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.

翻译：不同区域间的不确定性传播在随机模拟中具有基础性重要性。本文针对含随机输入的稳态偏微分方程，提出了一种新型随机区域分解方法。变量分离方法作为求解随机偏微分方程最精确高效的方法之一，本文将其推广至随机代数系统，并融合确定性区域分解方法的核心理念，由此发展出本文研究的基于变量分离的随机区域分解方法。该方法的一个显著优势在于：通过物理系统推导的随机函数显式表示，能够有效缓解"维数灾难"。SDD-VS方法旨在获得随机界面问题解的分离表示，为此引入了离线-在线计算分解以提升效率。离线阶段的核心挑战在于获取随机代数系统的仿射表示——这对SDD-VS方法至关重要，该目标通过连续灵活运用变量分离方法实现。在线阶段则利用准最优分离表示估计随机偏微分方程的界面未知量，从而简化子问题高效代理模型的构建。最后通过三个具体算例验证了所提方法的有效性。