Rational Krylov methods for fractional diffusion problems on graphs

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) b , where f is a non-analytic function involving fractional powers and b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for f(L^T) 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.