A comparison of limited-memory Krylov methods for Stieltjes functions of Hermitian matrices

Given a limited amount of memory and a target accuracy, we propose and compare several polynomial Krylov methods for the approximation of f(A)b, the action of a Stieltjes matrix function of a large Hermitian matrix on a vector. Using new error bounds and estimates, as well as existing results, we derive predictions of the practical performance of the methods, and rank them accordingly. As by-products, we derive new results on inexact Krylov iterations for matrix functions in order to allow for a fair comparison of rational Krylov methods with polynomial inner solves.

[1]  Stefan Güttel,et al.  Scaled and Squared Subdiagonal Padé Approximation for the Matrix Exponential , 2016, SIAM J. Matrix Anal. Appl..

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

[3]  Stefan Güttel,et al.  A black-box rational Arnoldi variant for Cauchy–Stieltjes matrix functions , 2013 .

[4]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[5]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[6]  Andreas Frommer,et al.  MONOTONE CONVERGENCE OF THE LANCZOS APPROXIMATIONS TO MATRIX FUNCTIONS OF HERMITIAN MATRICES , 2009 .

[7]  Christian Berg,et al.  Stieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity , 2007 .

[8]  Valeria Simoncini,et al.  A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations , 2007, SIAM J. Sci. Comput..

[9]  Stefan Güttel,et al.  Convergence of Restarted Krylov Subspace Methods for Stieltjes Functions of Matrices , 2014, SIAM J. Matrix Anal. Appl..

[10]  Valeria Simoncini,et al.  Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices , 2016, BIT Numerical Mathematics.

[11]  Chad Lieberman,et al.  On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems , 2010, SIAM J. Sci. Comput..

[12]  Matthias Rottmann,et al.  Multigrid preconditioning for the overlap operator in lattice QCD , 2014, Numerische Mathematik.

[13]  H. V. D. Vorst,et al.  Numerical methods for the QCDd overlap operator. I. Sign-function and error bounds , 2002, hep-lat/0202025.

[14]  Vladimir Druskin,et al.  On monotonicity of the Lanczos approximation to the matrix exponential , 2008 .

[15]  Vipin Kerala Varma,et al.  Conformal map and harmonic measure of the Bunimovich stadium , 2014, 1410.4932.

[16]  Marcel Schweitzer.  Monotone convergence of the extended Krylov subspace method for Laplace–Stieltjes functions of Hermitian positive definite matrices☆ , 2016 .

[17]  C. D. Boor,et al.  Divided Differences , 2005, math/0502036.

[18]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[19]  Stefan Güttel,et al.  Superlinear convergence of the rational Arnoldi method for the approximation of matrix functions , 2012, Numerische Mathematik.

[20]  Artan Borici.  Fast Methods for Computing the Neuberger Operator , 2000 .

[21]  A. Pettitt,et al.  Fast sampling from a Gaussian Markov random field using Krylov subspace approaches , 2008 .

[22]  Lothar Reichel,et al.  The extended Krylov subspace method and orthogonal Laurent polynomials , 2009 .

[23]  Igor Moret.  Rational Lanczos approximations to the matrix square root and related functions , 2009, Numer. Linear Algebra Appl..

[24]  Oliver G. Ernst,et al.  A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions , 2006, SIAM J. Numer. Anal..

[25]  Igor Moret,et al.  The restarted shift‐and‐invert Krylov method for matrix functions , 2014, Numer. Linear Algebra Appl..

[26]  D. Kressner,et al.  Limited‐memory polynomial methods for large‐scale matrix functions , 2020, GAMM-Mitteilungen.

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

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

[29]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[30]  Lothar Reichel,et al.  Error Estimates and Evaluation of Matrix Functions via the Faber Transform , 2009, SIAM J. Numer. Anal..

[31]  S. Güttel,et al.  Topical Issue Applied and Numerical Linear Algebra (2/2) , 2020, GAMM-Mitteilungen.

[32]  Andreas Frommer,et al.  Fast CG-Based Methods for Tikhonov-Phillips Regularization , 1999, SIAM J. Sci. Comput..

[33]  I. Turner,et al.  A restarted Lanczos approximation to functions of a symmetric matrix , 2010 .

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

[35]  Lothar Reichel,et al.  Recursion relations for the extended Krylov subspace method , 2011 .

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

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

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

[39]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

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

[41]  Stefan Güttel,et al.  Generalized Rational Krylov Decompositions with an Application to Rational Approximation , 2015, SIAM J. Matrix Anal. Appl..

[42]  M. Schweitzer.  Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions , 2018 .

[43]  Igor Moret,et al.  Krylov subspace methods for functions of fractional differential operators , 2017, Math. Comput..

[44]  Christian Berg,et al.  Potential Theory on Locally Compact Abelian Groups , 1975 .

[45]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[46]  Erin Carson,et al.  On the cost of iterative computations , 2020, Philosophical Transactions of the Royal Society A.

[47]  Valeria Simoncini,et al.  A new investigation of the extended Krylov subspace method for matrix function evaluations , 2009, Numer. Linear Algebra Appl..

[48]  Bruno Lang,et al.  An iterative method to compute the sign function of a non-Hermitian matrix and its application to the overlap Dirac operator at nonzero chemical potential , 2007, Comput. Phys. Commun..

[49]  A. Pettitt,et al.  Bayesian computations and efficient algorithms for computing functions of large, sparse matrices , 2004 .