Stochastic Domain Decomposition Based on Variable-Separation Method

Uncertainty propagation across different domains is of fundamental importance in stochastic simulations. In this work, we develop a novel stochastic domain decomposition method for steady-state partial differential equations (PDEs) with random inputs. The Variable-separation (VS) method is one of the most accurate and efficient approaches to solving the stochastic partial differential equation (SPDE). We extend the VS method to stochastic algebraic systems, and then integrate its essence with the deterministic domain decomposition method (DDM). It leads to the stochastic domain decomposition based on the Variable-separation method (SDD-VS) that we investigate in this paper. A significant merit of the proposed SDD-VS method is that it is competent to alleviate the "curse of dimensionality", thanks to the explicit representation of stochastic functions deduced by physical systems. The SDD-VS method aims to get a separated representation of the solution to the stochastic interface problem. To this end, an offline-online computational decomposition is introduced to improve efficiency. The main challenge in the offline phase is to obtain the affine representation of stochastic algebraic systems, which is crucial to the SDD-VS method. This is accomplished through the successive and flexible applications of the VS method. In the online phase, the interface unknowns of SPDEs are estimated using the quasi-optimal separated representation, making it easier to construct efficient surrogate models of subproblems. At last, three concrete examples are presented to illustrate the effectiveness of the proposed method.