Lower Bounds for Weak Approximation Errors for Spatial Spectral Galerkin Approximations of 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.

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

“…Numerical simulations of these equations require a full discretization in space, time, and noise. Our result complements the results on spatial and temporal discretizations in [5,25,24] and thereby completes the weak error analysis of numerical discretizations of the above-mentioned class of equations, providing a full picture of their complexity (see Table 1). Weak (as opposed to strong) convergence rates offer a flexible way of measuring the quality of the approximation, as the class of test functions can be chosen to reflect the priorities at application level.…”

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

“…In the context of the numerical approximation of the solution processes of such equations, the quantity of interest is typically the expected value of some functional of the solution and one is thus interested in the weak convergence rate of the considered numerical scheme. While the weak convergence analysis for numerical approximations of SPDE with Gaussian noise is meanwhile relatively far developed, see, e.g., [1,2,3,7,8,9,10,11,12,14,15,16,17,18,23,30], available results for non-Gaussian Lévy noise have been restricted to linear equations so far [4,5,21,25]. In this article, we analyze for the first time the weak convergence rate of numerical approximations for a class of semi-linear SPDE with non-Gaussian Lévy noise.…”

Numerical approximation and simulation of the stochastic wave equation on the sphere

“…We end this section by observing that several strategies for proving weak rates of convergence of numerical solutions to SPDEs in the literature could be extended to the present setting or in the case of numerical discretizations of nonlinear stochastic wave equations on the sphere, see for instance [13,24,34,5,20,16,23] and references therein. This could be subject of future research.…”