Preconditioners for Diagonal-times-Toeplitz Matrices in Fractional Diffusion Equations

In last few decades, more and more anomalous diffusion phenomena have been found in the real world applications, which lead to fractional diffusion equations. The use of fractional diffusion equations was shown to provide an adequate and accurate description for these diffusion phenomena. One of the main characteristics of the fractional differential operator is non-local. Most numerical methods for fractional diffusion equations generate dense coefficient matrices which are different from sparse coefficient matrices arising from the standard second-order diffusion equations. However, such dense coefficient matrices have Toeplitz-like structure which can be written as the sum of the scaled identify matrix and two diagonal-times-Toeplitz matrices. Therefore, the storage requirement can be reduced to O(n) where n is the number of grid points. By using the fast Fourier transform, the matrix-vector multiplication of these dense coefficient matrices can be done in O(n log n) operations.

The main aim of this proposal is to develop preconditioning techniques for such diagonal-times-Toeplitz matrices arising from the discretization scheme of fractional diffusion equations. Our idea is to construct a preconditioner by approximating the inverse of the sum of the scaled identify matrix and two diagonal-times-Toeplitz matrices. One of the interesting property of these Toeplitz matrices is that their diagonals are the Fourier coefficients of a function behaving like |x|^a, where a is the fractional order of the derivative in the fractional diffusion equation. The preconditioning techniques for this kind of Toeplitz matrices has not been studied in the literature. We would like to establish new theoretical results for circulant preconditioners for this kind of Toeplitz matrices. By using the new proposed results of circulant approximation, we will study and analyze the spectral properties of the approximate inverse preconditioned matrices. In particular, we would like to show the preconditioned matrices are the sum of a matrix of low rank and a matrix of small norm. The performance of the proposed preconditioners will be analyzed and evaluated by testing for a wide range of examples including three-dimensional fractional diffusion equations.