Determining the space dependent coefficients in space-time fractional diffusion equations via Krylov preconditioning
Asim Ilyas, Muhammad Faisal Khan, Rosita L. Sormani, Giacomo Tento, Stefano Serra-Capizzano
TL;DR
The paper addresses an inverse source problem for a time-space fractional diffusion equation with a variable coefficient, regularized by a quasi-boundary value approach. It discretizes the regularized problem into a large all-at-once two-by-two block system and analyzes its spectral properties through Generalized Locally Toeplitz (GLT) theory, designing a preconditioner that yields eigenvalues clustered near 1 for the preconditioned matrix. The authors prove that the preconditioner shares the same GLT symbol as the original coefficient matrix, ensuring spectral efficiency, and validate the approach with numericalGMRES experiments showing substantial iteration reduction and robustness across fractional orders. They also discuss a Strang-type Toeplitz preconditioner for constant coefficients to further improve efficiency via FFTs. Overall, the work provides a theoretically grounded, scalable strategy to solve large-scale inverse fractional diffusion problems efficiently.
Abstract
We consider a time-space fractional diffusion equation with a variable coefficient and investigate the inverse problem of reconstructing the source term, after regularizing the problem with the quasiboundary value method to mitigate the ill-posedness. The equation involves a Caputo fractional derivative in the space variable and a tempered fractional derivative in the time variable, both of order in (0, 1). A finite difference approximation leads to a two-by-two block linear system of large dimensions. We conduct a spectral analysis of the associated matrix sequences, employing tools from Generalized Locally Toeplitz (GLT) theory, and construct the preconditioner guided by the GLT analysis. Numerical experiments are reported and commented, followed by concluding remarks.