Preconditioning Techniques for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations

Abstract

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-timesToeplitz 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.

On HSS-like iteration method for the space fractional coupled nonlinear Schrödinger equations

“…We remark that, HSS-like can also be used as a preconditioner for Krylov subspace methods. Instead of the HSS-like preconditioner, we can employ the approximate inverse circulant-plus-diagonal preconditioners [7,10,11,13] for solving the linear system (3.1) and (3.2).…”

Fast numerical solution for fractional diffusion equations by exponential quadrature rule

“…For steps 3 and 4 in Algorithm 1, note that A h in (6) is diagonally dominant Toeplitz-like. There are many fast algorithms for solving such a linear system in O(n log n) operations; see [22,25,30,32,36] for more discussions. In this paper, we employ the preconditioned GMRES method [38] with the generalized Strang's circulant preconditioner proposed in [22] to solve the linear system iteratively.…”

“…Therefore, the computational cost for such a matrix-vector multiplication can be carried out in O(n log n) operations using a fast algorithm based on the fast Fourier transform (FFT), and the storage requirement is reduced from O(n 2 ) to O(n), where n is the number of spatial grid points in the discretization. As the resulting coefficient matrix is still ill-conditioned [32], many fast iterative methods have been proposed to speed up the convergence rate; see [22,25,30,32,36]. The complexity by those methods for solving the resulting system at each time step is of order O(n log n).…”

Fast Numerical Contour Integral Method for Fractional Diffusion Equations

“…Recall that the resulting coefficient matrices of space-fractional diffusion equations possess Toeplitz-like structures [38,40]; see also (3.8) and (3.20). Many efficient strategies have been proposed and used for fast solving the resulting systems emerged in the time-stepping scheme, such as the fast conjugate gradient squared method [38], multigrid method [25], preconditioned Krylov subspace methods with circulant-type preconditioners [10,24,27], and band preconditioners [12], etc. In our numerical experiments, we adopt the preconditioning strategies proposed in [10] to solve the resulting shifted linear systems.…”