论文标题
线性GMRE收敛相对于紧凑扰动的稳定性
Stability of linear GMRES convergence with respect to compact perturbations
论文作者
论文摘要
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 $k\in\mathbb{N}$.我们证明，具有紧凑的操作员$ a = b+c $的gmres接纳$ \ | | r_k \ |/\ |/\ | r_0 \ | \ | \ | \ | \ leq \ prod_ {j = 1}^k \ bigl（m_b+（m_b+（1+m_b）即，奇异值$σ_J（c）$控制与未解决问题的界限。该结果可以看作是[I。 Moret，关于GMRES超线性收敛的注释，Siam J. Numer。肛门，34（1997），第513-516页，https：//doi.org/10.1137/s0036142993259792]，其中仅考虑$ b =λi$的情况。在此特殊情况下，$ m_b = 0 $，所产生的收敛是超线性。
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 $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B)\,\|A^{-1}\|\,σ_j(C)\bigr)$, i.e., the singular values $σ_j(C)$ control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513-516, https://doi.org/10.1137/S0036142993259792], where only the case $B=λI$ is considered. In this special case $M_B=0$ and the resulting convergence is superlinear.
