Inverse linear problems on Hilbert space and their Krylov solvability

TL;DR

This paper explores the theoretical foundations of solving inverse linear problems in Hilbert spaces using Krylov subspace methods, focusing on convergence and solvability in infinite-dimensional settings.

Contribution

It introduces a comprehensive analysis of Krylov subspace methods for inverse problems in Hilbert spaces, highlighting conditions for solvability and convergence.

Findings

Krylov subspace methods can effectively approximate solutions in infinite-dimensional Hilbert spaces.

Conditions for Krylov solvability of inverse problems are characterized.

Theoretical insights support the application of projection methods to infinite-dimensional inverse problems.

Abstract

This monograph is centred at the intersection of three mathematical topics, that are theoretical in nature, yet with motivations and relevance deep rooted in applications: the linear inverse problems on abstract, in general infinite-dimensional Hilbert space; the notion of Krylov subspace associated to an inverse problem, i.e., the cyclic subspace built upon the datum of the inverse problem by repeated application of the linear operator; the possibility to solve the inverse problem by means of Krylov subspace methods, namely projection methods where the finite-dimensional truncation is made with respect to the Krylov subspace and the approximants converge to an exact solution to the inverse problem.