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

TL;DR

This paper develops a numerical method for the 1D stochastic Allen-Cahn equation with white noise, involving noise regularization and Monte Carlo approximation, validated through theoretical benchmarks.

Contribution

It introduces a novel two-stage discretization approach combining noise regularization and Monte Carlo methods for the stochastic Allen-Cahn problem.

Findings

Effective noise regularization via piecewise constant approximation

Convergence of the regularized problem to the original stochastic problem

Numerical scheme matches theoretical benchmark results

Abstract

We address the numerical discretization of the Allen-Cahn prob- lem with additive white noise in one-dimensional space. 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.