Fast Preconditioned Iterative Methods For Fractional Sturm-Liouville Equations

In this paper, we have considered fast solutions of the linear system arising from the fractional Sturm-Liouville problem, whose coefficient matrix contains the product of Toeplitz-like matrices. Based on suitable circulant approximations of the related coefficient matrix, we establish a matching preconditioner of matrix-free form. In theory, the spectrum of the preconditioned matrix is shown to cluster around [1/2, 1), which suggests the fast convergence speed of the proposed preconditioner within Krylov subspace acceleration. In addition, to reduce the computation time and storage, we consider an all-at-once discretized system and explore its low-rank tensor structure and alternating iterative tensor algorithms. Numerical experiments are given to show the effectiveness of the proposed solution techniques compared with some existing methods.