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Stability of Linear GMRES Convergence with Respect to Compact Perturbations
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2021-03-23 , DOI: 10.1137/20m1340848
Jan Blechta

SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 436-447, January 2021.
Suppose that a linear bounded operator $B$ on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists $M_B<1$ such that the GMRES residuals fulfill $\|r_k\|\leq M_B\|r_{k-1}\|$ for every initial residual $r_0$ and step $k\in\mathbb{N}$. We prove that GMRES with a compactly perturbed operator $A=B+C$ admits the bound $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B)\,\|A^{-1}\|\,\sigma_j(C)\bigr)$, i.e., the singular values $\sigma_j(C)$ control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, SIAM J. Numer. Anal., 34 (1997), pp. 513--516], where only the case $B=\lambda I$ is considered. In this special case $M_B=0$ convergence is superlinear.


中文翻译:

关于紧凑扰动的线性 GMRES 收敛的稳定性

SIAM 矩阵分析与应用杂志,第 42 卷,第 1 期,第 436-447 页,2021 年 1 月。
假设希尔伯特空间上的线性有界算子 $B$ 至少表现出线性 GMRES 收敛,即存在 $M_B<1$ 使得 GMRES 残差满足 $\|r_k\|\leq M_B\|r_{k- 1}\|$ 对于每个初始残差 $r_0$ 和步骤 $k\in\mathbb{N}$。我们证明了具有紧扰动算子 $A=B+C$ 的 GMRES 承认有界 $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B )\,\|A^{-1}\|\,\sigma_j(C)\bigr)$,即奇异值 $\sigma_j(C)$ 控制未扰动问题的边界偏离。这个结果可以看作是[I. Moret, SIAM J. 数字。Anal., 34 (1997), pp. 513--516],其中仅考虑 $B=\lambda I$ 的情况。在这种特殊情况下,$M_B=0$ 收敛是超线性的。
更新日期:2021-03-23
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