@Article{NMTMA-15-3,
author = {Xue-Lei, Lin and Ng, Michael, K. and Xin, Huang and Sun, Hai-Wei and Xue-Lei, Lin and Ng, Michael, K. and Sun, Hai-Wei},
title = {Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2022},
volume = {15},
number = {3},
pages = {565--591},
abstract = {<p style="text-align: justify;">In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations.
The finite difference method is employed to approximate the multi-dimensional
Riesz fractional derivatives, which generates symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method
with a preconditioner based on the sine transform is employed to solve the resulting
linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval $(\frac{1}{2},\frac{3}{2})$ and thus the preconditioned
conjugate gradient method converges linearly within an iteration number independent of the discretization step-size. Moreover, the proposed method can be extended
to handle ill-conditioned multi-level Toeplitz matrices whose blocks are generated by
functions with zeros of fractional order. Our theoretical results fill in a vacancy in the
literature. Numerical examples are presented to show the convergence performance
of the proposed preconditioner that is better than other preconditioners.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2022-0032},
url = {https://global-sci.com/article/90263/spectral-analysis-for-preconditioning-of-multi-dimensional-riesz-fractional-diffusion-equations}
}