Commun. Comput. Phys., 6 (2009), pp. 342-353.


A Preconditioned Recycling GMRES Solver for Stochastic Helmholtz Problems

Chao Jin 1, Xiao-Chuan Cai 2*

1 Cadence Design Systems, 555 River Oaks Parkway, San Jose, CA 95134, USA.
2 Department of Computer Science, University of Colorado at Boulder, Boulder, CO 80309, USA.

Received 20 October 2007; Accepted (in revised version) 6 October 2008
Available online 15 December 2008

Abstract

We present a parallel Schwarz type domain decomposition preconditioned recycling Krylov subspace method for the numerical solution of stochastic indefinite elliptic equations with two random coefficients. Karhunen-Loeve expansions are used to represent the stochastic variables and the stochastic Galerkin method with double orthogonal polynomials is used to derive a sequence of uncoupled deterministic equations. We show numerically that the Schwarz preconditioned recycling GMRES method is an effective technique for solving the entire family of linear systems and, in particular, the use of recycled Krylov subspaces is the key element of this successful approach.

AMS subject classifications: 35R60, 60H15, 60H35, 65C30, 47B80, 65N55, 65M55

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Key words: Recycling GMRES, domain decomposition, additive Schwarz preconditioner, stochastic Helmholtz equation.

*Corresponding author.
Email: chao@cadence.com (C. Jin), cai@cs.colorado.edu (X.-C. Cai)
 

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