Approximation of SPDE covariance operators by finite elements: A semigroup approach

TL;DR

This paper develops a semigroup-based method to approximate covariance operators of solutions to linear SPDEs using finite element spatial discretization and rational time approximations, with proven convergence rates.

Contribution

It introduces a novel integral equation approach for covariance approximation of SPDE solutions and derives convergence rates for finite element and rational approximation methods.

Findings

Convergence rates established in trace class and Hilbert--Schmidt norms.

Numerical simulations confirm theoretical convergence results.

Applicable to stochastic advection-diffusion and wave equations.

Abstract

The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert--Schmidt norms with numerical simulations illustrating the results.