Exploring preconditioning strategies for the Riesz operator in fractional models

In this talk, we introduce and analyze new preconditioning strategies for the Riesz operator depending on a parameter which is widely used in fractional models such as anomalous diffusion. For values of such parameter close to 2, various effective preconditioners with linear computational cost are available. However, as this parameter approaches 1, achieving linear complexity becomes more challenging. Previous work has tackled this by approximating the Riesz operator as a fractional power of a discretized Laplacian using the Gauss-Jacobi formula. In a more recent studies, this rational preconditioning approach has been extended by incorporating additional quadrature rules with exponential convergence. In more detail, the Gauss-Laguerre and sinc quadratures have been investigated, showing that after an optimal choice of parameters, both methods enable the construction of preconditioners based on a sum of shifted Laplacian inverses, ensuring high computational efficiency and numerical optimality. Numerical results reveal that the sinc-based preconditioner is more versatile than the Gauss-Laguerre preconditioner, and both outperform the Gauss-Jacobi preconditioner.