Stability of linear GMRES convergence with respect to compact perturbations
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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≤ M_Br_k-1 for every initial residual r_0 and step k∈N. We prove that GMRES with a compactly perturbed operator A=B+C admits the bound r_k/r_0≤∏_j=1^k(M_B+(1+M_B) A^-1 σ_j(C)), 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=λ I is considered. In this special case M_B=0 and the resulting convergence is superlinear.
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