Stochastic Domain Decomposition via Moment Minimization

Abstract

Propagating uncertainty accurately across different domains in multiscale physical systems with vastly different correlation lengths is of fundamental importance in stochastic simulations. We propose a new method to address this issue, namely, the stochastic domain decomposition via moment minimization (SDD-MM). Specifically, we develop a new moment minimizing interface condition to match the stochastic solutions at the interface of the nonoverlapping domains. Unlike other stochastic domain decomposition methods, the proposed method serves as a general framework that works with heterogeneous local stochastic solvers and does not rely on accessing global random trajectories, which are typically not available in realistic multiscale simulations. We analyze the computational complexity of the method and we quantify the contributing errors. The convergence property of SDD-MM is tested in several examples that include the stochastic reaction equation, Fisher's equation, as well as a two-dimensional Allen--Cahn equation. We observe good performance of the method for nonlinear problems as well as problems with different correlation lengths.