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.

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关于分数阶扩散的有理Krylov和简化基方法

我们在解决分数扩散问题的两类方法之间建立了等价关系，即简化的基础方法（RBM）和有理Krylov方法（RKM）。特别是，我们证明了一些最近提出的用于分数扩散的RBM可以解释为RKM。这种变化的观点使我们能够为以前没有可用方法的某些方法提供收敛证明。我们还提出了一种新的RKM，用于求解分数极点扩散问题，该极点使用在空间离散化矩阵的频谱间隔中的函数$ x ^ {-s} $的最佳有理逼近来选择。我们证明了该方法的收敛速度，并通过数值方法证明了与简化方法，有理Krylov和直接有理逼近类相比，该方法与许多方法都具有竞争优势或优于其他方法。