Fast approximate inversion of a block triangular Toeplitz matrix with applications to fractional sub‐diffusion equations

Abstract

SummaryA fast approximate inversion method is proposed for the block lower triangular Toeplitz with tri‐diagonal blocks (BL3TB) matrix. The BL3TB matrix is approximated by a block ϵ‐circulant matrix, which can be efficiently inverted using the fast Fourier transforms. The error estimation is given to show the high accuracy of the approximation. In applications, the proposed method is employed to solve the fractional sub‐diffusion equation whose discretized matrix by a finite difference method is a BL3TB matrix. Numerical experiments are carried out to demonstrate the efficiency of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.

Approximate inversion method for time‐fractional subdiffusion equations

“…Note that if the numerical solutions in all time steps are stacked in one vector, then the finite-difference discretization of Equation 1 can lead to the following block lower triangular Toeplitz (BLTT) system 16,17 :…”

“…Most recently, another type of method, named the approximate inversion method (AIM), has been proposed by Lu et al 17 to fast solve the BLTT system (2). Based on the BLTT structure, the authors approximated the coefficient matrix of Equation 2 by the block -circulant matrix, which can be block-diagonalized by the Fourier matrix and diagonal matrix in (mn log n) operations.…”

“…As a consequence, the total computational cost by the AIM is of (mn log n) and the storage requirement is of (mn). In addition, a sufficient condition has been given in the work by Lu et al 17 to guarantee the invertibility of the corresponding block -circulant matrix, and the error estimation of the method has been provided. Nevertheless, the sufficient condition given in the work by Lu et al 17 is not easy to verify in practice, and the range of its application is limited.…”

“…In addition, a sufficient condition has been given in the work by Lu et al 17 to guarantee the invertibility of the corresponding block -circulant matrix, and the error estimation of the method has been provided. Nevertheless, the sufficient condition given in the work by Lu et al 17 is not easy to verify in practice, and the range of its application is limited. In this paper, we revisit the AIM and provide a new sufficient condition, which is easier to check and can be applied to several existing numerical schemes.…”

Dedication to Robert J. Plemmons

“…Numerical results have shown that compared with Robust Principal Component Analysis (RPCA)‐based models, the proposed methods are much efficient and effective. In the fifth paper, Lu et al developed a fast approximate inversion method for solving block lower triangular Toeplitz with tridiagonal blocks systems. In applications, the proposed method is employed to solve the fractional subdiffusion equation very efficiently.…”