Analysis of a modified Euler scheme for parabolic semilinear stochastic PDEs

TL;DR

This paper introduces a modified Euler scheme for weak approximation of parabolic semilinear stochastic PDEs, improving qualitative properties, preserving invariants, and enabling applications in multiscale systems and MCMC methods.

Contribution

The paper proposes a new modified Euler scheme that preserves spatial regularity and invariant distributions, with improved weak approximation properties for stochastic PDEs.

Findings

Preserves spatial regularity at all times.

Maintains Gaussian invariant distribution for any time-step size.

Achieves a weak order of convergence of 1/2, with enhanced approximation for Gibbs measures.

Abstract

We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. The new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein--Uhlenbeck process obtained when the nonlinearity is absent, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to $1/2$ in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order $1/2$, which does not hold for the standard method. This is the first result of this type in the literature. A key point in the analysis is the interpretation of the proposed modified Euler scheme as the accelerated exponential Euler scheme applied to a modified stochastic evolution equation. Finally, it is shown that the proposed method can be applied to design an asymptotic preserving scheme for a class of slow-fast multiscale systems, and to construct a Markov Chain Monte Carlo method which is well-defined in infinite dimension. We also revisit the analysis of the standard and the accelerated exponential Euler scheme, and we prove new results with approximation in the total variation distance, which serve to illustrate the behavior of the proposed modified Euler scheme.