Preconditioning of domain decomposition methods for stochastic elliptic equations ; Préconditionnement de méthodes de décomposition de domaine pour les équations elliptiques stochastiques

This thesis presents a new numerical method to efficiently generate samples of the solution of stochastic elliptical equations with random coefficients. Particular emphasis is placed on coefficients with high variance and short correlation length.This work concerns the adaptation of some classical Domain Decomposition (DD) Methods to the sampling of stochastic problems.Classical deterministic DD methods are based on iterative approaches which require preconditioning strategies capable of maintaining a high rate of convergence when the number of subdomains increases. In our stochastic context, determining a classical preconditioner suitable for each sample can be expensive, and alternative strategies can be more efficient. Each sample amounts to solving a reduced linear system for the values of the solution at the interfaces of the subdomains, according to a finite element discretization. This reduced system is then solved by an iterative method. This thesis proposed three main contributions to efficient preconditioning, by introducing surrogates of 1) the reduced global operator, 2) the contribution of each subdomain to the reduced global operator, and 3) local preconditioners (multi-preconditioning).The first contribution focuses on the iterative Schwarz method and introduces a stochastic preconditioner consisting of a surrogate of the Schwarz system for the unknown values on the interface of the subdomains. In a preprocessing stage, a truncated Karhunen-Loève (KL) expansion of the coefficient field and a Polynomial Chaos (PC) expansion of the Schwarz system are constructed to form the stochastic preconditioner. At the sampling stage, the preconditioner of each sample is recovered thanks to the very efficient evaluation of the PC expansions. Numerical simulations on a one-dimensional problem illustrate the rapid convergence of the resulting approach, provided that the number of KL modes and the PC degree are both sufficiently large.The second contribution extends the previous idea to non-overlapping DD methods ...