Open problems and perspectives on solving Friedrichs systems by Krylov approximation
Noe Angelo Caruso, Alessandro Michelangeli
TL;DR
The paper investigates Krylov solvability for inverse linear problems $A f=g$ in a Hilbert space, with a focus on operators of Friedrichs type. It develops a compact, abstract framework for Krylov subspaces $\mathcal{K}(A,g)$ and their finite-dimensional truncations, linking solvability to whether a solution lies in $\overline{\mathcal{K}(A,g)}$ and to algorithmic schemes like GMRES and CG. It then situates these ideas within the Friedrichs systems framework, where the operator pair $(A_0,\widetilde{A}_0)$ and its realizations yield well-posed inverse problems under boundary-conditions choices, and outlines key questions (Q1–Q6) about when Krylov solvability holds, how it behaves under perturbations, and how to relate abstract results to concrete PDEs. The authors provide partial results indicating, for example, that when $A_1+\widetilde{A}_1=\alpha I$ and the smooth vectors align, Krylov solvability can be established via containment relations involving $A^*A$; they also discuss a prototypical unbounded Friedrichs operator $A f = -f'+f$ and related conjectures. Overall, the work aims to furnish structural criteria guiding numerical Krylov approximations for a broad Friedrichs class of differential inverse problems, offering a priori insights into solvability and convergence of truncation-based methods.
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.