Approximation of random/stochastic partial differential equations

Abstract

The overarching interest of this thesis lies in approximations of partial differential equations (PDEs) with randomness or stochasticity. We focus on three rather different problems: a study of random fields on spherical shells, and its applications to PDE problems; quasi-Monte Carlo (QMC) methods for a class of PDEs with random coefficients; and a discretisation for the solution of stochastic PDEs. First, we consider Gaussian random fields on spherical shells that are radially anisotropic and rotationally isotropic. The smoothness of the covariance function is connected to the sample continuity, partial differentiability, and the Sobolev smoothness. Based on the regularity results, convergence rates of filtered approximations are established: Gaussian and log-normal random fields approximated with filtering, and a class of elliptic PDEs with approximated random coefficients, are considered. Second, we consider QMC integration of output functionals of solutions of a class of PDEs with a log-normal random coefficient. The coefficient is assumed to be given by an exponential of a Gaussian random field that is represented by a series expansion in terms of some system of functions with local supports. A quadrature error decay rate almost 1 is established, and the theory developed here is applied to a wavelet stochastic model. It is shown that a wide class of path smoothness can be treated with this framework. Finally, we turn our attention to an approximation of stochastic parabolic PDEs. We consider three discretisations: temporal, spatial, and the truncation of the infinite-dimensional space-valued Wiener process. Temporally, we consider the implicit Euler–Maruyama method with a non-uniform time step. For the spatial discretisation, we consider the spectral method. Further, we truncate the Wiener process, which is assumed to admit a series representation. We establish a time discrete error estimate for this algorithm. Further, a discrete analogue of maximal L2-regularity of the scheme is established, which has the same form as their continuous counterpart.