Stochastic Domain Decomposition Based on Variable-Separation Method

Abstract: In this work, we propose a new stochastic domain decomposition method for solving steady-state partial differential equations (PDEs) with random inputs. 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 apply it to stochastic domain decomposition. The resulting Stochastic Domain Decomposition based on the Variable-separation method (SDD-VS) effectively addresses the ``curse of dimensionality" by leveraging the explicit representation of stochastic functions derived from physical systems. The SDD-VS method aims to obtain a separated representation of the solution for the stochastic interface problem. To enhance efficiency, an offline-online computational decomposition is introduced. In the offline phase, the affine representation of stochastic algebraic systems is obtained through the successive application of the VS method. This serves as a crucial foundation for the SDD-VS method. In the online phase, the interface unknowns of SPDEs are estimated using a quasi-optimal separated representation, enabling the construction of efficient surrogate models for subproblems. The effectiveness of the proposed method is demonstrated via the numerical results of three concrete examples.