On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

Krylov solvability under perturbations of abstract inverse linear problems

“…At this stage, this preliminary investigation completes a first cycle of study on abstract inverse linear problems, their finite-dimensional truncations and approximations, their Krylov solvability in the bounded and unbounded case, and the stability of Krylov solvability under perturbations, that we developed in our previous recent works [5,6,4,3] and in the present one.…”

“…The elements of such limit subspace would provide exploitable approximants for the solution to the original problem Af = g. One last remark concerns the topologies underlying all the questions above. We explicitly formulated them in terms of the operator norm and Hilbert norm, but alternatively there is a variety of weaker notions of convergence that are still highly informative for the solution to the considered inverse problem -we discussed this point extensively in [6]. Thus, the "weaker" counterpart of the above questions represents equally challenging and potentially useful problems to address.…”

“…To a lesser degree one has a theory of inverse linear problems governed by bounded operators on infinite-dimensional spaces: it consists of a more limited amount of sophisticated results, only few of which have a counterpart in the unbounded case. The literature is obviously enormous: we refer to that body of ideas and tools generally called iterative methods [29], Petrov-Galerkin methods [9,28], generalised projection methods [6], Krylov projection methods [29,24], in which as said the complex of knowledges when A is bounded on an infinite-dimensional H is certainly less systematic, let alone when A itself is unbounded.…”