Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs

TL;DR

This paper develops and analyzes deterministic numerical methods for computing the first and second moments of solutions to stochastic PDEs, focusing on stability, convergence, and well-posedness of these methods.

Contribution

It introduces Petrov-Galerkin discretizations for second moment equations of stochastic PDEs and proves their stability, convergence, and well-posedness, advancing numerical analysis in this area.

Findings

Stable and convergent Petrov-Galerkin methods for second moments

Well-posedness of second moment equations in tensor product spaces

Numerical examples demonstrating method effectiveness

Abstract

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the first and second moments, as well as the covariance. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein-Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective-injective tensor product spaces as trial-test spaces. We construct Petrov-Galerkin discretizations based on tensor product piecewise polynomials and analyze their stability and convergence in these natural norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov-Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are illustrated by numerical examples.