Section: New Results

Qualitative methods for inverse scattering problems

Sampling methods with time dependent data

Participant : Houssem Haddar.

Together with A. Lechleiter and S. Marmorat we proposed and analyzed a time domain linear sampling method as an algorithm to solve the inverse scattering problem of reconstructing an obstacle with Robin or Neumann boundary condition from time-dependent near-field measurements of scattered waves. Our algorithm is based on our earlier work to solve a similar inverse scattering problem for obstacles with Dirichlet boundary conditions. In addition to the analysis of a different scattering problem, we provided a substantial improvement of the method on both theoretical and numerical levels. More specifically, we analyzed the method for incident waves generated by pulses with bounded spectrum. Moreover, adapting the function space setting to this type of data allowed us to provide a simpler analysis. On the numerical side, we presented a fast implementation of the inversion algorithm that relies on a FFT-based evaluation of the near-field operator [34] .

Inverse problems for periodic penetrable media

Participant : Dinh Liem Nguyen.

Imaging periodic penetrable scattering objects is of interest for non-destructive testing of photonic devices. The problem is motivated by the decreasing size of periodic structures in photonic devices, together with an increasing demand in fast non-destructive testing. In this project, we considered the problem of imaging a periodic penetrable structure from measurements of scattered electromagnetic waves. As a continuation of earlier work jointly with A. Lechleiter [24] , [25] , [23] , we considered an electromagnetic problem for transverse magnetic waves (previous work treats transverse electric fields), and also the full Maxwell equations. In both cases, we treat the direct problem by a volumetric integral equation approach and construct a Factorization method [4] , [44] , [43] , [48] .

Transmission Eigenvalues and their application to the identification problem

Participant : Houssem Haddar.

The so-called interior transmission problem plays an important role in the study of inverse scattering problems from (anisotropic) inhomogeneities. Solutions to this problem associated with singular sources can be used for instance to establish uniqueness for the imaging of anisotropic inclusions from muti-static data at a fixed frequency. It is also well known that the injectivity of the far field operator used in sampling methods is related to the uniqueness of solutions to this problem. The frequencies for which this uniqueness fails are called transmission eigenvalues. We are currently developing approaches where these frequencies can be used in identifying (qualitative informations on) the medium properties. Our research on this topic is mainly done in the framework of the associate team ISIP http://www-direction.inria.fr/international/PHP/Networks/LiEA.php with the University of Delaware. A review article on the state of art concerning the transmission eigenvalue problem has been written in collaboration with F. Cakoni [32] . We are also in the process of editing a spacial issue of the journal Inverse Problems dedicated to the use of these transmission eigenvalues in inverse problems. Our recent contributions are the following:

• In collaboration with M. Fares and F. Collino from CERFACS and A. Cossonnière from ENSIEETA we finalized our work on the use of a surface integral equation approach to numerically compute transmission eigenvalues for inclusions with piecewise constant index. The main difficulty behind this procedure is the compactness of the obtained integral operator in usual Sobolev spaces associated with the forward scattering problem. We solved this difficulty by introducing a preconditioning operator associated with a “coercive” transmission problem. On the theoretical side, together with A; Cossonnière we also finalyzed the analysis of the Fredholm properties of the interior transmission problem for the cases where the index contrast changes sign outside the boundary by using the surface integral equation approach [16] .

• With G. Giorgi, we developed a method that give estimates on the material properties using the first transmission eigenvalue. This method is based on reformulating the interior transmission eigenvalue problem into an eigenvalue problem for the material coefficients. We validated our methodology for homogeneous and inhomogeneous inclusions and backgrounds. We also treated the case of a background with absorption and the case of scatterers with multiple connected components of different refractive indexes [21] .

• With F. Cakoni and D. Colton we initiated the study of transmission eigenvalues for absorbing media. In particular, we showed that, in the case of absorbing media, transmission eigenvalues form a discrete set, exist for sufficiently small absorption and for spherically stratified media exist without this assumption. For constant index of refraction, we also obtained regions in the complex plane where the transmission eigenvalues cannot exist and obtain a priori estimates for real transmission eigenvalues [14] .

• With F. Cakoni and A. Cossonnière we considered the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this work we employed a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle [15] .

The factorization method for inverse scattering problems

The factorization method for cracks with impedance boundary conditions

Participants : Yosra Boukari, Houssem Haddar.

We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from farfields associated with incident plane waves at a fixed fre- quency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks [38] .

The factorization method for EIT with uncertain background

Participants : Giovanni Migliorati, Houssem Haddar.

We extended the Factorization Method for Electrical Impedance Tomography to the case of background featuring uncertainty. This work is based on our earlier algorithm for known but inhomogeneous backgrounds. We developed three methodologies to apply the Factorization Method to the more difficult case of piece-wise constant but uncertain background. The first one is based on a recovery of the background through an optimization scheme and is well adapted to relatively low dimensional random variables describing the background. The second one is based on a weighted combination of the indicator functions provided by the Factorization Method for different realiza- tions of the random variables describing the uncertain background. We show through numerical experiments that this procedure is well suited to the case where many real- izations of the measurement operators are available. The third strategy is a variant of the previous one when measurements for the inclusion-free background are available. In that case, a single pair of measurements is sufficient to achieve comparable accuracy to the deterministic case [42] .

The factorization method for GIBC

Participants : Mathieu Chamaillard, Nicolas Chaulet, Houssem Haddar.

We are concerned with the identification of some obstacle and some Generalized Impedance Boundary Conditions (GIBC) on the boundary of such obstacle from far field measurements generated by the scattering of harmonic incident waves. The GIBCs are approximate models for thin coatings, corrugated surfaces, rough surfaces or imperfectly conducting media.

We justified the use of the Factorization method to solve the inverse obstacle problem in the presence of GIBCs. This method gives a uniqueness proof as well as a fast algorithm to reconstruct the obstacle from the knowledge of the far field produced by incident plane waves for all the directions of incidence at a given frequency. We also provided some numerical reconstructions of obstacles for several impedance operators.