Department of Mathematics
Preconditioning techniques for diagonal-times-toeplitz matrices in fractional diffusion equations
© 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.
Approximate inverse, Circulant matrix, Fast Fourier transform, Fractional diffusion equation, Krylov subspace methods, Toeplitz matrix
Source Publication Title
SIAM Journal on Scientific Computing
Society for Industrial and Applied Mathematics
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.