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A Convergence Analysis of the Inexact Simplified jacobi-davidson algorithm for Polynomial Eigenvalue Problems

JOURNAL OF SCIENTIFIC COMPUTING 2018年第3期75卷 1207-1228页

作者： Zhao, Tao Chinese Acad Sci Shenzhen Inst Adv Technol Shenzhen 518055 Peoples R China

The simplified jacobi-davidson (JD) method is a variant of the JD method without subspace acceleration. If the correction equation is solved approximately, the inexact simplified JD method is obtained. In this paper, we present a new convergence analysis of the inexact simplified JD method. The analysis applies to polynomial eigenvalue problems with simple eigenpairs. We first establish a relationship between the solution of the correction equation and the residual of the approximate eigenpair. From this relationship, we find the difference of two adjacent approximate eigenvalues bounded in terms of the residual norm of the approximate eigenpair. Then we prove the convergence of the inexact simplified JD method in terms of the residual norm of the approximate eigenpair. Depending on how accurately we solve the correction equation, the convergence rate of the inexact simplified JD may take several different forms. Numerical experiments confirm the convergence analysis.

关键词： Polynomial eigenvalue problem jacobi-davidson algorithm Convergence analysis Inexact correction equation solver Simple eigenvalue

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Parallel two-level domain decomposition based jacobi-davidson algorithms for pyramidal quantum dot simulation

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COMPUTER PHYSICS COMMUNICATIONS 2016年 204卷 74-81页

作者： Zhao, Tao Hwang, Feng-Nan Cai, Xiao-Chuan Chinese Acad Sci Shenzhen Inst Adv Technol Shenzhen 518055 Peoples R China Natl Cent Univ Dept Math Jhongli 320 Taiwan Univ Colorado Dept Comp Sci Boulder CO 80309 USA

We consider a quintic polynomial eigenvalue problem arising from the finite volume discretization of a quantum dot simulation problem. The problem is solved by the jacobi-davidson (JD) algorithm. Our focus is on how to achieve the quadratic convergence of JD in a way that is not only efficient but also scalable when the number of processor cores is large. For this purpose, we develop a projected two-level Schwarz preconditioned JD algorithm that exploits multilevel domain decomposition techniques. The pyramidal quantum dot calculation is carefully studied to illustrate the efficiency of the proposed method. Numerical experiments confirm that the proposed method has a good scalability for problems with hundreds of millions of unknowns on a parallel computer with more than 10,000 processor cores. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Polynomial eigenvalue problem Two-level Schwarz preconditioner jacobi-davidson algorithm Parallel performance Quantum dot simulation

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On large-scale diagonalization techniques for the Anderson model of localization

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SIAM REVIEW 2008年第1期50卷 91-112页

作者： Schenk, Olaf Bollhoefer, Matthias Roemer, Rudolf A. Univ Basel Dept Comp Sci CH-4056 Basel Switzerland Tech Univ Carolo Wilhelmina Braunschweig Inst Computat Math D-38106 Braunschweig Germany Univ Warwick Dept Phys Coventry CV4 7AL W Midlands England Univ Warwick Ctr Comp Sci Coventry CV4 7AL W Midlands England

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques ill the implicitly restarted Lanczos method and in the jacobi-davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

关键词： Anderson model of localization large-scale eigenvalue problem Lanczos algorithm jacobi-davidson algorithm Cullum-Willoughby implementation symmetric indefinite matrix multilevel preconditioning maximum weighted matching

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On large-scale diagonalization techniques for the Anderson model of localization

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2006年第3期28卷 963-983页

作者： Schenk, Olaf Bollhofer, Matthias Romer, Rudolf A. Univ Basel Dept Comp Sci CH-4056 Basel Switzerland Tech Univ Berlin Dept Math D-10623 Berlin Germany Univ Warwick Ctr Comp Sci Coventry CV4 7AL W Midlands England Univ Warwick Dept Phys Coventry CV4 7AL W Midlands England

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the jacobi-davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.

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On a parallel multilevel preconditioned Maxwell eigensolver

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PARALLEL COMPUTING 2006年第2期32卷 157-165页

作者： Arbenz, P Becka, M Geus, R Hetmaniuk, U Mengotti, T ETH Swiss Fed Inst Technol Inst Computat Sci CH-8092 Zurich Switzerland Paul Scherrer Inst CH-5232 Villigen Switzerland Sandia Natl Labs Albuquerque NM 87185 USA

We report on a parallel implementation of the jacobi-davidson algorithm to compute a few eigenvalues and corresponding eigenvectors of a large real symmetric generalized matrix eigenvalue problem Ax = lambda Mx, C(T)x = 0. The eigenvalue problem stems from the design of cavities of particle accelerators. It is obtained by the finite element discretization of the time-harmonic Maxwell equation in weak form by a combination of Nedelec (edge) and Lagrange (node) elements. We found the jacobi-davidson (JD) method to be a very effective solver provided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration. The preconditioner of our choice is a combination of a hierarchical basis preconditioner and the ML smoothed aggregation AMG preconditioner. It is close to optimal regarding iteration count. The parallel code makes extensive use of the Trilinos software framework. In our examples from accelerator physics, we observe satisfactory speedups and efficiencies. (c) 2005 Elsevier B.V. All rights reserved.

关键词： Maxwell equation finite element method generalized eigenvalue problem jacobi-davidson algorithm smoothed aggregation AMG preconditioning

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On a parallel multilevel preconditioned Maxwell eigensolver

On a parallel multilevel preconditioned Maxwell eigensolver

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3rd Workshop on Parallel Matrix algorithms and Applications (PMAA 2004)

关键词： Maxwell equation finite element method generalized eigenvalue problem jacobi-davidson algorithm smoothed aggregation AMG preconditioning

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Computation of optical modes in axisymmetric open cavity resonators

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FUTURE GENERATION COMPUTER SYSTEMS-THE INTERNATIONAL JOURNAL OF ESCIENCE 2005年第8期21卷 1263-1274页

作者： Chinellato, O Arbenz, P Streiff, M Witzig, A Swiss Fed Inst Technol Inst Computat Sci CH-8092 Zurich Switzerland Swiss Fed Inst Technol Integrated Syst Lab CH-8092 Zurich Switzerland

The computation of optical modes inside axisymmetric cavity resonators with a general spatial permittivity profile is a formidable computational task. In order to avoid spurious modes the vector Helmholtz equations are discretised by a mixed finite element approach. We formulate the method for first and second order Nedelec edge and Lagrange nodal elements. We discuss how to accurately compute the element matrices and solve the resulting large sparse complex symmetric eigenvalue problems. We validate our approach by three numerical examples that contain varying material parameters and absorbing boundary conditions (ABC). (c) 2004 Elsevier B.V. All rights reserved.

关键词： optical mode computation numerical device simulation axisymmetric cavity body of revolution semiconductor laser finite element method large sparse complex symmetric eigenproblem jacobi-davidson algorithm Nedelecedge elements

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Computation of optical modes in axisymmetric open cavity resonators

Computation of optical modes in axisymmetric open cavity res...

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European Grid Conference

作者： Chinellato, O Arbenz, P Streiff, M Witzig, A Swiss Fed Inst Technol Inst Computat Sci CH-8092 Zurich Switzerland Swiss Fed Inst Technol Integrated Syst Lab CH-8092 Zurich Switzerland

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A jacobi-davidson method for solving complex symmetric eigenvalue problems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2004年第5期25卷 1655-1673页

作者： Arbenz, P Hochstenbach, ME ETH Inst Computat Sci CH-8092 Zurich Switzerland Univ Utrecht Dept Math NL-3508 TA Utrecht Netherlands

We discuss variants of the jacobi-davidson method for solving the generalized complex symmetric eigenvalue problem. The jacobi-davidson algorithm can be considered as an accelerated inexact Rayleigh quotient iteration. We show that it is appropriate to replace the Euclidean inner product in C-n with an indefinite inner product. The Rayleigh quotient based on this indefinite inner product leads to an asymptotically cubically convergent Rayleigh quotient iteration. Advantages of the method are illustrated by numerical examples. We deal with problems from electromagnetics that require the computation of interior eigenvalues. The main drawback that we experience in these particular examples is the lack of efficient preconditioners.

关键词： generalized complex symmetric eigenvalue problem interior eigenvalues jacobi-davidson algorithm Rayleigh quotient iteration cubic convergence

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A comparison of solvers for large eigenvalue problems occuring in the design of resonant cavities

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 1999年第1期6卷 3-16页

作者： Arbenz, P Geus, R Swiss Fed Inst Technol Inst Comp Sci CH-8092 Zurich Switzerland

We present experiments with various solvers for large sparse generalized symmetric matrix eigenvalue problems. These problems occur in the computation of a few of the lowest frequencies of standing electromagnetic waves in resonant cavities with the finite element method. The solvers investigated are (1) subspace iteration, (2) block Lanczos algorithm, (3) implicitly restarted Lanczos algorithm and (4) jacobi-davidson algorithm. The experiments have been conducted on a Hewlett-Packard Exemplar S-Class system. Copyright (C) 1999 John Wiley & Sons, Ltd.

关键词： Maxwell equations cavity resonator large sparse eigenvalue problem subspace iteration Lanczos algorithm restarted Lanczos algorithm jacobi-davidson algorithm

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