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

Journal Article · Tue Oct 02 00:00:00 EDT 2018 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.2215· OSTI ID:1475222

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

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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

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 of 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.

• Sponsoring Organization: USDOE
• Grant/Contract Number: AC02-06CH11357
• OSTI ID: 1475222
• Journal Information: Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Vol. 26 Journal Issue: 1; ISSN 1070-5325
• Publisher: Wiley Blackwell (John Wiley & Sons)
• Country of Publication: United Kingdom
• Language: English