An efficient reduced-order approximation for the stochastic Allen-Cahn equation

Abstract

In this paper, we propose and analyze an efficient numerical method for solving the stochastic Allen-Cahn equation with additive noise. The method combines a stabilized semi-implicit time discretization scheme with a reduced-order finite element spatial discretization method. The core idea is to approximate the original high-dimensional solution space via a low-dimensional subspace, constructed by the Proper Orthogonal Decomposition (POD) method based on an ensemble of snapshots from the full-order model at selected time instances. First, we rigorously establish the spatio-temporal strong convergence rates between the mild solution and the reduced-order solution. Second, in large-sample simulations, the reduced-order basis exhibits a certain generalization capability in capturing the average behavior of the numerical solutions. Numerical experiments are provided to verify the theoretical error estimates and to demonstrate the effectiveness of the proposed method.