On Second Order Shape Optimization Methods for Electrical Impedance Tomography

Abstract

This paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion ω inside a body Ω 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 ∂ω, then we choose a cost function in order to recover the geometry of ∂ω 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.

On the ersatz material approximation in level-set methods

“…While the derivative of the normal vector is obtained by a straightforward calculus, we have to transport from ∂Ω t to ∂Ω the Laplace-Beltrami operator and the tangential gradient in order to compute the other derivatives. We recall here facts proved in [2].…”

“…In [1,2,14], more situations have been studied. The shape hessian at the global minimum is compact and the optimization procedures are severely ill-posed.…”

“…In the sequel, we will use some technical formulae given in [2]. Given a smooth vector field h, we denote…”

Computing quantities of interest for random domains with second order shape sensitivity analysis

“…Due to their much higher complexity, shape Hessians are not so often applied and the literature about computations with second order shape derivatives is restricted (see, e.g., [11,12,14,23]). Nevertheless, the shape Hessian underlies the study of stability issues in shape optimization (see [1,3,4,6,7,8,13,21,25,28,31] for examples in imaging, tomography, fluid mechanics, aircraft construction, etc.). For the sake of readability, we present two academic but representative examples for which the expressions of the shape derivatives remains simple.…”

Stability of critical shapes for the drag minimization problem in Stokes flow

“…This point is particularly crucial in order to make numerical simulations: if the problem is unstable, regularization is required in the numerical minimization of the functional (see for example [2,3]). The question of stability is addressed with the second order derivative of the functional at a critical point.…”