Model order reduction for fractional diffusion problems

Abstract

In this thesis we present a unified framework to efficiently approximate solutions to fractional diffusion problems of elliptic and parabolic type. After finite element discretization, we take the point of view that the solution is obtained by a matrix-vector product of the form f(L)b, where L is the discretization matrix of the spatial operator, b a prescribed vector, and f a parametric function. To alleviate the computational expenses, a model order reduction strategy in the form of a rational Krylov method is applied which projects the matrix to a low-dimensional space where a direct evaluation of the eigensystem is feasible. The particular choice of the subspace depends on a collection of parameters, the so-called poles. On the basis of the third Zolotar\"ev problem, we propose a variety of attractive pole selection strategies which allow us to efficiently query the solution map for multiple instances of the parameter. We either prove exponential convergence rates or provide the description of a computable error certificate to assess the quality of several poles where no analytical results are available.The analytical findings are confirmed by numerical experiments, including a systematic comparison of the presented schemes and a parameter study which provides deep insights in the effect of the fractional parameters.