Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations

TL;DR

This paper establishes lower bounds for weak approximation errors in spectral Galerkin methods applied to stochastic wave equations, confirming the optimality of previously known upper bounds in a general setting.

Contribution

It provides the first lower bounds for weak errors in spectral Galerkin approximations of stochastic wave equations, demonstrating the tightness of existing upper bounds.

Findings

Lower bounds match known upper bounds, indicating optimality.

Weak errors cannot be improved beyond established bounds in the general framework.

Results apply to a broad class of semilinear stochastic wave equations.

Abstract

Although for a number of semilinear stochastic wave equations existence and uniqueness results for corresponding solution processes are known from the literature, these solution processes are typically not explicitly known and numerical approximation methods are needed in order for mathematical modelling with stochastic wave equations to become relevant for real world applications. This, in turn, requires the numerical analysis of convergence rates for such numerical approximation processes. A recent article by the authors proves upper bounds for weak errors for spatial spectral Galerkin approximations of a class of semilinear stochastic wave equations. The findings there are complemented by the main result of this work, that provides lower bounds for weak errors which show that in the general framework considered the established upper bounds can essentially not be improved.