Krylov solvability under perturbations of abstract inverse linear problems

TL;DR

This paper investigates how Krylov solvability of inverse linear problems in Hilbert spaces is affected by small perturbations, with implications for the stability of Krylov subspace methods in noisy or uncertain environments.

Contribution

It provides a detailed analysis of the stability and behavior of Krylov solvability under perturbations, including a framework to assess Krylov solvability via perturbed problems.

Findings

Krylov solvability can persist, be gained, or lost under small perturbations.

The weak gap metric effectively monitors the distance between perturbed and original Krylov subspaces.

The study offers criteria to predict Krylov solvability stability in inverse problems.

Abstract

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution, i.e., it belongs to the Krylov subspace of the problem. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study here the possible behaviours of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the inverse problem -- the underlying motivations being the stability or instability of Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. We present a whole scenario of occurrences in the first part of the work. In the second, we exploit the weak gap metric induced, in the sense of Hausdorff distance, by the Hilbert weak topology, in order to conveniently monitor the distance between perturbed and unperturbed Krylov subspaces.