Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus

TL;DR

This paper provides weak error estimates for the exponential Euler scheme applied to semi-linear SPDEs, introducing a new error representation formula that simplifies analysis without using Malliavin calculus.

Contribution

It derives a novel weak error representation formula for exponential integrators in SPDEs, enabling easier analysis without Malliavin calculus and applying it to full discretizations.

Findings

Weak error estimates are established for the exponential Euler scheme.

The new error formula simplifies analysis by avoiding irregular terms.

Application to spectral Galerkin discretization demonstrates practical effectiveness.

Abstract

This paper deals with the weak error estimates of the exponential Euler method for semi-linear stochastic partial differential equations (SPDEs). A weak error representation formula is first derived for the exponential integrator scheme in the context of truncated SPDEs. The obtained formula that enjoys the absence of the irregular term involved with the unbounded operator is then applied to a parabolic SPDE. Under certain mild assumptions on the nonlinearity, we treat a full discretization based on the spectral Galerkin spatial approximation and provide an easy weak error analysis, which does not rely on Malliavin calculus.