Analysis of a HMM time-discretization scheme for a system of Stochastic PDE's

TL;DR

This paper develops and analyzes a time-discretization scheme for a coupled system of stochastic PDEs with slow and fast components, using Heterogeneous Multiscale Methods and semi-implicit Euler schemes, providing error bounds for both trajectory and law approximations.

Contribution

It introduces a novel HMM-based discretization approach for stochastic PDE systems with multiple time scales, extending existing results to infinite-dimensional processes.

Findings

Derived strong and weak error bounds for the discretization scheme.

Extended previous finite-dimensional results to infinite-dimensional stochastic PDEs.

Validated the effectiveness of the method through theoretical analysis.

Abstract

We consider the discretization in time of a system of parabolic stochastic partial differential equations with slow and fast components; the fast equation is driven by an additive space-time white noise. The numerical method is inspired by the Averaging Principle satisfied by this system, and fits to the framework of Heterogeneous Multiscale Methods.The slow and the fast components are approximated with two coupled numerical semi-implicit Euler schemes depending on two different timestep sizes. We derive bounds of the approximation error on the slow component in the strong sense - approximation of trajectories - and in the weak sense - approximation of the laws. The estimates generalize the results of \cite{E-L-V} in the case of infinite dimensional processes.