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

在本文中，我们提出了一种计算有向网络上分数扩散方程的解的方法，该方法可以用图拉普拉斯算子L 表示为乘积\(f(L^T) \varvec{b}\)，其中f是一个涉及分数幂的非解析函数，\(\varvec{b}\)是一个给定的向量。图拉普拉斯算子是一个奇异矩阵，导致\(f(L^T) \varvec{b}\) 的Krylov 方法收敛得更慢。为了克服这个困难并实现更快的收敛，我们使用有理 Krylov 方法应用于图拉普拉斯算子的去奇异化版本，通过一级移位或在子空间上的投影获得。