On the Properties for Iteration of a Compact Operator with Unstructured Perturbation

We consider certain speed estimates for Krylov subspace methods (such as GMRES) when applied upon systems consisting of a compact operator K with small unstructured perturbation B. Information about the decay of singular values of K is also assumed. Our main result is that the Krylov method will perform initially at superlinear speed when applied upon such pre-conditioned system. However, with large iteration numbers the effect of B is seen to be dominant. As a byproduct, we present several upper speed estimates with explicit constant that should be useful for numerical purposes. Given a bounded linear operator L ∈ L(H), what kind of computational problem is the inversion of equation (1.1) x = Lx + g by iterative means? How fast can it be done by Krylov subspace methods? How to make good speed estimates for the iterations? The answers to these (and related) questions give us much information about L. In particular the properties for iteration of a compact operator are by now rather well known, see [1]. In the theory of preconditioning and iterative inversion of Toeplitz operators we meet another kind of problem, which is related to the case of the compact operator — but not quite, see [2]. We get linear operators consisting of a large compact part K and a small non-compact perturbation B to invert. In the Toeplitz context we have a complicated interaction between K and B that we cannot lay our hands on. However, we may always generalize and look at K with a small unstructured perturbation B, whose norm alone is known. The study of such systems is the subject of this paper. Another aspect on the properties for iteration of K + B can be found in [3]. We shall see that an abstract Krylov subspace method will start with a su-perlinear convergence rate corresponding to the structure of the compact part K. When the compact part has been depleted, solver will " attack " the unstruc-tured small perturbation, and convergence rate is now linear. In applications the small perturbation part could be a preconditioning error (see [2]) or a linear operator arising from noise. In both cases, the compact part is what matters and iteration should be stopped once it enters the linear phase-at least if the small perturbation is noise. So we may say, that the properties for iteration of K are in a sense preserved under small …