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Preprint 8-2007

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

Source file is available as :   Postscript Document

Author(s) : Y.A. Erlangga , R. Nabben

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 8-2007

MSC 2000

65F10 Iterative methods for linear systems
65F50 Sparse matrices

Abstract :
For quite some times, deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, convergence of CG combined with deflation has been analyzed and compared with other preconditioners, e.g. with the abstract balancing preconditioner~{\it [Nabben and 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 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 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.

Keywords : Deflation, balancing preconditioner, nonsymmetric matrix, GMRES, convection-diffusion

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