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 𝑧 −𝑠 with 𝑧 ranging over 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.

中文翻译：

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关于分数扩散的有理 Krylov 和约简基方法

我们建立了解决分数扩散问题的两类方法之间的等价性，即归约基方法 (RBM) 和有理 Krylov 方法 (RKM)。特别是，我们证明了最近提出的几种用于分数扩散的 RBM 可以解释为 RKM。这种改变的观点使我们能够为一些以前没有可用的方法提供收敛性证明。我们还提出了一种新的 RKM，用于使用函数的最佳有理逼近选择极点的分数扩散问题𝑧 −𝑠和𝑧范围在空间离散矩阵的谱区间。我们证明了这种方法的收敛速度，并在数值上证明了它与约简基、有理 Krylov 和直接有理逼近类的许多方法相比具有竞争力或优越性。我们为一些椭圆分数扩散模型问题提供了数值测试。