Open problems and perspectives on solving Friedrichs systems by Krylov approximation

TL;DR

This paper investigates the conditions under which inverse problems involving Friedrichs operators can be solved within Krylov subspaces, providing insights into the effectiveness of Krylov-based algorithms for differential inverse problems.

Contribution

It establishes a general framework for understanding Krylov solvability of Friedrichs systems in abstract Hilbert spaces, linking it to practical solution methods for differential inverse problems.

Findings

Krylov solvability depends on specific properties of Friedrichs operators.

The framework predicts when Krylov truncation algorithms can accurately approximate solutions.

Provides theoretical criteria for the success of Krylov methods in inverse problems.

Abstract

We set up, at the abstract Hilbert space setting, the general question on when an inverse linear problem induced by an operator of Friedrichs type admits solutions belonging to (the closure of) the Krylov subspace associated to such operator. Such Krylov solvability of abstract Friedrichs systems allows to predict when, for concrete differential inverse problems, truncation algorithms can or cannot reproduce the exact solutions in terms of approximants from the Krylov subspace.