Robust Schur complement preconditioner for block-Toeplitz system and its application in image restoration

In this paper, we are mainly concerned with the iterative method for solving the block-Toeplitz linear system. Based on the exact block Schur complement factorization of the block-Toeplitz coefficient matrix, a class of parameterized robust Schur complement preconditioner is constructed, which relies on Sherman-Morrison-Woodbury inversion formula of the Schur complement of the partitioned coefficient matrix. Meanwhile, we show that the new preconditioned system can result in much better spectrum distributions than some existing preconditioners, and deflate some small eigenvalues of the preconditioned matrix. The convergence analysis of our proposed method is established, and a strategy for the practical choice of an optimal parameter is given. Numerical experiments arising from image restoration are provided, which show the proposed preconditioner is effective and confirm our theoretical results are correct.