Finite element approximation of Lyapunov equations related to parabolic stochastic PDEs

TL;DR

This paper develops a finite element and semi-implicit Euler scheme to approximate Lyapunov equations related to linear SPDEs with multiplicative noise, providing convergence rates and a deterministic method for path-dependent functionals.

Contribution

It introduces the first weak convergence analysis for fully discrete finite element approximations of SPDEs driven by multiplicative noise, achieving double the strong rate of convergence.

Findings

Derived convergence rates in operator norm for the discretization

Established a deterministic method for computing quadratic path-dependent functionals

Numerical experiments show advantages over Monte Carlo methods in stability

Abstract

A numerical analysis for the fully discrete approximation of an operator Lyapunov equation related to linear SPDEs (stochastic partial differential equations) driven by multiplicative noise is considered. The discretization of the Lyapunov equation in space is given by finite elements and in time by a semiimplicit Euler scheme. The main result is the derivation of the rate of convergence in operator norm. Moreover, it is shown that the solution of the equation provides a representation of a quadratic and path dependent functional of the SPDE solution. This fact yields a deterministic numerical method to compute such functionals. As a secondary result, weak error rates are established for a fully discrete finite element approximation of the SPDE with respect to this functional. This is obtained as a consequence of the approximation analysis of the Lyapunov equation. It is the first weak convergence analysis for fully discrete finite element approximations of SPDEs driven by multiplicative noise that obtains double the strong rate of convergence, especially for path dependent functionals and smooth spatial noise. Numerical experiments illustrate the results empirically and it is demonstrated that the deterministic method has advantages over Monte Carlo sampling in a stability context.