An Optimal Preconditioned MINRES Method for Symmetrized Multilevel Block Toeplitz Systems With Applications

In this work, we propose a novel preconditioned minimal residual method for a class of real, nonsymmetric multilevel block Toeplitz systems, which generalizes an ideal preconditioner established in [J. Pestana. Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices. SIAM Journal on Matrix Analysis and Applications, 40(3):870–887, 2019]. First, we symmetrize a certain real, nonsymmetric multilevel block Toeplitz system with symmetric blocks using a simple permutation matrix. We then show that its Hermitian part serves as an ideal preconditioner. Subsequently, we develop a novel preconditioning strategy for the aforementioned systems, with applications to solving nonlocal evolutionary fractional diffusion equations. Specifically, we first transform the nonsymmetric block Toeplitz all-at-once system arising from the equations into a symmetric block Hankel system via a symmetrization technique. Then, we propose a symmetric positive definite block Tau preconditioner for the symmetrized system, which can be implemented efficiently using the discrete sine transform. We prove that mesh-size-independent convergence can be achieved using the preconditioned minimal residual method. Theoretically, we show that the eigenvalues of the preconditioned matrices under consideration are bounded within disjoint intervals containing (Formula presented.), without outliers. The effectiveness of the proposed preconditioning strategy, in terms of iterations and CPU times, is validated through numerical examples.