In this talk we present a unified framework to efficiently approximate solutions to elliptic and parabolic fractional diffusion problems. After a brief classification of existing model order reduction schemes, we interpret the discrete solutions as matrix-vector products of the form g^T(L)b, where L is the discretization matrix of the spatial operator, b a prescribed vector, and g^T a parametric function, such as the fractional power or the Mittag-Leffler function. To alleviate the computational expenses, we apply a rational Krylov method to project 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. In the abstract framework of Stieltjes and complete Bernstein functions, we prove that the approximation error can be bounded by the third Zolotarëv problem. Inspired by these results, it is shown that the rational Krylov surrogate converges exponentially in the subspace dimension and uniformly in T when choosing the poles according to solutions of the third Zolotarëv problem. These poles are typically not nested which is inconvenient if one wishes to incrementally increase the accuracy of the surrogate. To address this difficulty, we provide the description of a weak greedy algorithm which allows us to sample the poles in a nested fashion. Our analytical and experimental findings confirm that this choice is competitive with the one based on the third Zolotarëv problem and allows for an adaptive enrichment of the search space.

Joint work with Joachim Schöberl.