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.

}, keywords = {Compact operator, Lanczos bidiagonalization, Linear ill-posed problem, LSQR}, issn = {0024-3795}, doi = {https://doi.org/10.1016/j.laa.2019.08.024}, url = {https://www.sciencedirect.com/science/article/pii/S0024379519303714}, author = {Noe Caruso and Paolo Novati} } @article {2019, title = {On Krylov solutions to infinite-dimensional inverse linear problems}, journal = {Calcolo}, volume = {56}, year = {2019}, pages = {1{\textendash}25}, abstract = {We 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.

}, author = {Noe Caruso and Alessandro Michelangeli and Paolo Novati} } @article {2018, title = {Truncation and convergence issues for bounded linear inverse problems in Hilbert space}, number = {SISSA;50/2018/MATE}, year = {2018}, institution = {SISSA}, abstract = {We 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.}, url = {http://preprints.sissa.it/handle/1963/35326}, author = {Noe Caruso and Alessandro Michelangeli and Paolo Novati} }