2020/02/04 by Xue-lei Lin, Lin, X. -L., M. Ng +1 · 4 citations
Computer Science · Engineering · Physics and Astronomy · #15B05 #65F08 #65F10 #65M22 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Scattering and Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2002.01108
openalex publication_date 2020/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [McDonald, Pestana and Wathen, SIAM J. Sci. Comput., 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter ε>0 into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one.Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small ε; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when ε is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.