In this paper, we propose a reduced-order method (ROM) based on the Monte Carlo finite difference method (FDM) or Monte Carlo finite element method (FEM) for the stochastic Allen–Cahn (SAC) equation with multiplicative noise. The reduction method is a combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). In the theoretical part, the error bounds of full-order method (FOM) and POD–DEIM are provided, as well as the computational complexity of FOM and POD–DEIM. In the numerical experiments section, several two-dimensional SAC examples with multiplicative noise are considered. Using the Monte Carlo FDM and Monte Carlo FEM as the reference methods, the POD–DEIM has obvious advantage in the computational efficiency. And with the encryption in the space or time direction, this advantage is more prominent.

在本文中，我们针对具有噪声的随机Allen-Cahn（SAC）方程，提出了一种基于Monte Carlo有限差分法（FDM）或Monte Carlo有限元方法（FEM）的降阶方法（ROM）。归约方法是适当的正交分解（POD）和离散经验插值方法（DEIM）的组合。在理论部分，提供了全阶法（FOM）和POD–DEIM的误差范围，以及FOM和POD–DEIM的计算复杂性。在数值实验部分中，考虑了几个具有乘法噪声的二维SAC示例。使用Monte Carlo FDM和Monte Carlo FEM作为参考方法，POD-DEIM在计算效率上具有明显的优势。而且在空间或时间方向上进行加密时，此优势更加突出。