Increasing resolution and instability for linear inverse scattering problems

TL;DR

This paper analyzes how increasing frequency in linear inverse scattering problems enhances the stable recovery of features, while also highlighting the persistent instability for other features, supported by rigorous mathematical proofs.

Contribution

It provides a rigorous analysis of the resolution and stability trade-offs in high-frequency linear inverse scattering problems, using advanced mathematical tools.

Findings

Number of stably recoverable features increases with frequency

Singular values remain constant in stable regions and decay exponentially in unstable regions

Mathematical proofs based on spectral analysis and kernel computations

Abstract

In this work we study the increasing resolution of linear inverse scattering problems at a large fixed frequency. We consider the problem of recovering the density of a Herglotz wave function, and the linearized inverse scattering problem for a potential. It is shown that the number of features that can be stably recovered (stable region) becomes larger as the frequency increases, whereas one has strong instability for the rest of the features (unstable region). To show this rigorously, we prove that the singular values of the forward operator stay roughly constant in the stable region and decay exponentially in the unstable region. The arguments are based on structural properties of the problems and they involve the Courant min-max principle for singular values, quantitative Agmon-H\"ormander estimates, and a Schwartz kernel computation based on the coarea formula.