Rational Krylov methods for fractional diffusion problems on graphs

TL;DR

This paper introduces rational Krylov methods with desingularization techniques to efficiently solve fractional diffusion equations on directed graphs, addressing convergence issues caused by the singular graph Laplacian.

Contribution

It presents a novel approach using rational Krylov methods combined with desingularization to improve convergence in fractional diffusion problems on graphs.

Findings

Enhanced convergence rates demonstrated with desingularized Krylov methods.

Effective handling of singular graph Laplacian in fractional diffusion computations.

Applicable to directed networks with fractional diffusion dynamics.

Abstract

In this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian $L$ as a product $f(L^T) \boldsymbol{b}$, where $f$ is a non-analytic function involving fractional powers and $\boldsymbol{b}$ is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $f(L^T) \boldsymbol{b}$ to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.