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We address the numerical discretization of the Allen-Cahn prob- lem withadditive white noise in one-dimensional space. The discretization is conductedin two stages: (1) regularize the white noise and study the regularizedproblem, (2) approximate the regularized problem. We address (1) by introducinga piecewise constant random approximation of the white noise with respect to aspace-time mesh. We analyze the regularized problem and study its relation toboth the original problem and the deterministic Allen-Cahn problem. Step (2) isthen performed leading to a practical Monte-Carlo method combined with a FiniteElement-Implicit Euler scheme. The resulting numerical scheme is tested againsttheoretical benchmark results.

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