# On second order shape optimization methods for electrical impedance tomography

## TL;DR

This paper introduces a second order method for shape optimization in electrical impedance tomography, analyzing derivatives, stability, and ill-posedness to improve shape recovery accuracy.

## Contribution

It is the first to develop a second order approach for shape optimization in electrical impedance tomography, including derivative analysis and stability assessment.

## Key findings

- Derived second order derivatives of the state with respect to shape perturbations.
- Proposed a Newton method for shape reconstruction.
- Proved the ill-posedness via Hessian compactness at the solution.

## Abstract

This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partial\omega$, then we choose a cost function in order to recover the geometry of $\partial \omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.