Numerical Approximation of Solutions to Stochastic Partial Differential Equations and Their Moments

Sammanfattning: The first part of this thesis focusses on the numerical approximation of the first two moments of solutions to parabolic stochastic partial differential equations (SPDEs) with additive or multiplicative noise. More precisely, in Paper I an earlier result (A. Lang, S. Larsson, and Ch. Schwab, Covariance structure of parabolic stochastic partial differential equations, Stoch. PDE: Anal. Comp., 1(2013), pp. 351–364), which shows that the second moment of the solution to a parabolic SPDE driven by additive Wiener noise solves a well-posed deterministic space-time variational problem, is extended to the class of SPDEs with multiplicative Lévy noise. In contrast to the additive case, this variational formulation is not posed on Hilbert tensor product spaces as trial–test spaces, but on projective–injective tensor product spaces, i.e., on non-reflexive Banach spaces. Well-posedness of this variational problem is derived for the case when the multiplicative noise term is sufficiently small. This result is improved in Paper II by disposing of the smallness assumption. Furthermore, the deterministic equations in variational form are used to derive numerical methods for approximating the first and the second moment of solutions to stochastic ordinary and partial differential equations without Monte Carlo sampling. Petrov–Galerkin discretizations are proposed and their stability and convergence are analyzed. In the second part the numerical solution of fractional order elliptic SPDEs with spatial white noise is considered. Such equations are particularly interesting for applications in statistics, as they can be used to approximate Gaussian Matérn fields. Specifically, in Paper III a numerical scheme is proposed, which is based on a finite element discretization in space and a quadrature for an integral representation of the fractional inverse involving only non-fractional inverses. For the resulting approximation, an explicit rate of convergence to the true solution in the strong mean-square sense is derived. Subsequently, in Paper IV weak convergence of this approximation is established. Finally, in Paper V a similar method, which exploits a rational approximation of the fractional power operator instead of the quadrature, is introduced and its performance with respect to accuracy and computing time is compared to the quadrature approach from Paper III and to existing methods for inference in spatial statistics.