Speaker
Description
In this talk, we investigate preconditioning strategies 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, it is well known that the Laplacian itself serves as an effective preconditioner with linear computational cost. However, as $\alpha$ decreases toward 1, its performance deteriorates, requiring more specialized approaches.
Our approach consists of the following steps:
- approximate $-(-\Delta)^{\frac{\alpha}{2}}$ as the fractional power of a given discretization of the Laplacian, using the Matrix Transfer Technique;
- represent the resulting fractional power operator in integral form, adopting the Dunford-Taylor integral representation;
- explore various quadrature rules to approximate the integral, leading to rational approximations of the fractional power operator.
As a result, we define and analyze preconditioners for the Riesz operator, expressed as a sum of $ m$ inverses of shifted Laplacian matrices, where $ m$ corresponds to or is closely related to the number of quadrature points.
Using a Gauss-Jacobi quadrature approach, we show that while the resulting preconditioner performs comparably to the Laplacian for $\alpha$ close to 2, it yields significantly better results for $\alpha$ close to 1, even with a reasonably small $ m $, still retaining the same computational complexity as the Laplacian. To further enhance these results, we investigate the feasibility of using exponentially convergent quadrature rules, like sinc and Gauss-Laguerre, to ensure the number of inverses remains small while achieving numerical optimality.
