Asymptotic error distribution of numerical methods for parabolic SPDEs with multiplicative noise
Jialin Hong, Diancong Jin, Xu Wang
TL;DR
The paper addresses the problem of understanding the asymptotic error distribution for numerical methods applied to parabolic SPDEs with multiplicative noise. It introduces the normalized temporal exponential Euler error $U^m(t)=m^{1/2}(X^m(t)-X(t))$ and shows its convergence in distribution in $\dot{H}^\eta$ to a linear SPDE-driven limit $U(t)$, driven by infinitely many independent $Q$-Wiener processes. The framework is extended to fully discrete schemes, and concrete SPDE examples (notably the stochastic heat equation) illustrate pointwise and functional convergence; it also reveals that spatial semi-discretizations can exhibit highly problem-dependent convergence speeds with degenerate limit distributions. The results highlight a richer and more delicate error-structure for SPDEs with multiplicative noise than in finite dimensions or additive-noise cases, informing both theory and practice for numerical analysis of SPDEs. The work opens avenues for further exploration of nontrivial spatial discretization limits and sharper problem-dependent error characterizations.
Abstract
This paper aims to investigate the asymptotic error distribution of several numerical methods for stochastic partial differential equations (SPDEs) with multiplicative noise. Firstly, we give the limit distribution of the normalized error process of the exponential Euler method in $\dot{H}^η$ for some $η>0$. A key finding is that the asymptotic error in distribution of the exponential Euler method is governed by a linear SPDE driven by infinitely many independent $Q$-Wiener processes. This characteristic represents a significant difference from numerical methods for both stochastic ordinary differential equations and SPDEs with additive noise. Secondly, as applications of the above result, we derive the asymptotic error distribution of a full discretization based on the temporal exponential Euler method and the spatial finite element method. As a concrete illustration, we provide the pointwise limit distribution of the normalized error process when the exponential Euler method is applied to a specific class of stochastic heat equations. Finally, by studying the asymptotic error of the spatial semi-discrete spectral Galerkin method, we demonstrate that the actual strong convergence speed of spatial semi-discrete numerical methods may be highly problem-dependent, rather than universally predictable.