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.
翻译:在本文中,我们提出一种方法来计算定向网络上分解扩散方程式的解决方案,可以用Laplacian $Log 表示,用Laplacian $(LQT)\boldsymbol{b}$(美元)表示,美元是一种非分析功能,涉及分数功率,美元\boldsymbol{b}$(美元)是给定矢量。 Laplacian 图形是一个单质矩阵,使Krylov 方法对 $f(LQQT)\boldsylmbol{b}(美元) 更为缓慢地趋同。为了克服这一困难并实现更快的趋同,我们使用理性的 Krylov 方法对Laplacian 图形的淡化版本应用, 以一级变化或对子空间的投影。