Exploring rational approximations of fractional power operators for preconditioning

We define and analyze preconditioners for the Riesz operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-(- \Delta )<^>{\frac{\alpha }{2}}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1,2]$$\end{document} commonly used in fractional models, such as anomalous diffusion. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} close to 2 there are various effective preconditioners at disposal with linear computational cost. Seminal results on treatment of the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} near 1 that still maintains linear computational complexity has been obtained approximating the Riesz operator as a fractional power of a discretized Laplacian, using Gauss-Jacobi formula. In this work, we extend this rational preconditioning approach by leveraging additional quadrature rules with exponential convergence. More precisely, we investigate both sinc and Gauss-Laguerre quadratures and show that, after an opportune choice of the involved parameters, both allow us to construct preconditioners based on a sum of a few shifted Laplacian inverses, and achieve high computational efficiency, ensuring numerical optimality. Several numerical results show that the sinc-based preconditioner is more versatile than the Gauss-Laguerre preconditioner, and that both outperform the Gauss-Jacobi one.