Parallel Domain Decomposition Methods for Stochastic Elliptic Equations

We present parallel Schwarz-type domain decomposition preconditioned recycling Krylov subspace methods for the numerical solution of stochastic elliptic problems, whose coefficients are assumed to be a random field with finite variance. Karhunen-Loeve (KL) expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations. The key to an efficient algorithm lies in “recycling computed subspaces.” Based on a careful analysis of the KL expansion, we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation. We show theoretically and experimentally that the Schwarz preconditioned recycling GMRES method is optimal for the entire family of linear systems. A fully parallel implementation is provided, and scalability results are reported in the paper.

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