An Accurate Restarting for Shift-and-Invert Krylov Subspaces Computing Matrix Exponential Actions of Nonsymmetric Matrices

An accurate residual--time (AccuRT) restarting for computing matrix exponential actions of nonsymmetric matrices by the shift-and-invert (SAI) Krylov subspace method is proposed. The proposed restarting method is an extension of the recently proposed RT (residual--time) restarting and it is designed to avoid a possible accuracy loss in the conventional RT restarting. An expensive part of the SAI Krylov method is solution of linear systems with the shifted matrix. Since the AccuRT algorithm adjusts the shift value, we discuss how the proposed restarting can be implemented with just a single LU~factorization (or a preconditioner setup) of the shifted matrix. Numerical experiments demonstrate an improved accuracy and efficiency of the approach.

[1]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[2]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[3]  Awad H. Al-Mohy,et al.  Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators , 2011, SIAM J. Sci. Comput..

[4]  Hillel Tal-Ezer,et al.  On Restart and Error Estimation for Krylov Approximation of w=f(A)v , 2007, SIAM J. Sci. Comput..

[5]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[6]  J. Verwer,et al.  Unconditionally stable integration of Maxwell's equations , 2009 .

[7]  I. Moret,et al.  RD-Rational Approximations of the Matrix Exponential , 2004 .

[8]  L. Knizhnerman,et al.  Adaptive residual-time restarting for Krylov subspace matrix exponential evaluations , 2019, Keldysh Institute Preprints.

[9]  Vladimir Druskin,et al.  Solution of Large Scale Evolutionary Problems Using Rational Krylov Subspaces with Optimized Shifts , 2009, SIAM J. Sci. Comput..

[10]  Anne Greenbaum,et al.  Using Nonorthogonal Lanczos Vectors in the Computation of Matrix Functions , 1998, SIAM J. Sci. Comput..

[11]  Stefan Güttel,et al.  Efficient and Stable Arnoldi Restarts for Matrix Functions Based on Quadrature , 2014, SIAM J. Matrix Anal. Appl..

[12]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[13]  Mike A. Botchev,et al.  ART: adaptive residual-time restarting for Krylov subspace matrix exponential evaluations , 2018, J. Comput. Appl. Math..

[14]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[15]  Stefan Güttel,et al.  Deflated Restarting for Matrix Functions , 2011, SIAM J. Matrix Anal. Appl..

[16]  S. Güttel.  Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection , 2013 .

[17]  L. Knizhnerman,et al.  Two polynomial methods of calculating functions of symmetric matrices , 1991 .

[18]  Roger B. Sidje,et al.  Expokit: a software package for computing matrix exponentials , 1998, TOMS.

[19]  Allen Taflove,et al.  Computational Electrodynamics the Finite-Difference Time-Domain Method , 1995 .

[20]  M. Eiermann,et al.  Implementation of a restarted Krylov subspace method for the evaluation of matrix functions , 2008 .

[21]  Christian Wieners,et al.  Efficient time integration for discontinuous Galerkin approximations of linear wave equations , 2015 .

[22]  Mike A. Botchev,et al.  Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources , 2017, J. Comput. Appl. Math..

[23]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[24]  Marlis Hochbruck,et al.  Residual, Restarting, and Richardson Iteration for the Matrix Exponential , 2010, SIAM J. Sci. Comput..

[25]  E. Celledoni,et al.  A Krylov projection method for systems of ODEs , 1997 .

[26]  Stefan Güttel,et al.  Rational Krylov Methods for Operator Functions , 2010 .

[27]  Hans De Raedt,et al.  New Unconditionally Stable Algorithms to Solve the Time-Dependent Maxwell Equations , 2002, International Conference on Computational Science.

[28]  Jörg Niehoff,et al.  Projektionsverfahren zur Approximation von Matrixfunktionen mit Anwendungen auf die Implementierung exponentieller Integratoren , 2007 .

[29]  Stefan Güttel,et al.  Three-Dimensional Transient Electromagnetic Modeling Using Rational Krylov Methods , 2015 .

[30]  M. Hochbruck,et al.  Exponential integrators , 2010, Acta Numerica.

[31]  L. Knizhnerman.  Calculation of functions of unsymmetric matrices using Arnoldi's method , 1991 .

[32]  VALERIA SIMONCINI,et al.  MATRIX FUNCTIONS , 2006 .

[33]  W. Auzinger,et al.  Computable upper error bounds for Krylov approximations to matrix exponentials and associated φ -functions. , 2018, BIT. Numerical mathematics.

[34]  W. Auzinger,et al.  Computable upper error bounds for Krylov approximations to matrix exponentials and associated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{- , 2018, BIT Numerical Mathematics.

[35]  Marlis Hochbruck,et al.  Exponential Integrators for Quantum-Classical Molecular Dynamics , 1999 .

[36]  Mike A. Botchev,et al.  Krylov subspace exponential time domain solution of Maxwell's equations in photonic crystal modeling , 2016, J. Comput. Appl. Math..

[37]  Marlis Hochbruck,et al.  Preconditioning Lanczos Approximations to the Matrix Exponential , 2005, SIAM J. Sci. Comput..

[38]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[39]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[40]  Marco Vianello,et al.  Efficient computation of the exponential operator for large, sparse, symmetric matrices , 2000, Numer. Linear Algebra Appl..