Rational Krylov methods for fractional diffusion problems on graphs

Michele Benzi; Igor Simunec

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arXiv · 2012.08389

Rational Krylov methods for fractional diffusion problems on graphs

Michele Benzi·Igor Simunec

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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.

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Michele Benzi, Igor Simunec. 2020-12-15. Rational Krylov methods for fractional diffusion problems on graphs. https://arxiv.org/abs/2012.08389

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