NDLI: Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Erlangga, Yogi A. Nabben, Reinhard
Copyright Year	2008
Abstract	For quite some time, the deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, the convergence of conjugate gradient methods combined with deflation has been analyzed and compared with other preconditioners, e.g., with the abstract balancing preconditioner [R. Nabben and C. Vuik, SIAM J. Sci. Comput., 27 (2006), pp. 17421759]. In this paper, we extend the convergence analysis to nonsymmetric linear systems in the context of GMRES iteration and compare it with the abstract nonsymmetric balancing preconditioner. We are able to show that many results for symmetric positive definite matrices carry over to arbitrary nonsymmetric matrices. First we establish that the spectra of the preconditioned systems are similar. Moreover, we show that under certain conditions, the 2-norm of residuals produced by GMRES combined with deflation is never larger than the 2-norm of residuals produced by GMRES combined with the abstract balancing preconditioner. Numerical experiments are done to nonsymmetric linear systems arising from a finite volume discretization of the convection-diffusion equation, and the numerical results confirm our theoretical results.
Starting Page	684
Ending Page	699
Page Count	16
File Format	PDF
ISSN	08954798
DOI	10.1137/060678257
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	2
Volume Number	30
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2008-06-18
Access Restriction	Subscribed
Subject Keyword	balancing preconditioner Solution of discretized equations Iterative methods for linear systems GMRES deflation nonsymmetric matrix Sparse matrices Multigrid methods; domain decomposition convection-diffusion
Content Type	Text
Resource Type	Article
Subject	Analysis

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