Preconditioners for Fractional Diffusion Equations Based on the Spectral Symbol

TL;DR

This paper introduces a new spectral symbol-based preconditioner for fractional diffusion equations, demonstrating improved performance in multi-dimensional cases compared to existing methods.

Contribution

A novel preconditioner based on the spectral symbol for fractional diffusion equations, effective especially in multi-dimensional problems.

Findings

The new preconditioner belongs to the τ algebra.

It outperforms existing preconditioners in multi-dimensional settings.

Low band structure preconditioners are more effective in 1D, but less so in higher dimensions.

Abstract

It is well known that the discretization of fractional diffusion equations (FDEs) with fractional derivatives $\alpha\in(1,2)$, using the so-called weighted and shifted Gr\"unwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called Generalized Locally Toeplitz (GLT) class. Conversely, when the given FDE have constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function $\mathcal{F}_\alpha$, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one and two dimensional cases. In this note we propose a new preconditioner denoted by $\mathcal{P}_{\mathcal{F}_\alpha}$ which belongs to the $\tau$ algebra and it is based on the spectral symbol $\mathcal{F}_\alpha$. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one-dimensional case, the new preconditioner performs better in the more challenging multi-dimensional setting.