Rational Krylov methods for fractional diffusion problems on graphs

12/15/2020

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by   Michele Benzi, et al.

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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.