In this work, we develop a novel preconditioned method for solving space-fractional diffusion equations, which both accounts for and improves upon an ideal preconditioner pioneered in [J. Pestana. Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices. SIAM Journal on Matrix Analysis and Applications, 40(3):870-887, 2019]. Following standard discretization on the equation, the resultant linear system is a non-symmetric, multilevel Toeplitz system. Through a simple symmetrization strategy, we transform the original linear system into a symmetric multilevel Hankel system. Subsequently, we propose a symmetric positive definite multilevel Tau preconditioner for the symmetrized system, which can be efficiently implemented using discrete sine transforms. Theoretically, we demonstrate that mesh-independent convergence can be achieved when employing the minimal residual method. In particular, we prove that the eigenvalues of the preconditioned matrix are bounded within disjoint intervals containing $\pm 1$, without any outliers. Numerical examples are provided to critically discuss the results, showcase the spectral distribution, and support the efficacy of our preconditioning strategy.

翻译：本文针对空间分数阶扩散方程的求解，提出了一种新颖的预条件方法。该方法不仅借鉴了[J. Pestana. Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices. SIAM Journal on Matrix Analysis and Applications, 40(3):870-887, 2019]中开创的理想预条件子思想，并对其进行了改进。通过对该方程进行标准离散化，所得线性系统为非对称的多层Toeplitz系统。通过一种简单的对称化策略，我们将原线性系统转化为对称的多层Hankel系统。随后，我们针对该对称化系统提出了一种对称正定的多层Tau预条件子，该预条件子可利用离散正弦变换高效实现。理论上，我们证明了当采用最小残差法时，可实现与网格无关的收敛性。特别地，我们证明了预条件后矩阵的特征值被限制在包含$\pm 1$的不相交区间内，且不存在异常特征值。文中提供了数值算例，用于深入讨论相关结果、展示谱分布，并验证我们所提预条件策略的有效性。