SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2766-A2784, January 2021.
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 $\epsilon>0$ 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 $\epsilon$; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when $\epsilon$ 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.

中文翻译：

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进化偏微分方程的一次性预处理器

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2766-A2784 页，2021 年 1 月。
在 [McDonald、Pestana 和 Wathen，SIAM J. Sci。Comput., 40 (2018), pp. A1012--A1033]，为演化偏微分方程产生的一次性线性系统提出了一个块循环预处理器，其中预处理矩阵被证明是可对角化的并且具有在热方程的情况下，恒等加低秩分解。在本文中，我们通过在块循环预处理器的右上角块中引入一个小参数 $\epsilon>0$ 来推广块循环预处理器。广义预处理器的实现需要与块循环相同的计算复杂度。理论上，我们证明了 (i) 泛化保留了对角化和恒等加低秩分解；(ii) 对于足够小的 $\epsilon$，新的预处理矩阵的所有特征值都聚集在 1 处；(iii) 当 $\epsilon$ 小于或等于时间步长的平方根时，预处理系统的 GMRES 方法具有与线性系统大小无关的线性收敛率。报告的数值结果证实了所提出的预处理器的效率，并表明泛化提高了块循环预处理器的性能。