INdAM Workshop: "Fast Methods for Isogeometric Analysis"

In this talk, we present a comparative analysis of reliable quadrature techniques for approximating fractional operators, emphasizing error estimates and their effectiveness in preconditioning the Riesz operator. Such operator, essential in fractional models like anomalous diffusion, depends on a parameter which lies between 1 and 2. When the parameter is close to 2, several robust preconditioning methods with linear computational cost exist. However, as the parameter approaches 1, achieving efficient preconditioners with linear complexity becomes more challenging. Previous work approximated the Riesz operator as a fractional power of a discretized Laplacian using the Gauss-Jacobi rule. Recent studies have enhanced this by using advanced quadrature rules, such as Gauss-Laguerre and sinc quadratures, which provide faster convergence. By appropriately selecting the number of quadrature points, both methods generate preconditioners based on sums of a few shifted Laplacian inverses, ensuring high efficiency and accuracy. Numerical tests show that the sinc-based preconditioner is more versatile than Gauss-Laguerre and both outperform the Gauss-Jacobi approach.