On [formula omitted] matrix-based approximate inverse preconditioning technique for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations

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In this paper, we firstly explore the special structure of the discretized linear systems from the spatial fractional diffusion equations. The coefficient matrices of the resulting discretized systems have a diagonal-plus-Toeplitz structure. Because the resulting Toeplitz matrix is symmetric positive definite (SPD), then we can employ the τ matrix to approximate it. By making use of the piecewise interpolation polynomials, we propose a new approximate inverse preconditioner to handle the diagonal-plus-Toeplitz coefficient matrices. The τ matrix-based approximate inverse (TAI) preconditioning technique can be implemented very efficiently by using discrete sine transforms(DST). Theoretically, we have proved that the spectrum of the resulting preconditioned matrices are clustered around one. Thus, Krylov subspace methods with the proposed preconditioners converge very fast. To demonstrate the efficiency of the new preconditioners, numerical experiments are implemented. The numerical results show that with the proper interpolation node numbers, the performance of the τ-matrix based preconditioning technique is better than the other tested preconditioners.