Approximation with matrix-valued kernels and highly effective error estimators for reduced basis approximations

Abstract

This thesis can be summarized under the aspect of surrogate modelling for vector-valued functions and error quantification for those surrogate models. The thesis, in a broad sense, is split into two different parts. The first aspect deals with constructing surrogate models via matrix-valued kernels using both interpolation and regularization procedures. For this purpose, a new class of so called uncoupled separable matrix-valued kernels is introduced and heavy emphasis is placed on how suitable sample points for the construction of the surrogate can be chosen in such a way that quasi-optimal convergence rates can be achieved. In the second part, the focus does not lie on the construction of the surrogate itself, but on how existing a-posteriori error estimation can be improved to result in highly efficient error bounds. This is done in the context of reduced basis methods, which similar to the kernel surrogates, construct surrogate models by using data acquired from samples of the desired target function. Both parts are accompanied by numerical experiments which illustrate the effectiveness as well as verify the analytically derived properties of the presented methods.