Rational Krylov methods for fractional diffusion problems on graphs

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 )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.

A Unified Rational Krylov Method for Elliptic and Parabolic Fractional Diffusion Problems

“…as the unique rational function of degree (n, k) with denominator q k that interpolates f in Λ. The following two results establishes a close relation between the rational Ritz values and functions of type (11), which is instrumental in the analysis of RKMs. For a proof we refer to [34, Lemma 4.6, Theorem 4.8].…”

“…Let V be a basis of Q Ξ k+1 (L, b), L k+1 = V † LV , and r k = p k /q k for some polynomial p k ∈ P k . Then the rational Krylov approximation of r k (L)b is exact, i.e., (11). The quality of approximation clearly depends on the rational Krylov space and the way it is extracted from it.…”

“…where r f Λ,Ξ ∈ R n−1,k is defined by (11) with f (λ) = (ζ + λ) −1 ; see e.g., [34]. Together with quasi-optimality (13), this yields for any matrix V whose columns form a basis of…”

“…To see this, we provide the following variant of Hermite-Walsh's formula. (17), and r f Λ,Ξ as in (11)…”

“…Another class for approximating f τ (L)b are the rational Krylov methods (RKM) [54,2,53,11], to which, in view of [24], a variety of reduced basis methods [28,17,25,26] belong to. The RKM extracts a low-dimensional surrogate u k+1 ≈ f τ (L)b from a search space of the form Q Ξ k+1 (L, b) := span{q k (L) −1 b, .…”