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International audienceThe aim of our work is to reconstruct an inclusion ω immersed in a fluid flowing in a larger bounded domain Ω via a boundary measurement on ∂Ω. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing ω thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method. © 2013 American Institute of Mathematical Sciences

The paper extends investigations of identification problems by shape optimization methods for perfectly conducting inclusions to the case of perfectly insulating material. The Kohn and Vogelius criteria as well as a tracking type objective are considered for a variational formulation. In case of problems in dimension two, the necessary condition implies immediately a perfectly matching situation for both formulations. Similar to the perfectly conducting case, the compactness of the shape Hessian is shown and the illposedness of the identification problem follows. That is, the second order quadratic form is no longer coercive. We illustrate the general results by some explicit examples and we present some numerical results.

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.

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