On Rational Krylov and Reduced Basis Methods for Fractional Diffusion

TL;DR

This paper reveals an equivalence between Reduced Basis Methods and Rational Krylov Methods for fractional diffusion, providing new convergence proofs and proposing a novel RKM with competitive performance demonstrated through numerical tests.

Contribution

It establishes an equivalence between RBM and RKM for fractional diffusion, introduces a new RKM with optimal pole selection, and proves its convergence with numerical validation.

Findings

New equivalence between RBM and RKM for fractional diffusion

Proposed a new RKM with best rational approximation poles

Numerical results show the new RKM is competitive or superior

Abstract

We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function $x^{-s}$ in the spectral interval of the spatial discretization matrix. We prove convergence rates for this method and demonstrate numerically that it is competitive with or superior to many methods from the reduced basis, rational Krylov, and direct rational approximation classes. We provide numerical tests for some elliptic fractional diffusion model problems.