider several iterative methods for solving the nonlinear system arising from an enthalpy formulation of a phase change problem. We present the formulation of the problem. Implicit discretization of the governing equations results in a mildly nonlinear system at each time step. We discuss solving this system using Jacobi, Gauss-Seidel, and SOR iterations and a new modified preconditioned conjugate gradient (MPCG) algorithm. The new MPCG algorithm and its properties are discussed in detail. Numerical results are presented comparing the performance of the SOR algorithm and the MPCG algorithm with 1-step SSOR preconditioning. The MPCG algorithm exhibits a superlinear rate of convergence. The SOR algorithm exhibits a linear rate of convergence. Thus, the MPCG algorithm requires fewer iterations to converge than the SOR algorithm. However in most cases, the SOR algorithm requires less total computation time than the MPCG algorithm. Hence, the SOR algorithm appears to be more appropriate for the class of problems considered. 27 refs., 11 figs

Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces

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Cho Yeol

2011-01-01

Full Text Available Abstract In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g. 2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.

New preconditioned conjugate gradient algorithms for nonlinear unconstrained optimization problems

International Nuclear Information System (INIS)

Al-Bayati, A.; Al-Asadi, N.

1997-01-01

This paper presents two new predilection conjugate gradient algorithms for nonlinear unconstrained optimization problems and examines their computational performance. Computational experience shows that the new proposed algorithms generally imp lone the efficiency of Nazareth's [13] preconditioned conjugate gradient algorithm. (authors). 16 refs., 1 tab

Improving stability and strength characteristics of framed structures with nonlinear behavior

Science.gov (United States)

Pezeshk, Shahram

1990-01-01

In this paper an optimal design procedure is introduced to improve the overall performance of nonlinear framed structures. The design methodology presented here is a multiple-objective optimization procedure whose objective functions involve the buckling eigenvalues and eigenvectors of the structure. A constant volume with bounds on the design variables is used in conjunction with an optimality criterion approach. The method provides a general tool for solving complex design problems and generally leads to structures with better limit strength and stability. Many algorithms have been developed to improve the limit strength of structures. In most applications geometrically linear analysis is employed with the consequence that overall strength of the design is overestimated. Directly optimizing the limit load of the structure would require a full nonlinear analysis at each iteration which would be prohibitively expensive. The objective of this paper is to develop an algorithm that can improve the limit-load of geometrically nonlinear framed structures while avoiding the nonlinear analysis. One of the novelties of the new design methodology is its ability to efficiently model and design structures under multiple loading conditions. These loading conditions can be different factored loads or any kind of loads that can be applied to the structure simultaneously or independently. Attention is focused on optimal design of space framed structures. Three-dimensional design problems are more complicated to carry out, but they yield insight into real behavior of the structure and can help avoiding some of the problems that might appear in planar design procedure such as the need for out-of-plane buckling constraint. Although researchers in the field of structural engineering generally agree that optimum design of three-dimension building frames especially in the seismic regions would be beneficial, methods have been slow to emerge. Most of the research in this area has dealt

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.

Bayesian nonlinear regression for large small problems

KAUST Repository

Chakraborty, Sounak; Ghosh, Malay; Mallick, Bani K.

2012-01-01

Statistical modeling and inference problems with sample sizes substantially smaller than the number of available covariates are challenging. This is known as large p small n problem. Furthermore, the problem is more complicated when we have multiple correlated responses. We develop multivariate nonlinear regression models in this setup for accurate prediction. In this paper, we introduce a full Bayesian support vector regression model with Vapnik's ε-insensitive loss function, based on reproducing kernel Hilbert spaces (RKHS) under the multivariate correlated response setup. This provides a full probabilistic description of support vector machine (SVM) rather than an algorithm for fitting purposes. We have also introduced a multivariate version of the relevance vector machine (RVM). Instead of the original treatment of the RVM relying on the use of type II maximum likelihood estimates of the hyper-parameters, we put a prior on the hyper-parameters and use Markov chain Monte Carlo technique for computation. We have also proposed an empirical Bayes method for our RVM and SVM. Our methods are illustrated with a prediction problem in the near-infrared (NIR) spectroscopy. A simulation study is also undertaken to check the prediction accuracy of our models. © 2012 Elsevier Inc.

Bayesian nonlinear regression for large small problems

KAUST Repository

Chakraborty, Sounak

2012-07-01

Statistical modeling and inference problems with sample sizes substantially smaller than the number of available covariates are challenging. This is known as large p small n problem. Furthermore, the problem is more complicated when we have multiple correlated responses. We develop multivariate nonlinear regression models in this setup for accurate prediction. In this paper, we introduce a full Bayesian support vector regression model with Vapnik\\'s ε-insensitive loss function, based on reproducing kernel Hilbert spaces (RKHS) under the multivariate correlated response setup. This provides a full probabilistic description of support vector machine (SVM) rather than an algorithm for fitting purposes. We have also introduced a multivariate version of the relevance vector machine (RVM). Instead of the original treatment of the RVM relying on the use of type II maximum likelihood estimates of the hyper-parameters, we put a prior on the hyper-parameters and use Markov chain Monte Carlo technique for computation. We have also proposed an empirical Bayes method for our RVM and SVM. Our methods are illustrated with a prediction problem in the near-infrared (NIR) spectroscopy. A simulation study is also undertaken to check the prediction accuracy of our models. © 2012 Elsevier Inc.

A Smooth Newton Method for Nonlinear Programming Problems with Inequality Constraints

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Vasile Moraru

2012-02-01

Full Text Available The paper presents a reformulation of the Karush-Kuhn-Tucker (KKT system associated nonlinear programming problem into an equivalent system of smooth equations. Classical Newton method is applied to solve the system of equations. The superlinear convergence of the primal sequence, generated by proposed method, is proved. The preliminary numerical results with a problems test set are presented.

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)

Periodic solutions for one dimensional wave equation with bounded nonlinearity

Science.gov (United States)

Ji, Shuguan

2018-05-01

This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u (x) satisfies ess infηu (x) > 0 with ηu (x) = 1/2 u″/u - 1/4 (u‧/u)2, which actually excludes the classical constant coefficient model. For the case ηu (x) = 0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T = 2p-1/q (p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu (x) > 0.

Nonlinear programming for classification problems in machine learning

Science.gov (United States)

Astorino, Annabella; Fuduli, Antonio; Gaudioso, Manlio

2016-10-01

We survey some nonlinear models for classification problems arising in machine learning. In the last years this field has become more and more relevant due to a lot of practical applications, such as text and web classification, object recognition in machine vision, gene expression profile analysis, DNA and protein analysis, medical diagnosis, customer profiling etc. Classification deals with separation of sets by means of appropriate separation surfaces, which is generally obtained by solving a numerical optimization model. While linear separability is the basis of the most popular approach to classification, the Support Vector Machine (SVM), in the recent years using nonlinear separating surfaces has received some attention. The objective of this work is to recall some of such proposals, mainly in terms of the numerical optimization models. In particular we tackle the polyhedral, ellipsoidal, spherical and conical separation approaches and, for some of them, we also consider the semisupervised versions.

On a mixed problem for a coupled nonlinear system

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Marcondes R. Clark

1997-03-01

Full Text Available In this article we prove the existence and uniqueness of solutions to the mixed problem associated with the nonlinear system $$ u_{tt}-M(int_Omega |abla u|^2dxDelta u+|u|^ ho u+heta =f $$ $$ heta _t -Delta heta +u_{t}=g $$ where $M$ is a positive real function, and $f$ and $g$ are known real functions.

Green's function-stochastic methods framework for probing nonlinear evolution problems: Burger's equation, the nonlinear Schroedinger's equation, and hydrodynamic organization of near-molecular-scale vorticity

International Nuclear Information System (INIS)

Keanini, R.G.

2011-01-01

Research highlights: → Systematic approach for physically probing nonlinear and random evolution problems. → Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. → Organization of near-molecular scale vorticity mediated by hydrodynamic modes. → Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the

Global gradient estimates for divergence-type elliptic problems involving general nonlinear operators

Science.gov (United States)

Cho, Yumi

2018-05-01

We study nonlinear elliptic problems with nonstandard growth and ellipticity related to an N-function. We establish global Calderón-Zygmund estimates of the weak solutions in the framework of Orlicz spaces over bounded non-smooth domains. Moreover, we prove a global regularity result for asymptotically regular problems which are getting close to the regular problems considered, when the gradient variable goes to infinity.

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...

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

OpenAIRE

Leibov Roman

2017-01-01

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems ...

Quasi-stability of a vector trajectorial problem with non-linear partial criteria

Directory of Open Access Journals (Sweden)

Vladimir A. Emelichev

2003-10-01

Full Text Available Multi-objective (vector combinatorial problem of finding the Pareto set with four kinds of non-linear partial criteria is considered. Necessary and sufficient conditions of that kind of stability of the problem (quasi-stability are obtained. The problem is a discrete analogue of the lower semicontinuity by Hausdorff of the optimal mapping. Mathematics Subject Classification 2000: 90C10, 90C05, 90C29, 90C31.

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

Two novel classes of solvable many-body problems of goldfish type with constraints

Energy Technology Data Exchange (ETDEWEB)

Calogero, F [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , 00185 Rome (Italy); Gomez-Ullate, D [Departamento de Fisica Teorica II, Universidad Complutense, 28040 Madrid (Spain)

2007-05-18

Two novel classes of many-body models with nonlinear interactions 'of goldfish type' are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints), i.e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited, i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.

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.

Determination of Nonlinear Stiffness Coefficients for Finite Element Models with Application to the Random Vibration Problem

Science.gov (United States)

Muravyov, Alexander A.

1999-01-01

In this paper, a method for obtaining nonlinear stiffness coefficients in modal coordinates for geometrically nonlinear finite-element models is developed. The method requires application of a finite-element program with a geometrically non- linear static capability. The MSC/NASTRAN code is employed for this purpose. The equations of motion of a MDOF system are formulated in modal coordinates. A set of linear eigenvectors is used to approximate the solution of the nonlinear problem. The random vibration problem of the MDOF nonlinear system is then considered. The solutions obtained by application of two different versions of a stochastic linearization technique are compared with linear and exact (analytical) solutions in terms of root-mean-square (RMS) displacements and strains for a beam structure.

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

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)

Nonlinear Preconditioning and its Application in Multicomponent Problems

KAUST Repository

Liu, Lulu

2015-12-07

The Multiplicative Schwarz Preconditioned Inexact Newton (MSPIN) algorithm is presented as a complement to Additive Schwarz Preconditioned Inexact Newton (ASPIN). At an algebraic level, ASPIN and MSPIN are variants of the same strategy to improve the convergence of systems with unbalanced nonlinearities; however, they have natural complementarity in practice. MSPIN is naturally based on partitioning of degrees of freedom in a nonlinear PDE system by field type rather than by subdomain, where a modest factor of concurrency can be sacrificed for physically motivated convergence robustness. ASPIN, originally introduced for decompositions into subdomains, is natural for high concurrency and reduction of global synchronization. The ASPIN framework, as an option for the outermost solver, successfully handles strong nonlinearities in computational fluid dynamics, but is barely explored for the highly nonlinear models of complex multiphase flow with capillarity, heterogeneity, and complex geometry. In this dissertation, the fully implicit ASPIN method is demonstrated for a finite volume discretization based on incompressible two-phase reservoir simulators in the presence of capillary forces and gravity. Numerical experiments show that the number of global nonlinear iterations is not only scalable with respect to the number of processors, but also significantly reduced compared with the standard inexact Newton method with a backtracking technique. Moreover, the ASPIN method, in contrast with the IMPES method, saves overall execution time because of the savings in timestep size. We consider the additive and multiplicative types of inexact Newton algorithms in the field-split context, and we augment the classical convergence theory of ASPIN for the multiplicative case. Moreover, we provide the convergence analysis of the MSPIN algorithm. Under suitable assumptions, it is shown that MSPIN is locally convergent, and desired superlinear or even quadratic convergence can be

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)

Fermat collocation method for the solutions of nonlinear system of second order boundary value problems

Directory of Open Access Journals (Sweden)

Salih Yalcinbas

2016-01-01

Full Text Available In this study, a numerical approach is proposed to obtain approximate solutions of nonlinear system of second order boundary value problem. This technique is essentially based on the truncated Fermat series and its matrix representations with collocation points. Using the matrix method, we reduce the problem system of nonlinear algebraic equations. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results.

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.

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...

Domain decomposition based iterative methods for nonlinear elliptic finite element problems

Energy Technology Data Exchange (ETDEWEB)

Cai, X.C. [Univ. of Colorado, Boulder, CO (United States)

1994-12-31

The class of overlapping Schwarz algorithms has been extensively studied for linear elliptic finite element problems. In this presentation, the author considers the solution of systems of nonlinear algebraic equations arising from the finite element discretization of some nonlinear elliptic equations. Several overlapping Schwarz algorithms, including the additive and multiplicative versions, with inexact Newton acceleration will be discussed. The author shows that the convergence rate of the Newton`s method is independent of the mesh size used in the finite element discretization, and also independent of the number of subdomains into which the original domain in decomposed. Numerical examples will be presented.

Numerical solution of non-linear diffusion problems

International Nuclear Information System (INIS)

Carmen, A. del; Ferreri, J.C.

1998-01-01

This paper presents a method for the numerical solution of non-linear diffusion problems using finite-differences in moving grids. Due to the presence of steep fronts in the solution domain and to the presence of advective terms originating in the grid movement, an implicit TVD scheme, first order in time and second order in space has been developed. Some algebraic details of the derivation are given. Results are shown for the pure advection of a scalar as a test case and an example dealing with the slow spreading of viscous fluids over plane surfaces. The agreement between numerical and analytical solutions is excellent. (author). 8 refs., 3 figs

Iterative Runge–Kutta-type methods for nonlinear ill-posed problems

International Nuclear Information System (INIS)

Böckmann, C; Pornsawad, P

2008-01-01

We present a regularization method for solving nonlinear ill-posed problems by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical regularization method. We prove that the developed iterative regularization method converges to a solution under certain conditions and with a general stopping rule. Some particular iterative regularization methods are numerically implemented. Numerical results of the examples show that the developed Runge–Kutta-type regularization methods yield stable solutions and that particular implicit methods are very efficient in saving iteration steps

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 ...

Efficient algorithm for bifurcation problems of variational inequalities

International Nuclear Information System (INIS)

Mittelmann, H.D.

1983-01-01

For a class of variational inequalities on a Hilbert space H bifurcating solutions exist and may be characterized as critical points of a functional with respect to the intersection of the level surfaces of another functional and a closed convex subset K of H. In a recent paper [13] we have used a gradient-projection type algorithm to obtain the solutions for discretizations of the variational inequalities. A related but Newton-based method is given here. Global and asymptotically quadratic convergence is proved. Numerical results show that it may be used very efficiently in following the bifurcating branches and that is compares favorably with several other algorithms. The method is also attractive for a class of nonlinear eigenvalue problems (K = H) for which it reduces to a generalized Rayleigh-quotient interaction. So some results are included for the path following in turning-point problems

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.

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.

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)

An Algorithmic Comparison of the Hyper-Reduction and the Discrete Empirical Interpolation Method for a Nonlinear Thermal Problem

Directory of Open Access Journals (Sweden)

Felix Fritzen

2018-02-01

Full Text Available A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems is presented. First, the Galerkin reduced basis (RB formulation is presented, which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renowned methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete Empirical Interpolation Method (EIM, DEIM. An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (DEIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations.

Polyharmonic boundary value problems positivity preserving and nonlinear higher order elliptic equations in bounded domains

CERN Document Server

Gazzola, Filippo; Sweers, Guido

2010-01-01

This monograph covers higher order linear and nonlinear elliptic boundary value problems in bounded domains, mainly with the biharmonic or poly-harmonic operator as leading principal part. Underlying models and, in particular, the role of different boundary conditions are explained in detail. As for linear problems, after a brief summary of the existence theory and Lp and Schauder estimates, the focus is on positivity or - since, in contrast to second order equations, a general form of a comparison principle does not exist - on “near positivity.” The required kernel estimates are also presented in detail. As for nonlinear problems, several techniques well-known from second order equations cannot be utilized and have to be replaced by new and different methods. Subcritical, critical and supercritical nonlinearities are discussed and various existence and nonexistence results are proved. The interplay with the positivity topic from the first part is emphasized and, moreover, a far-reaching Gidas-Ni-Nirenbe...

On the Cauchy problem for nonlinear Schrödinger equations with rotation

KAUST Repository

Antonelli, Paolo; Marahrens, Daniel; Sparber, Christof

2011-01-01

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superuid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable in fiuence in proving finite time blow-up in the focusing case.

On the Cauchy problem for nonlinear Schrödinger equations with rotation

KAUST Repository

Antonelli, Paolo

2011-10-01

We consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superuid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in [11, 12]. Moreover, we find that the rotation term has a considerable in fiuence in proving finite time blow-up in the focusing case.

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...

Complex energy eigenvalues of a linear potential with a parabolical barrier

International Nuclear Information System (INIS)

Malherbe, J.B.

1978-01-01

The physical meaning and restrictions of complex energy eigenvalues are briefly discussed. It is indicated that a quasi-stationary phase describes an idealised disintegration system. Approximate resonance-eigenvalues of the one dimensional Schrodinger equation with a linear potential and parabolic barrier are calculated by means of Connor's semiclassical method. This method is based on the generalized WKB-method of Miller and Good. The results obtained confirm the correctness of a model representation which explains the unusual distribution of eigenvalues by certain other linear potentials in a complex energy level [af

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.)

Existence of bounded solutions of Neumann problem for a nonlinear degenerate elliptic equation

Directory of Open Access Journals (Sweden)

Salvatore Bonafede

2017-10-01

Full Text Available We prove the existence of bounded solutions of Neumann problem for nonlinear degenerate elliptic equations of second order in divergence form. We also study some properties as the Phragmen-Lindelof property and the asymptotic behavior of the solutions of Dirichlet problem associated to our equation in an unbounded domain.

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

KAUST Repository

Yang, Haijian

2016-07-26

Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

KAUST Repository

Yang, Haijian; Yang, Chao; Sun, Shuyu

2016-01-01

Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

Spectral methods for a nonlinear initial value problem involving pseudo differential operators

International Nuclear Information System (INIS)

Pasciak, J.E.

1982-01-01

Spectral methods (Fourier methods) for approximating the solution of a nonlinear initial value problem involving pseudo differential operators are defined and analyzed. A semidiscrete approximation to the nonlinear equation based on an L 2 projection is described. The semidiscrete L 2 approximation is shown to be a priori stable and convergent under sufficient decay and smoothness assumptions on the initial data. It is shown that the semidiscrete method converges with infinite order, that is, higher order decay and smoothness assumptions imply higher order error bounds. Spectral schemes based on spacial collocation are also discussed

Initial-value problem for the Gardner equation applied to nonlinear internal waves

Science.gov (United States)

Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim

2017-04-01

The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of

A quadratic approximation-based algorithm for the solution of multiparametric mixed-integer nonlinear programming problems

KAUST Repository

Domínguez, Luis F.

2012-06-25

An algorithm for the solution of convex multiparametric mixed-integer nonlinear programming problems arising in process engineering problems under uncertainty is introduced. The proposed algorithm iterates between a multiparametric nonlinear programming subproblem and a mixed-integer nonlinear programming subproblem to provide a series of parametric upper and lower bounds. The primal subproblem is formulated by fixing the integer variables and solved through a series of multiparametric quadratic programming (mp-QP) problems based on quadratic approximations of the objective function, while the deterministic master subproblem is formulated so as to provide feasible integer solutions for the next primal subproblem. To reduce the computational effort when infeasibilities are encountered at the vertices of the critical regions (CRs) generated by the primal subproblem, a simplicial approximation approach is used to obtain CRs that are feasible at each of their vertices. The algorithm terminates when there does not exist an integer solution that is better than the one previously used by the primal problem. Through a series of examples, the proposed algorithm is compared with a multiparametric mixed-integer outer approximation (mp-MIOA) algorithm to demonstrate its computational advantages. © 2012 American Institute of Chemical Engineers (AIChE).

A Simple FEM Formulation Applied to Nonlinear Problems of Impact with Thermomechanical Coupling

Directory of Open Access Journals (Sweden)

João Paulo de Barros Cavalcante

Full Text Available Abstract The thermal effects of problems involving deformable structures are essential to describe the behavior of materials in feasible terms. Verifying the transformation of mechanical energy into heat it is possible to predict the modifications of mechanical properties of materials due to its temperature changes. The current paper presents the numerical development of a finite element method suitable for nonlinear structures coupled with thermomechanical behavior; including impact problems. A simple and effective alternative formulation is presented, called FEM positional, to deal with the dynamic nonlinear systems. The developed numerical is based on the minimum potential energy written in terms of nodal positions instead of displacements. The effects of geometrical, material and thermal nonlinearities are considered. The thermodynamically consistent formulation is based on the laws of thermodynamics and the Helmholtz free-energy, used to describe the thermoelastic and the thermoplastic behaviors. The coupled thermomechanical model can result in secondary effects that cause redistributions of internal efforts, depending on the history of deformation and material properties. The numerical results of the proposed formulation are compared with examples found in the literature.

Nonlinear vibrations of thin arbitrarily laminated composite plates subjected to harmonic excitations using DKT elements

Science.gov (United States)

Chiang, C. K.; Xue, David Y.; Mei, Chuh

1993-04-01

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite thin plates using the Discrete Kirchhoff Theory (DKT) triangular elements. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite triangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, triangular plates, different boundary conditions, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Adaptive discretizations for the choice of a Tikhonov regularization parameter in nonlinear inverse problems

International Nuclear Information System (INIS)

Kaltenbacher, Barbara; Kirchner, Alana; Vexler, Boris

2011-01-01

Parameter identification problems for partial differential equations usually lead to nonlinear inverse problems. A typical property of such problems is their instability, which requires regularization techniques, like, e.g., Tikhonov regularization. The main focus of this paper will be on efficient methods for determining a suitable regularization parameter by using adaptive finite element discretizations based on goal-oriented error estimators. A well-established method for the determination of a regularization parameter is the discrepancy principle where the residual norm, considered as a function i of the regularization parameter, should equal an appropriate multiple of the noise level. We suggest to solve the resulting scalar nonlinear equation by an inexact Newton method, where in each iteration step, a regularized problem is solved at a different discretization level. The proposed algorithm is an extension of the method suggested in Griesbaum A et al (2008 Inverse Problems 24 025025) for linear inverse problems, where goal-oriented error estimators for i and its derivative are used for adaptive refinement strategies in order to keep the discretization level as coarse as possible to save computational effort but fine enough to guarantee global convergence of the inexact Newton method. This concept leads to a highly efficient method for determining the Tikhonov regularization parameter for nonlinear ill-posed problems. Moreover, we prove that with the so-obtained regularization parameter and an also adaptively discretized Tikhonov minimizer, usual convergence and regularization results from the continuous setting can be recovered. As a matter of fact, it is shown that it suffices to use stationary points of the Tikhonov functional. The efficiency of the proposed method is demonstrated by means of numerical experiments. (paper)

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.

A Kind of Nonlinear Programming Problem Based on Mixed Fuzzy Relation Equations Constraints

Science.gov (United States)

Li, Jinquan; Feng, Shuang; Mi, Honghai

In this work, a kind of nonlinear programming problem with non-differential objective function and under the constraints expressed by a system of mixed fuzzy relation equations is investigated. First, some properties of this kind of optimization problem are obtained. Then, a polynomial-time algorithm for this kind of optimization problem is proposed based on these properties. Furthermore, we show that this algorithm is optimal for the considered optimization problem in this paper. Finally, numerical examples are provided to illustrate our algorithms.

Nonlinear Fourier transform for dual-polarization optical communication system

OpenAIRE

Gaiarin, Simone

2018-01-01

New services and applications are causing an exponential increase in the internet traffic. In a few years, the current fiber-optic communication system infrastructure will not be able to meet this demand because fiber nonlinearity dramatically limits the information transmission rate. Eigenvalue communication is considered an emerging paradigm in fiber-optic communications that could potentially overcome these limitations. It relies on a mathematical technique called “inverse scattering trans...

Numerical methods for the design of large-scale nonlinear discrete ill-posed inverse problems

International Nuclear Information System (INIS)

Haber, E; Horesh, L; Tenorio, L

2010-01-01

Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled as indirect, noisy observations of the model collected via a set of plausible experiments. An inversion estimate based on these data is obtained by a weighted Tikhonov regularization whose weights control the contribution of the different experiments to the data misfit term. These weights are selected by minimization of an empirical estimate of the Bayes risk that is penalized to promote sparsity. This formulation entails a bilevel optimization problem that is solved using a simple descent method. We demonstrate the viability of our design with a problem in electromagnetic imaging based on direct current resistivity and magnetotelluric data

Optimization for nonlinear inverse problem

International Nuclear Information System (INIS)

Boyadzhiev, G.; Brandmayr, E.; Pinat, T.; Panza, G.F.

2007-06-01

The nonlinear inversion of geophysical data in general does not yield a unique solution, but a single model, representing the investigated field, is preferred for an easy geological interpretation of the observations. The analyzed region is constituted by a number of sub-regions where the multi-valued nonlinear inversion is applied, which leads to a multi-valued solution. Therefore, combining the values of the solution in each sub-region, many acceptable models are obtained for the entire region and this complicates the geological interpretation of geophysical investigations. In this paper are presented new methodologies, capable to select one model, among all acceptable ones, that satisfies different criteria of smoothness in the explored space of solutions. In this work we focus on the non-linear inversion of surface waves dispersion curves, which gives structural models of shear-wave velocity versus depth, but the basic concepts have a general validity. (author)

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...

Adaptive wavelet collocation methods for initial value boundary problems of nonlinear PDE's

Science.gov (United States)

Cai, Wei; Wang, Jian-Zhong

1993-01-01

We have designed a cubic spline wavelet decomposition for the Sobolev space H(sup 2)(sub 0)(I) where I is a bounded interval. Based on a special 'point-wise orthogonality' of the wavelet basis functions, a fast Discrete Wavelet Transform (DWT) is constructed. This DWT transform will map discrete samples of a function to its wavelet expansion coefficients in O(N log N) operations. Using this transform, we propose a collocation method for the initial value boundary problem of nonlinear PDE's. Then, we test the efficiency of the DWT transform and apply the collocation method to solve linear and nonlinear PDE's.

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.

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

Energy Technology Data Exchange (ETDEWEB)

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

1993-12-31

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

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)

Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities

Indian Academy of Sciences (India)

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all R R . Assuming the existence of an upper and of a lower ...

A New Spectral Local Linearization Method for Nonlinear Boundary Layer Flow Problems

Directory of Open Access Journals (Sweden)

S. S. Motsa

2013-01-01

Full Text Available We propose a simple and efficient method for solving highly nonlinear systems of boundary layer flow problems with exponentially decaying profiles. The algorithm of the proposed method is based on an innovative idea of linearizing and decoupling the governing systems of equations and reducing them into a sequence of subsystems of differential equations which are solved using spectral collocation methods. The applicability of the proposed method, hereinafter referred to as the spectral local linearization method (SLLM, is tested on some well-known boundary layer flow equations. The numerical results presented in this investigation indicate that the proposed method, despite being easy to develop and numerically implement, is very robust in that it converges rapidly to yield accurate results and is more efficient in solving very large systems of nonlinear boundary value problems of the similarity variable boundary layer type. The accuracy and numerical stability of the SLLM can further be improved by using successive overrelaxation techniques.

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)

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)

Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

Directory of Open Access Journals (Sweden)

J. Gwinner

2013-01-01

Full Text Available The purpose of this paper is twofold. Firstly we consider nonlinear nonsmooth elliptic boundary value problems, and also related parabolic initial boundary value problems that model in a simplified way steady-state unilateral contact with Tresca friction in solid mechanics, respectively, stem from nonlinear transient heat conduction with unilateral boundary conditions. Here a recent duality approach, that augments the classical Babuška-Brezzi saddle point formulation for mixed variational problems to twofold saddle point formulations, is extended to the nonsmooth problems under consideration. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities in the time-dependent case. Secondly we are concerned with the stability of the solution set of a general class of differential mixed variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated nonlinear maps, the nonsmooth convex functionals, and the convex constraint set. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. We impose weak convergence assumptions on the perturbed maps using the monotonicity method of Browder and Minty.

Special function solutions of a spectral problem for a nonlinear quantum oscillator

International Nuclear Information System (INIS)

Schulze-Halberg, A; Morris, J R

2012-01-01

We construct exact solutions of a spectral problem involving the Schrödinger equation for a nonlinear, one-parameter oscillator potential. In contrast to a previous analysis of the problem (Carinena et al 2007 Ann. Phys. 322 434–59), where solutions were given through a Rodrigues-type formula, our approach leads to closed-form representations of the solutions in terms of special functions, not containing any derivative operators. We show normalizability and orthogonality of our solutions, as well as correct reduction of the problem to the harmonic oscillator model, if the parameter in the potential gets close to zero. (paper)

Lamé Parameter Estimation from Static Displacement Field Measurements in the Framework of Nonlinear Inverse Problems

DEFF Research Database (Denmark)

Hubmer, Simon; Sherina, Ekaterina; Neubauer, Andreas

2018-01-01

. The main result of this paper is the verification of a nonlinearity condition in an infinite dimensional Hilbert space context. This condition guarantees convergence of iterative regularization methods. Furthermore, numerical examples for recovery of the Lam´e parameters from displacement data simulating......We consider a problem of quantitative static elastography, the estimation of the Lam´e parameters from internal displacement field data. This problem is formulated as a nonlinear operator equation. To solve this equation, we investigate the Landweber iteration both analytically and numerically...... a static elastography experiment are presented....

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 ...

Studies in nonlinear problems of energy

Energy Technology Data Exchange (ETDEWEB)

Matkowsky, B.J.

1992-07-01

Emphasis has been on combustion and flame propagation. The research program was on modeling, analysis and computation of combustion phenomena, with emphasis on transition from laminar to turbulent combustion. Nonlinear dynamics and pattern formation were investigated in the transition. Stability of combustion waves, and transitions to complex waves are described. Combustion waves possess large activation energies, so that chemical reactions are significant only in thin layers, or reaction zones. In limit of infinite activation energy, the zones shrink to moving surfaces, (fronts) which must be found during the analysis, so that (moving free boundary problems). The studies are carried out for limiting case with fronts, while the numerical studies are carried out for finite, though large, activation energy. Accurate resolution of the solution in the reaction zones is essential, otherwise false predictions of dynamics are possible. Since the the reaction zones move, adaptive pseudo-spectral methods were developed. The approach is based on a synergism of analytical and computational methods. The numerical computations build on and extend the analytical information. Furthermore, analytical solutions serve as benchmarks for testing the accuracy of the computation. Finally, ideas from analysis (singular perturbation theory) have induced new approaches to computations. The computational results suggest new analysis to be considered. Among the recent interesting results, was spatio-temporal chaos in combustion. One goal is extension of the adaptive pseudo-spectral methods to adaptive domain decomposition methods. Efforts have begun to develop such methods for problems with multiple reaction zones, corresponding to problems with more complex, and more realistic chemistry. Other topics included stochastics, oscillators, Rysteretic Josephson junctions, DC SQUID, Markov jumps, laser with saturable absorber, chemical physics, Brownian movement, combustion synthesis, etc.

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

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 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.

A MODIFIED DECOMPOSITION METHOD FOR SOLVING NONLINEAR PROBLEM OF FLOW IN CONVERGING- DIVERGING CHANNEL

Directory of Open Access Journals (Sweden)

MOHAMED KEZZAR

2015-08-01

Full Text Available In this research, an efficient technique of computation considered as a modified decomposition method was proposed and then successfully applied for solving the nonlinear problem of the two dimensional flow of an incompressible viscous fluid between nonparallel plane walls. In fact this method gives the nonlinear term Nu and the solution of the studied problem as a power series. The proposed iterative procedure gives on the one hand a computationally efficient formulation with an acceleration of convergence rate and on the other hand finds the solution without any discretization, linearization or restrictive assumptions. The comparison of our results with those of numerical treatment and other earlier works shows clearly the higher accuracy and efficiency of the used Modified Decomposition Method.

Strongly nonlinear nonhomogeneous elliptic unilateral problems with L^1 data and no sign conditions

Directory of Open Access Journals (Sweden)

Elhoussine Azroul

2012-05-01

Full Text Available In this article, we prove the existence of solutions to unilateral problems involving nonlinear operators of the form: $$ Au+H(x,u,abla u=f $$ where $A$ is a Leray Lions operator from $W_0^{1,p(x}(Omega$ into its dual $W^{-1,p'(x}(Omega$ and $H(x,s,xi$ is the nonlinear term satisfying some growth condition but no sign condition. The right hand side $f$ belong to $L^1(Omega$.

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

Stochastic Fixed Points and Nonlinear Perron-Frobenius Theorem

OpenAIRE

Babaei, E.; Evstigneev, I. V.; Pirogov, S. A.

2016-01-01

We provide conditions for the existence of measurable solutions to the equation $\\xi(T\\omega)=f(\\omega,\\xi(\\omega))$, where $T:\\Omega \\rightarrow\\Omega$ is an automorphism of the probability space $\\Omega$ and $f(\\omega,\\cdot)$ is a strictly non-expansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron-Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping $D(\\omega)$ of a random closed cone $K(\\omeg...

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.

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

International Nuclear Information System (INIS)

Manakov, S V; Santini, P M

2008-01-01

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev-Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann-Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann-Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

Energy Technology Data Exchange (ETDEWEB)

Manakov, S V [Landau Institute for Theoretical Physics, Moscow (Russian Federation); Santini, P M [Dipartimento di Fisica, Universita di Roma ' La Sapienza' , and Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1, Piazz.le Aldo Moro 2, I-00185 Rome (Italy)

2008-02-08

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev-Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann-Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann-Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.

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)

Finite element solution of nonlinear eddy current problems with periodic excitation and its industrial applications.

Science.gov (United States)

Bíró, Oszkár; Koczka, Gergely; Preis, Kurt

2014-05-01

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three-dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady-state periodic solution is of interest only. This is represented either in the frequency domain as a finite Fourier series or in the time domain as a set of discrete time values within one period for each finite element degree of freedom. The former approach is the (continuous) harmonic balance method and, in the latter one, discrete Fourier transformation will be shown to lead to a discrete harmonic balance method. Due to the nonlinearity, all harmonics, both continuous and discrete, are coupled to each other. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a time-independent permeability distribution, the so-called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps. As industrial applications, analyses of large power transformers are presented. The first example is the computation of the electromagnetic field of a single-phase transformer in the time domain with the results compared to those obtained by traditional time-stepping techniques. In the second application, an advanced model of the same transformer is analyzed in the frequency domain by the harmonic balance method with the effect of the presence of higher harmonics on the losses investigated. Finally a third example tackles the case of direct current (DC) bias in the coils of a single-phase transformer.

State and parameter estimation in nonlinear systems as an optimal tracking problem

International Nuclear Information System (INIS)

Creveling, Daniel R.; Gill, Philip E.; Abarbanel, Henry D.I.

2008-01-01

In verifying and validating models of nonlinear processes it is important to incorporate information from observations in an efficient manner. Using the idea of synchronization of nonlinear dynamical systems, we present a framework for connecting a data signal with a model in a way that minimizes the required coupling yet allows the estimation of unknown parameters in the model. The need to evaluate unknown parameters in models of nonlinear physical, biophysical, and engineering systems occurs throughout the development of phenomenological or reduced models of dynamics. Our approach builds on existing work that uses synchronization as a tool for parameter estimation. We address some of the critical issues in that work and provide a practical framework for finding an accurate solution. In particular, we show the equivalence of this problem to that of tracking within an optimal control framework. This equivalence allows the application of powerful numerical methods that provide robust practical tools for model development and validation

A demonstration of the improved efficiency of the canonical coordinates method using nonlinear combined heat and power economic dispatch problems

Science.gov (United States)

Chang, Hung-Chieh; Lin, Pei-Chun

2014-02-01

Economic dispatch is the short-term determination of the optimal output from a number of electricity generation facilities to meet the system load while providing power. As such, it represents one of the main optimization problems in the operation of electrical power systems. This article presents techniques to substantially improve the efficiency of the canonical coordinates method (CCM) algorithm when applied to nonlinear combined heat and power economic dispatch (CHPED) problems. The improvement is to eliminate the need to solve a system of nonlinear differential equations, which appears in the line search process in the CCM algorithm. The modified algorithm was tested and the analytical solution was verified using nonlinear CHPED optimization problems, thereby demonstrating the effectiveness of the algorithm. The CCM methods proved numerically stable and, in the case of nonlinear programs, produced solutions with unprecedented accuracy within a reasonable time.

Group-invariant solutions of nonlinear elastodynamic problems of plates and shells

International Nuclear Information System (INIS)

Dzhupanov, V.A.; Vassilev, V.M.; Dzhondzhorov, P.A.

1993-01-01

Plates and shells are basic structural components in nuclear reactors and their equipment. The prediction of the dynamic response of these components to fast transient loadings (e.g., loadings caused by earthquakes, missile impacts, etc.) is a quite important problem in the general context of the design, reliability and safety of nuclear power stations. Due to the extreme loading conditions a more adequate treatment of the foregoing problem should rest on a suitable nonlinear shell model, which would allow large deflections of the structures regarded to be taken into account. Such a model is provided in the nonlinear Donnell-Mushtari-Vlasov (DMV) theory. The governing system of equations of the DMV theory consists of two coupled nonlinear fourth order partial differential equations in three independent and two dependent variables. It is clear, as the case stands, that the obtaining solutions to this system directly, by using any of the general analytical or numerical techniques, would involve considerable difficulties. In the present paper, the invariance of the governing equations of DMV theory for plates and cylindrical shells relative to local Lie groups of local point transformations will be employed to get some advantages in connection with the aforementioned problem. First, the symmetry of a functional, corresponding to the governing equations of DMV theory for plates and cylindrical shells is studied. Next, the densities in the corresponding conservation laws are determined on the basis of Noether theorem. Finally, we study a class of invariant solutions of the governing equations. As is well known, group-invariant solutions are often intermediate asymptotics for a wider class of solutions of the corresponding equations. When such solutions are considered, the number of the independent variables can be reduced. For the class of invariant solutions studied here, the system of governing equations converts into a system of ordinary differential equations

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...

On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation

Directory of Open Access Journals (Sweden)

Mesloub Said

2008-01-01

Full Text Available This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.

Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations

Directory of Open Access Journals (Sweden)

M. G. Crandall

1999-07-01

Full Text Available We study existence of continuous weak (viscosity solutions of Dirichlet and Cauchy-Dirichlet problems for fully nonlinear uniformly elliptic and parabolic equations. Two types of results are obtained in contexts where uniqueness of solutions fails or is unknown. For equations with merely measurable coefficients we prove solvability of the problem, while in the continuous case we construct maximal and minimal solutions. Necessary barriers on external cones are also constructed.

On the solvability of initial-value problems for nonlinear implicit difference equations

Directory of Open Access Journals (Sweden)

Ha Thi Ngoc Yen

2004-07-01

Full Text Available Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differential-algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems.

Exact solution to the problem of nonlinear pulse propagation through random layered media and its connection with number triangles

International Nuclear Information System (INIS)

Sokolow, Adam; Sen, Surajit

2007-01-01

An energy pulse refers to a spatially compact energy bundle. In nonlinear pulse propagation, the nonlinearity of the relevant dynamical equations could lead to pulse propagation that is nondispersive or weakly dispersive in space and time. Nonlinear pulse propagation through layered media with widely varying pulse transmission properties is not wave-like and a problem of broad interest in many areas such as optics, geophysics, atmospheric physics and ocean sciences. We study nonlinear pulse propagation through a semi-infinite sequence of layers where the layers can have arbitrary energy transmission properties. By assuming that the layers are rigid, we are able to develop exact expressions for the backscattered energy received at the surface layer. The present study is likely to be relevant in the context of energy transport through soil and similar complex media. Our study reveals a surprising connection between the problem of pulse propagation and the number patterns in the well known Pascal's and Catalan's triangles and hence provides an analytic benchmark in a challenging problem of broad interest. We close with comments on the relationship between this study and the vast body of literature on the problem of wave localization in disordered systems

Admissible solutions for a class of nonlinear parabolic problem with non-negative data

Czech Academy of Sciences Publication Activity Database

Feireisl, Eduard; Petzeltová, Hana; Simondon, F.

2001-01-01

Roč. 131, č. 5 (2001), s. 857-883 ISSN 0308-2105 R&D Projects: GA AV ČR IAA1019703 Keywords : admissible solutions%nonlinear parabolic problem * admissible solutions * comparison principle * non-negative data Subject RIV: BA - General Mathematics Impact factor: 0.441, year: 2001

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.)

Multi-atom Jaynes-Cummings model with nonlinear effects

International Nuclear Information System (INIS)

Aleixo, Armando Nazareno Faria; Balantekin, Akif Baha; Ribeiro, Marco Antonio Candido

2001-01-01

The standard Jaynes-Cummings (JC) model and its extensions, normally used in quantum optics, idealizes the interaction of matter with electromagnetic radiation by a simple Hamiltonian of a two-level atom coupled to a single bosonic mode. This Hamiltonian has a fundamental importance to the field of quantum optics and it is a central ingredient in the quantized description of any optical system involving the interaction between light and atoms. The JC Hamiltonian defines a molecule, a composite system formed from the coupling of a two-state system and a quantized harmonic oscillator. For this Hamiltonian, mostly the single-particle situation has been studied. This model can also be extended for the situation where one has N two-level systems, which interact only with the electromagnetic radiation. In this case the effects of the spatial distribution of the particles it is not taken into account and the spin angular momentum S-circumflex i of each particle contributes to form a total angular momentum J-circumflex of the system. When one considers the effects due to the spatial variation in the field intensity in a nonlinear medium it is necessary to further add a Kerr term to the standard JC Hamiltonian. This kind of nonlinear JC Hamiltonian is used in the study of micro masers. Another nonlinear variant of the JC model takes the coupling between matter and the radiation to depend on the intensity of the electromagnetic field. This model is interesting since this kind of interaction means that effectively the coupling is proportional to the amplitude of the field representing a very simple case of a nonlinear interaction corresponding to a more realistic physical situation. In this work we solve exactly the problem of the interaction of a N two-level atoms with an electromagnetic radiation when nonlinear effects due to the spatial variation in the field intensity in a nonlinear Kerr medium and the dependence on the intensity of the electromagnetic field on the matter

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.

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.

Iterative Methods for Solving Nonlinear Parabolic Problem in Pension Saving Management

Science.gov (United States)

Koleva, M. N.

2011-11-01

In this work we consider a nonlinear parabolic equation, obtained from Riccati like transformation of the Hamilton-Jacobi-Bellman equation, arising in pension saving management. We discuss two numerical iterative methods for solving the model problem—fully implicit Picard method and mixed Picard-Newton method, which preserves the parabolic characteristics of the differential problem. Numerical experiments for comparison the accuracy and effectiveness of the algorithms are discussed. Finally, observations are given.

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

Inverse scattering transform for the vector nonlinear Schroedinger equation with nonvanishing boundary conditions

International Nuclear Information System (INIS)

Prinari, Barbara; Ablowitz, Mark J.; Biondini, Gino

2006-01-01

The inverse scattering transform for the vector defocusing nonlinear Schroedinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system

TRUMP3-JR: a finite difference computer program for nonlinear heat conduction problems

International Nuclear Information System (INIS)

Ikushima, Takeshi

1984-02-01

Computer program TRUMP3-JR is a revised version of TRUMP3 which is a finite difference computer program used for the solution of multi-dimensional nonlinear heat conduction problems. Pre- and post-processings for input data generation and graphical representations of calculation results of TRUMP3 are avaiable in TRUMP3-JR. The calculation equations, program descriptions and user's instruction are presented. A sample problem is described to demonstrate the use of the program. (author)

Comparative Study of Evolutionary Multi-objective Optimization Algorithms for a Non-linear Greenhouse Climate Control Problem

DEFF Research Database (Denmark)

Ghoreishi, Newsha; Sørensen, Jan Corfixen; Jørgensen, Bo Nørregaard

2015-01-01

Non-trivial real world decision-making processes usually involve multiple parties having potentially conflicting interests over a set of issues. State-of-the-art multi-objective evolutionary algorithms (MOEA) are well known to solve this class of complex real-world problems. In this paper, we...... compare the performance of state-of-the-art multi-objective evolutionary algorithms to solve a non-linear multi-objective multi-issue optimisation problem found in Greenhouse climate control. The chosen algorithms in the study includes NSGAII, eNSGAII, eMOEA, PAES, PESAII and SPEAII. The performance...... of all aforementioned algorithms is assessed and compared using performance indicators to evaluate proximity, diversity and consistency. Our insights to this comparative study enhanced our understanding of MOEAs performance in order to solve a non-linear complex climate control problem. The empirical...

Nonlinear model order reduction for flexible multibody dynamics: a modal derivatives approach

Energy Technology Data Exchange (ETDEWEB)

Wu, Long, E-mail: L.Wu-1@tudelft.nl [Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering (Netherlands); Tiso, Paolo, E-mail: ptiso@ethz.ch [ETH Zürich, Institute for Mechanical Systems (Switzerland)

2016-04-15

An effective reduction technique is presented for flexible multibody systems, for which the elastic deflection could not be considered small. We consider here the planar beam systems undergoing large elastic rotations, in the floating frame description. The proposed method enriches the classical linear reduction basis with modal derivatives stemming from the derivative of the eigenvalue problem. Furthermore, the Craig–Bampton method is applied to couple the different reduced components. Based on the linear projection, the configuration-dependent internal force can be expressed as cubic polynomials in the reduced coordinates. Coefficients of these polynomials can be precomputed for efficient runtime evaluation. The numerical results show that the modal derivatives are essential for the correct approximation of the nonlinear elastic deflection with respect to the body reference. The proposed reduction method constitutes a natural and effective extension of the classical linear modal reduction in the floating frame.

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

International Nuclear Information System (INIS)