Preconditioning for the high-order sampling of the invariant distribution of parabolic semilinear SPDEs

TL;DR

This paper introduces a preconditioning technique and high-order integrators for efficiently sampling the invariant distribution of ergodic parabolic semilinear SPDEs, improving accuracy and regularity while maintaining computational feasibility.

Contribution

The authors develop a novel preconditioning method that enhances temporal regularity and enables high-order integrators for invariant distribution sampling in semilinear SPDEs.

Findings

New integrators with orders 1 and 2 for SPDE invariant distribution sampling.

Improved temporal regularity without altering the invariant distribution.

Numerical experiments demonstrating efficiency and theoretical validation.

Abstract

For a class of ergodic parabolic semilinear stochastic partial differential equations (SPDEs) with gradient structure, we introduce a preconditioning technique and design high-order integrators for the approximation of the invariant distribution. The preconditioning yields improved temporal regularity of the dynamics while preserving the invariant distribution and allows the application of postprocessed integrators. For the semilinear heat equation driven by space-time white noise in dimension $1$, we obtain new temporal integrators with orders $1$ and $2$ for sampling the invariant distribution with a minor overcost compared to the standard semilinear implicit Euler method of order $1/2$. Numerical experiments confirm the theoretical findings and illustrate the efficiency of the approach.