In this paper, we study a $\tau$-matrix approximation based preconditioner for the linear systems arising from discretization of unsteady state Riesz space fractional diffusion equation with non-separable variable coefficients. The structure of coefficient matrices of the linear systems is identity plus summation of diagonal-times-multilevel-Toeplitz matrices. In our preconditioning technique, the diagonal matrices are approximated by scalar identity matrices and the Toeplitz matrices are approximated by {\tau}-matrices (a type of matrices diagonalizable by discrete sine transforms). The proposed preconditioner is fast invertible through the fast sine transform (FST) algorithm. Theoretically, we show that the GMRES solver for the preconditioned systems has an optimal convergence rate (a convergence rate independent of discretization stepsizes). To the best of our knowledge, this is the first preconditioning method with the optimal convergence rate for the variable-coefficients space fractional diffusion equation. Numerical results are reported to demonstrate the efficiency of the proposed method.

翻译：本文研究了一类由非稳态Riesz空间分数阶扩散方程（具有不可分离变系数）离散化所得线性系统的基于$\tau$-矩阵近似的预条件子。该线性系统的系数矩阵结构为单位矩阵加上对角矩阵乘多层Toeplitz矩阵之和。在我们的预条件技术中，对角矩阵被近似为标量单位矩阵，Toeplitz矩阵被近似为$\tau$-矩阵（一类可通过离散正弦变换对角化的矩阵）。所提出的预条件子可通过快速正弦变换算法实现快速求逆。理论上，我们证明了预处理系统的GMRES求解器具有最优收敛速度（即收敛速度与离散步长无关）。据我们所知，这是首个针对变系数空间分数阶扩散方程具有最优收敛速度的预条件方法。数值结果验证了所提方法的有效性。