In this paper we analyze the behavior of the LSQR algorithm for the solution of compact operator equations in Hilbert spaces. We present results concerning existence of Krylov solutions and the rate of convergence in terms of an ℓp sequence where p depends on the summability of the singular values of the operator. Under stronger regularity requirements we also consider the decay of the error. Finally we study the approximation of the dominant singular values of the operator attainable with the bidiagonal matrices generated by the Lanczos bidiagonalization and the arising low rank approximations. Some numerical experiments on classical test problems are presented.

10aCompact operator10aLanczos bidiagonalization10aLinear ill-posed problem10aLSQR1 aCaruso, Noe1 aNovati, Paolo uhttps://www.sciencedirect.com/science/article/pii/S002437951930371400835nas a2200145 4500008004100000245007200041210006800113300001100181490000700192520032300199100001600522700002900538700001800567856010400585 2019 eng d00aOn Krylov solutions to infinite-dimensional inverse linear problems0 aKrylov solutions to infinitedimensional inverse linear problems a1–250 v563 aWe discuss, in the context of inverse linear problems in Hilbert space, the notion of the associated infinite-dimensional Krylov subspace and we produce necessary and sufficient conditions for the Krylov-solvability of a given inverse problem, together with a series of model examples and numerical experiments.

1 aCaruso, Noe1 aMichelangeli, Alessandro1 aNovati, Paolo uhttps://www.math.sissa.it/publication/krylov-solutions-infinite-dimensional-inverse-linear-problems01137nas a2200133 4500008004100000245009100041210006900132260001000201520068100211100001600892700002900908700001800937856004800955 2018 en d00aTruncation and convergence issues for bounded linear inverse problems in Hilbert space0 aTruncation and convergence issues for bounded linear inverse pro bSISSA3 aWe present a general discussion of the main features and issues that (bounded) inverse linear problems in Hilbert space exhibit when the dimension of the space is infinite. This includes the set-up of a consistent notation for inverse problems that are genuinely infinite-dimensional, the analysis of the finite-dimensional truncations, a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.1 aCaruso, Noe1 aMichelangeli, Alessandro1 aNovati, Paolo uhttp://preprints.sissa.it/handle/1963/35326