Numerically Computable A Posteriori-Bounds for stochastic Allen-Cahn equation

TL;DR

This paper develops a numerically computable a-posteriori error bound for a stochastic Allen-Cahn equation, enabling error verification based solely on numerical data without prior solution knowledge.

Contribution

It introduces a rigorous, data-driven a-posteriori error estimate for a stochastic PDE, applicable even when global solution existence is unknown.

Findings

Derived a computable mean square error bound

Validated the approach on a 1D stochastic Allen-Cahn equation

Provides a tool for error verification without a-priori solution info

Abstract

The aim of this paper is the derivation of an a-posteriori error estimate for a numerical method based on an exponential scheme in time and spectral Galerkin methods in space. We obtain analytically a rigorous bound on the mean square error conditioned to the calculated data, which is numerically computable and uses the given numerical approximation. Thus one can check a-posteriori the error for a given numerical computation without relying on an asymptotic result. All estimates are only based on the numerical data and the structure of the equation, but they do not use any a-priori information of the solution, which makes the approach applicable to equations where global existence of solutions is not known. For simplicity of presentation, we develop the method here in a relatively simple situation of a stable one-dimensional Allen-Cahn equation with additive forcing.