Stochastic preconditioning of domain decomposition methods for elliptic equations with random coefficients

This paper aims at developing an efficient preconditioned iterative domain decomposition (DD) method for the sampling of linear stochastic elliptic equations. To this end, we consider a non-overlapping DD method resulting in a Symmetric Positive Definite (SPD) Schur system for almost every sampled problem. To accelerate the iterative solution of the Schur system, we propose a new stochastic preconditioning strategy that produces a preconditioner adapted to each sampled problem and converges toward the ideal preconditioner (i.e., the Schur operator itself) when the numerical parameters increase. The construction of the stochastic preconditioner is trivially parallel and takes place in an off-line stage, while the evaluation of the sample's preconditioner during the sampling stage has a low and fixed cost. One key feature of the proposed construction is a factorized form combined with Polynomial Chaos expansions of local operators. The factorized form guarantees the SPD character of the sampled preconditioners while the local character of the PC expansions ensures a low computational complexity. The stochastic preconditioner is tested on a model problem in 2 space dimensions. In these tests, the preconditioner is very robust and significantly more efficient than the deterministic median-based preconditioner, requiring, on average, up to 7 times fewer iterations to converge. Complexity analysis suggests the scalability of the preconditioner with the number of subdomains.