Jin, Chao; Cai, Xiao-Chuan; Li, Congming Parallel domain decomposition methods for stochastic elliptic equations. (English) Zbl 1149.65007 SIAM J. Sci. Comput. 29, No. 5, 2096-2114 (2007). Summary: 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-Loève (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. Cited in 13 Documents MSC: 65C30 Numerical solutions to stochastic differential and integral equations 65Y05 Parallel numerical computation 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) Keywords:stochastic elliptic equations; domain decomposition; recycling Krylov subspace method; parallel scalability; parallel computation; numerical examples; algorithm; preconditioning; Karhunen-Loève expansion; GMRES method Software:PETSc PDFBibTeX XMLCite \textit{C. Jin} et al., SIAM J. Sci. Comput. 29, No. 5, 2096--2114 (2007; Zbl 1149.65007) Full Text: DOI Link