This cumulative thesis contains several contributions to the numerical analysis of stochastic partial differential equations. The main focus lies on investigating and improving numerical methods with respect to the qualitative properties stability and efficiency. Here the term stability of a numerical method denotes a measure for the approximation quality of the considered numerical method based on fixed refinement parameters in space and time or on a finite number of independent realisations for Monte Carlo estimators. In contrast, the efficiency of a numerical method for approximating the solution process of SPDEs measures the computational work needed to obtain a certain accuracy. Besides the separate discussion of these two qualitative properties, the emphasis is laid on investigating their interplay and their practical relevance for numerical experiments. This thesis consists of an introductory essay followed by 4 chapters based on 4 scientific articles. The purpose of the introduction is to provide an overview of relevant results on SPDEs and their approximations as well as discussing the existing literature on stability theory and efficiency investigations of numerical methods for stochastic differential equations. In Chapter 2 we investigate the performance of Monte Carlo methods for linear stochastic differential equations with an asymptotically almost surely stable, but mean-square unstable equilibrium solution. It is illustrated that under this specific stability setting standard Monte Carlo estimators fail to reproduce the qualitative behaviour of the second (or higher) moment(s) of the solution process. As a remedy an importance sampling technique focusing on the simulation of rarely occurring realisations of the solution process is proposed and numerically tested.
In Chapter 3 we develop importance sampling techniques for SPDEs based on an infinite-dimensional version of the well-known Girsanov transformation. An optimality result that provides the existence of a measure transformation, for which the Monte Carlo error vanishes completely, is used as guidance for constructing measure transformations that can be easily implemented. In Chapter 4 a structural mean-square stability theory for approximations of SPDEs is developed. For this we extend well-known mean-square stability results for approximations of finite-dimensional SDEs to an abstract tensor product-space setting and we derive necessary and sufficient conditions for the asymptotic mean-square stability of the zero solution of approximations of linear SPDEs. For a comparative study of numerical methods we investigate various combinations of rational approximations (of the semigroup) with either Maruyama or Milstein time integration schemes. Furthermore, results connecting the stability properties of the zero solution of the SPDE and of its numerical approximations are derived. In Chapter 5 we combine space-time multigrid techniques for deterministic partial differential equations with multilevel Monte Carlo methods for stochastic differential equations with additive noise. This coupling provides a new class of algorithms that are fully parallelisable, i.e., they can be computed in parallel with respect to space, time and probability. Extensive numerical experiments in Chapters 2-5 illustrate the theoretical results.