AbstractIn 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 $$f(L^T) \varvec{b}$$
f
(
L
T
)
b
, where f is a non-analytic function involving fractional powers and $$\varvec{b}$$
b
is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$
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.
Rational Krylov methods for fractional diffusion problems on graphs
