Document Type
Journal Article
Department/Unit
Department of Mathematics
Title
Preconditioning techniques for diagonal-times-toeplitz matrices in fractional diffusion equations
Language
English
Abstract
© 2014 Society for Industrial and Applied Mathematics. The fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable. The coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal-times-Toeplitz matrices. Standard circulant preconditioners may not work for such Toeplitz-like linear systems. The main aim of this paper is to propose and develop approximate inverse preconditioners for such Toeplitz-like matrices. An approximate inverse preconditioner is constructed to approximate the inverses of weighted Toeplitz matrices by circulant matrices, and then combine them together rowby-row. Because of Toeplitz structure, both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast Fourier transforms. Theoretically, we show that the spectra of the resulting preconditioned matrices are clustered around one. Thus Krylov subspace methods with the proposed preconditioner converge very fast. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show that its performance is better than the other testing preconditioners.
Keywords
Approximate inverse, Circulant matrix, Fast Fourier transform, Fractional diffusion equation, Krylov subspace methods, Toeplitz matrix
Publication Date
2014
Source Publication Title
SIAM Journal on Scientific Computing
Volume
36
Issue
6
Start Page
A2698
End Page
A2719
Publisher
Society for Industrial and Applied Mathematics
DOI
10.1137/130931795
Link to Publisher's Edition
http://dx.doi.org/10.1137/130931795
ISSN (print)
10648275
ISSN (electronic)
10957197
Recommended Citation
Pan, Jianyu, Rihuan Ke, Michael K. Ng, and Hai-Wei Sun. "Preconditioning techniques for diagonal-times-toeplitz matrices in fractional diffusion equations." SIAM Journal on Scientific Computing 36.6 (2014): A2698-A2719.