Exploring rational approximations of fractional power operators for preconditioning

Abstract We define and analyze preconditioners for the Riesz operator $$-(- \Delta )^{\frac{\alpha }{2}}$$ , $$\alpha \in (1,2]$$ commonly used in fractional models, such as anomalous diffusion. For $$\alpha$$ close to 2 there are various effective preconditioners at disposal with linear computational cost. Seminal results on treatment of the case $$\alpha$$ 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.