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

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 mean and the spatio-temporal covariance structure of the solution process. 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 then propose numerical approximations based on Petrov–Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor 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 validated by numerical examples.