Stability properties for a class of inverse problems

TL;DR

This paper proves Lipschitz stability for a class of nonlinear inverse problems involving integral operators, with applications to seismology and electromagnetic scattering, and demonstrates neural network-based reconstruction.

Contribution

It establishes Lipschitz stability for inverse problems where the direct problem involves a nonlinear integral operator, extending understanding of solution stability.

Findings

Inverse problems are solvable with finite data points.

Solutions exhibit Lipschitz stability in the data.

Neural networks can be used for reconstruction.

Abstract

We establish Lipschitz stability properties for a class of inverse problems. In that class, the associated direct problem is formulated by an integral operator Am depending non-linearly on a parameter m and operating on a function u. In the inversion step both u and m are unknown but we are only interested in recovering m. We discuss examples of such inverse problems for the elasticity equation with applications to seismology and for the inverse scattering problem in electromagnetic theory. Assuming a few injectivity and regularity properties for Am, we prove that the inverse problem with a finite number of data points is solvable and that the solution is Lipschitz stable in the data. We show a reconstruction example illustrating the use of neural networks.