Reduced basis methods for partial differential equations with stochastic influences

Abstract

This thesis is concerned with the development of reduced basis methods for parametrized partial differential equations (PPDEs) with stochastic influences. We consider uncertainties in the operator, right-hand side, boundary conditions and in the underlying domain. We are particularly interested in situations where the PPDE has to be evaluated quite often for various instances of the deterministic parameters and the stochastic influences. In the stochastic framework, such a situation occurs, e.g., in Monte Carlo simulations to compute statistical quantities such as mean, variance, or other moments. For the efficient application of the reduced basis method, it is necessary to develop affine decompositions with respect to the stochastic influences. We therefore extend the methodology of the empirical interpolation for the application in the stochastic setting, in particular for noisy input data. Alternatively, we also use a truncated Karhunen-Loe`ve (KL) expansion to resolve and affinely decompose the stochasticity. We derive a-posteriori error bounds for the state variable and output functionals, including also the KL-truncation errors. Non-standard dual problems are introduced for the approximation and analysis of special quadratic outputs which can in particular be applied to efficiently approximate statistical quantities such as mean and moments. We provide new error bounds for such outputs, outperforming standard approximations. To reduce the number of affine terms and hence for the improvement of the efficiency of the reduced simulations, we generalize the partitioning concepts for explicitly given deterministic parameter domains to arbitrary input functions with possibly unknown, high-dimensional, or even without direct parameter dependencies. No a-priori information about the input is necessary. We use all the presented methods for the application to PPDEs with stochastic influences on stochastic and additionally parametrized domains.