Interfaces and Free Boundaries
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Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem
Markos A. Katsoulakis (1), Georgios T. Kossioris (2) and Omar Lakkis (3)
(1) Department of Mathematics and Statistics, University of Massachusetts, 154 Hicks Way, MA 01003-3110, AMHERST, UNITED STATES(2) Institute of Applied and Comp. Maths, Research Centre of Crete, Foundation for Research and Technology-Hellas, G-711 10, VASILIKA VOUTON, GREECE
(3) Department of Mathematics, University of Sussex, Falmer, BN1 9RF, BRIGHTON, UNITED KINGDOM
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.
Keywords: Allen-Cahn, Stochastic PDE, Finite Elements, Regularity, Mean Curvature Flow