We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function $x^{-s}$ in the spectral interval of the spatial discretization matrix. We prove convergence rates for this method and demonstrate numerically that it is competitive with or superior to many methods from the reduced basis, rational Krylov, and direct rational approximation classes. We provide numerical tests for some elliptic fractional diffusion model problems.

翻译：特别是,我们证明最近提出的几项关于分解扩散的成果管理制建议可以被解释为RKM。这种改变的观点使我们能够为以前没有的各种方法提供趋同证据。我们还针对在空间分解矩阵的光谱间隔中使用最合理近似值$x ⁇ -s}函数选择的极点的分解扩散问题提出了一个新的RKM。我们证明了这一方法的趋同率,并用数字方式表明,它从减少的基础、理性Krylov和直接合理近似等级来看,与许多方法具有竞争力或优越性。我们为一些椭圆分解模型问题提供了数字测试。