High-order integrator for sampling the invariant distribution of a class of parabolic SPDEs with additive space-time noise

TL;DR

This paper presents a high-order time-integrator for accurately sampling the invariant distribution of certain semilinear SPDEs with additive noise, improving upon standard methods with minimal computational overhead.

Contribution

The authors develop a modified integrator that achieves second-order accuracy for invariant distribution approximation in finite dimensions and higher order in the SPDE context, with theoretical analysis and numerical validation.

Findings

The integrator attains order 2 for invariant distribution approximation in SDEs.

Numerical experiments confirm the higher order convergence and efficiency.

The method has negligible overhead compared to standard Euler-Maruyama.

Abstract

We introduce a time-integrator to sample with high order of accuracy the invariant distribution for a class of semilinear SPDEs driven by an additive space-time noise. Combined with a postprocessor, the new method is a modification with negligible overhead of the standard linearized implicit Euler-Maruyama method. We first provide an analysis of the integrator when applied for SDEs (finite dimension), where we prove that the method has order $2$ for the approximation of the invariant distribution, instead of $1$. We then perform a stability analysis of the integrator in the semilinear SPDE context, and we prove in a linear case that a higher order of convergence is achieved. Numerical experiments, including the semilinear heat equation driven by space-time white noise, confirm the theoretical findings and illustrate the efficiency of the approach.