Title: A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Abstract

Summary Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice ofmore » coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.« less

Authors:

Ghai, Aditi ^[1]; Lu, Cao ^[1]; Jiao, Xiangmin  ^[1]

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+ Show Author Affiliations

1. Department of Applied Mathematics &, Statistics Stony Brook University Stony Brook New York, Institute for Advanced Computational Science Stony Brook University Stony Brook New York

Publication Date:

Tue Oct 02 00:00:00 EDT 2018

Sponsoring Org.:

USDOE

OSTI Identifier:

1475222

Grant/Contract Number:  

AC02-06CH11357

Resource Type:

Publisher's Accepted Manuscript

Journal Name:

Numerical Linear Algebra with Applications

Additional Journal Information:

Journal Name: Numerical Linear Algebra with Applications Journal Volume: 26 Journal Issue: 1; Journal ID: ISSN 1070-5325

Publisher:

Wiley Blackwell (John Wiley & Sons)

Country of Publication:

United Kingdom

Language:

English