A Unified Rational Krylov Method for Elliptic and Parabolic Fractional Diffusion Problems

Abstract

We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form $f^{\boldsymbol{\tau}}(L)\mathbf{b}$, where $L$ is the discretization matrix of the spatial operator, $\mathbf{b}$ a prescribed vector, and $f^{\boldsymbol{\tau}}$ a parametric function, such as the fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotar\"ev's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map $\boldsymbol{\tau}\mapsto f^{\boldsymbol{\tau}}(L)\mathbf{b}$ and do not degenerate as the fractional parameters approach zero. We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space-time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate.