On instability mechanisms for inverse problems

TL;DR

This paper introduces three robust instability mechanisms for inverse problems based on compression properties, with applications to unique continuation, heat equations, and Calderón problems, potentially impacting control theory.

Contribution

It presents new instability mechanisms for inverse problems using smoothing properties, applicable to various equations and geometries, advancing theoretical understanding.

Findings

New instability arguments for unique continuation.

Instability mechanisms for the backward heat equation.

Applications to Calderón type problems in general geometries.

Abstract

In this article we present three robust instability mechanisms for linear and nonlinear inverse problems. All of these are based on strong compression properties (in the sense of singular value or entropy number bounds) which we deduce through either strong global smoothing, only weak global smoothing or microlocal smoothing for the corresponding forward operators, respectively. As applications we for instance present new instability arguments for unique continuation, for the backward heat equation and for linear and nonlinear Calder\'on type problems in general geometries, possibly in the presence of rough coefficients. Our instability mechanisms could also be of interest in the context of control theory, providing estimates on the cost of (approximate) controllability in rather general settings. This is a revised version of the article ``On instability mechanisms for inverse problems'' Ars Inveniendi Analytica (2021), Paper No. 7, 93 pp by the same authors.