Geodesic
Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem
Markos A. Katsoulakis1 ; Georgios T. Kossioris2 ; Omar Lakkis3
1University of Massachusetts, Amherst, USA
2Research Centre of Crete, Vasilika Vouton, Greece
3University of Sussex, Brighton, United Kingdom
Interfaces and free boundaries, Tome 9 (2007) no. 1, pp. 1-30

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DOI

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.
DOI : 10.4171/ifb/154
Classification : 35-XX, 65-XX, 76-XX, 92-XX
Mots-clés : Allen-Cahn, Stochastic PDE, Finite Elements, Regularity, Mean Curvature Flow

Markos A. Katsoulakis  1   ; Georgios T. Kossioris  2   ; Omar Lakkis  3

1 University of Massachusetts, Amherst, USA
2 Research Centre of Crete, Vasilika Vouton, Greece
3 University of Sussex, Brighton, United Kingdom 
Markos A. Katsoulakis; Georgios T. Kossioris; Omar Lakkis. Noise regularization and computations for the 1-dimensional stochastic Allen–Cahn problem. Interfaces and free boundaries, Tome 9 (2007) no. 1, pp. 1-30. doi: 10.4171/ifb/154
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