Stability of Linear GMRES Convergence with Respect to Compact Perturbations
Jan Blechta · SIAM Journal on Matrix Analysis and Applications · 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.