Weak approximation for stochastic reaction-diffusion equation near sharp interface limit

TL;DR

This paper develops a numerical approximation method for stochastic reaction-diffusion equations near the sharp interface limit, providing polynomial error bounds in terms of 1/ε and establishing a central limit theorem for the weak approximation.

Contribution

It introduces a regularized approach with exponential ergodicity and analyzes regularity to obtain polynomial error bounds, addressing an open problem in SPDE approximation near sharp interfaces.

Findings

Established weak error bounds depending polynomially on 1/ε.

Proved a central limit theorem for the weak approximation.

Demonstrated the regularization effect of noise on numerical SPDE solutions.

Abstract

It is known that when the diffuse interface thickness $\epsilon$ vanishes, the sharp interface limit of the stochastic reaction-diffusion equation is formally a stochastic geometric flow. To capture and simulate such geometric flow, it is crucial to develop numerical approximations whose error bounds depends on $\frac 1\epsilon$ polynomially. However, due to loss of spectral estimate of the linearized stochastic reaction-diffusion equation, how to get such error bound of numerical approximation has been an open problem. In this paper, we solve this weak error bound problem for stochastic reaction-diffusion equations near sharp interface limit. We first introduce a regularized problem which enjoys the exponential ergodicity. Then we present the regularity analysis of the regularized Kolmogorov and Poisson equations which only depends on $\frac 1{\epsilon}$ polynomially. Furthermore, we establish such weak error bound. This phenomenon could be viewed as a kind of the regularization effect of noise on the numerical approximation of stochastic partial differential equation (SPDE). As a by-product, a central limit theorem of the weak approximation is shown near sharp interface limit. Our method of proof could be extended to a number of other spatial and temporal numerical approximations for semilinear SPDEs.