  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.
