This work is on a fast and accurate reduced basis method for solving discretized fractional elliptic partial differential equations (PDEs) of the form $\mathcal{A}^su=f$ by rational approximation. A direct computation of the action of such an approximation would require solving multiple (20$\sim$30) large-scale sparse linear systems. Our method constructs the reduced basis using the first few directions obtained from the preconditioned conjugate gradient method applied to one of the linear systems. As shown in the theory and experiments, only a small number of directions (5$\sim$10) are needed to approximately solve all large-scale systems on the reduced basis subspace. This reduces the computational cost dramatically because: (1) We only use one of the large-scale problems to construct the basis; and (2) all large-scale problems restricted to the subspace have much smaller sizes. We test our algorithms for fractional PDEs on a 3d Euclidean domain, a 2d surface, and random combinatorial graphs. We also use a novel approach to construct the rational approximation for the fractional power function by the orthogonal greedy algorithm (OGA).

翻译：本文研究一种快速且精确的简化基方法，用于通过有理逼近求解形如 $\mathcal{A}^su=f$ 的离散化分数阶椭圆型偏微分方程。直接计算此类逼近的作用需要求解多个（20~30个）大规模稀疏线性系统。我们的方法利用预处理共轭梯度法应用于其中一个线性系统时得到的前几个方向构建简化基。理论和实验表明，仅需少量方向（5~10个）即可在简化基子空间上近似求解所有大规模系统。这大幅降低了计算成本，原因在于：(1) 我们仅使用一个大规模问题来构建基；(2) 所有限制在子空间上的大规模问题规模显著减小。我们在三维欧几里得域、二维曲面和随机组合图上测试了我们的分数阶偏微分方程算法。我们还采用了一种新方法，通过正交贪婪算法构建分数幂函数的有理逼近。