论文标题
进化部分微分方程的全面预处理
An all-at-once preconditioner for evolutionary partial differential equations
论文作者
论文摘要
在[麦当劳，佩斯塔纳和沃森，\ textit {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.在本文中，我们通过将一个小参数$ε> 0 $引入块循环预处理器的顶端块来概括块循环预处理。广义预处理的实现需要与块循环循环循环的计算复杂性相同。从理论上讲，我们证明（i）概括保留了对角度和身份加上 - 低率分解； （ii）新的预处理矩阵的所有特征值均以1的小$ $ε$聚类为1； （iii）预处理系统的gmres方法具有线性收敛速率，而当$ε$被视为小于或与时间步长的平方根相比时，与线性系统的大小无关。据报道，数值结果是为了确认所提出的预处理的效率，并表明概括提高了块循环前预处理的性能。
In [McDonald, Pestana and Wathen, \textit{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.
