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.

翻译：本文提出了一种新的随机区域分解方法，用于求解具有随机输入的稳态偏微分方程。基于变量分离方法在模拟随机偏微分方程中的高效性，我们将其扩展至随机代数系统，并应用于随机区域分解。由此产生的基于变量分离方法的随机区域分解有效解决了“维数灾难”问题，其关键在于利用物理系统导出的随机函数的显式表示。SDD-VS方法旨在获得随机界面问题解的分离表示。为提高效率，引入了离线-在线计算分解策略。在离线阶段，通过连续应用VS方法获得随机代数系统的仿射表示，这为SDD-VS方法奠定了关键基础。在线阶段，利用准最优分离表示估计SPDEs的界面未知量，从而为子问题构建高效的代理模型。三个具体算例的数值结果验证了所提方法的有效性。