Parallel Domain Decomposition Strategies for Stochastic Elliptic Equations Part B: Accelerated Monte Carlo Sampling with Local PC Expansions

Abstract

Solving stochastic partial differential equations (SPDEs) can be a computationally intensive task, particularly when the underlying parametrization of the stochastic input field involves a large number of random variables. Direct Monte Carlo (MC) sampling methods are well suited for this type of situation since their cost is independent of the input complexity. Unfortunately, MC sampling methods suffer from slow convergence. In this manuscript, we propose an acceleration framework for elliptic SPDEs that relies on domain decomposition techniques and polynomial chaos (PC) expansions of local operators to reduce the cost of solving a SPDE via MC sampling. The approach exploits the fact that, at the subdomain level, the number of variables required to accurately parametrize the input stochastic field can be significantly reduced, as covered in detail in the prequel (Part A) to this paper. This makes it feasible to construct PC expansions of the local contributions to the condensed problem (i.e., the Schur complement of the discretized operator). The approach basically consists of two main stages: (1) a preprocessing stage in which PC expansions of the condensed problem are computed and (2) a Monte Carlo sampling stage where random samples of the solution are computed. The proposed method its naturally parallelizable. Extensive numerical tests are used to validate the methodology and assess its serial and parallel performance.

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