Deflation and balancing preconditioners for Krylov subspace methods applied to nonsymmetric matrices

Yogi A. Erlangga, Reinhard Nabben

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)

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. 1742-1759]. 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.

Original languageEnglish
Pages (from-to)684-699
Number of pages16
JournalSIAM Journal on Matrix Analysis and Applications
Volume30
Issue number2
DOIs
Publication statusPublished - 2008

Keywords

  • Balancing preconditioner
  • Convectiondiffusion
  • Deflation
  • Gmres
  • Nonsymmetric matrix

ASJC Scopus subject areas

  • Analysis

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