Stability of Coupled-Physics Inverse Problems with internal measurements

TL;DR

This paper establishes stability results for a class of hybrid inverse problems involving internal measurements, specifically for recovering conductivity-like parameters from internal data governed by elliptic PDEs.

Contribution

It introduces a general framework to prove stability for the nonlinear step in hybrid inverse problems with functionals of the form σ|∇u|^p, extending previous results to a broader class of problems.

Findings

Proves stability of the linearized inverse problem.

Establishes Hölder conditional stability for the nonlinear problem.

Applies to functionals with 0 < p ≤ 1 in elliptic PDE settings.

Abstract

In this paper, we develop a general approach to prove stability for the non linear second step of hybrid inverse problems. We work with general functionals of the form $\sigma|\nabla u|^p$, $0 < p \leq 1$, where $u$ is the solution of the elliptic partial differential equation $\nabla\cdot \sigma \nabla u =0$ on a bounded domain $\Omega$ with boundary conditions $u|_{\partial \Omega} = f$. We prove stability of the linearization and H\"older conditional stability for the non-linear problem of recovering $\sigma$ from the internal measurement.