Stochastic domain decomposition based on variable-separation method

Abstract

This work proposes a stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. Specifically, based on the efficiency of the variable-separation (VS) method in simulating stochastic partial differential equations (SPDEs), we extend it to stochastic algebraic systems and employ it in the context of stochastic domain decomposition. The resulting method, termed stochastic domain decomposition based on variable-separation (SDD-VS) alleviates the challenge commonly known as the “curse of dimensionality” notably by leveraging explicit representations of stochastic functions derived from physical systems. The primary objective of the proposed SDD-VS method is to obtain a separate representation of the solution for the stochastic interface problem. To enhance computational efficiency, we introduce a two-phase approach consisting of offline and online computation. In the offline phase, we establish an affine representation of stochastic algebraic systems by systematically applying the VS method. During the online phase, we estimate the interface unknowns of SPDEs using a quasi-optimal separated representation, facilitating the construction of efficient surrogate models for subproblems. We substantiate the effectiveness of our proposed approach through numerical experiments involving three specific instances, demonstrating its capability to provide accurate solutions.