Multilevel Tau preconditioners for symmetrized multilevel Toeplitz systems with applications to solving space fractional diffusion equations

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.