Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem

We address the numerical discretization of the Allen-Cahn problem with additive white noise in one-dimensional space. Our main focus is to understand the behavior of the discretized equation with respect to a small "interface thickness" parameter and the noise intensity. The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem. We address (1) by introducing a piecewise constant random approximation of the white noise with respect to a space-time mesh. We analyze the regularized problem and study its relation to both the original problem and the deterministic Allen-Cahn problem. Step (2) is then performed leading to a practical Monte Carlo method combined with a finite element-implicit Euler scheme. The resulting numerical scheme is tested against theoretical benchmark results concerning the behavior of the solution as the interface thickness goes to zero.