Asymptotic error distribution of accelerated exponential Euler method for parabolic SPDEs

TL;DR

This paper investigates the asymptotic error distribution of the accelerated exponential Euler method for parabolic stochastic partial differential equations, providing new theoretical insights into its error behavior in infinite-dimensional settings.

Contribution

The paper establishes the limit distribution of the normalized error for the AEE method applied to SPDEs, extending understanding of error distributions in infinite-dimensional stochastic numerical analysis.

Findings

Derived the asymptotic error distribution for the AEE method on SPDEs.

Established a uniform approximation theorem for convergence in distribution.

Extended results to fully discrete schemes and stochastic ODEs.

Abstract

The asymptotic error distribution of numerical methods applied to stochastic ordinary differential equations has been well studied, which characterizes the evolution pattern of the error distribution in the small step-size regime. It is still open for stochastic partial differential equations whether the normalized error process of numerical methods admits a nontrivial limit distribution. We answer this question by presenting the asymptotic error distribution of the temporal accelerated exponential Euler (AEE) method when applied to parabolic stochastic partial differential equations. In order to overcome the difficulty caused by the infinite-dimensional setting, we establish a uniform approximation theorem for convergence in distribution. Based on it, we derive the limit distribution of the normalized error process of the AEE method by studying the limit distribution of its certain appropriate finite-dimensional approximation process. As applications of our main result, the asymptotic error distribution of a fully discrete AEE method for the original equation and that of the AEE method for a stochastic ordinary differential equation are also obtained.