## Section: New Results

### Sampling methods for inverse scattering problems

#### Sampling methods with time dependent data

Participants : Houssem Haddar, Armin Lechleiter.

We considered the extension of the so-called Factorization method to far-field data in the time domain. For a Dirichlet scattering object and incident wave fronts, the inverse problem under investigation consists in characterizing the shape of the scattering object from the behaviour of the scattered field far from the obstacle (far-field measurements). We derive a self-adjoint factorization of the time-domain far-field operator and show that the middle operator of this factorization possesses a weak type of coercivity. This allows to prove range inclusions between the far-field operator and the time-domain Herglotz operator. This project continues joint work with P. Monk and Q. Chen from the University of Delaware, for time-domain inverse scattering with near-field measurements [17] and we aim to extend this theory to electromagnetic problems.

#### Inverse problems for periodic penetrable media

Participants : Armin Lechleiter, 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, linked to the thesis project of Dinh Liem Nguyen, we considered the problem of imaging a periodic penetrable structure from measurements of scattered electromagnetic waves. As a continuation of earlier work, we considered an electromagnetic problem for transverse magnetic waves (earlier work treats transverse electric fields). We treat the direct problem by a volumetric integral equation approach and construct a Factorization method [40] . In the next step, we aim to tackle the full 3D-Maxwell problem.

#### The RG-LSM method applied to urban infrastructure imaging

Participant : Houssem Haddar.

The RG-LSM algorithm has been introduced by Colton-Haddar as a reformulation of the linear sampling method in the cases where measurements consist of Cauchy data at a given surface, by using the concept of reciprocity gap. The main advantage of this algorithm is to avoid the need of computing the background Green tensor (as required by classical sampling methods) as well as the Dirichlet-to-Neumann map for the probed medium (as required by sampling methods for impedance tomography problems). This method is for instance well suited for medical imaging techniques using microwaves (to detect tumors and malignancies characterized by strong variation in dielectric properties). However, in many other practical applications, like imaging of embedded facilities in the soil or mine detection, the required data at the interface cannot be easily obtained and one has only access to measurements of the scattered wave in the air. In order to overcome this limitation we proposed to couple the RG-LSM algorithm with a continuation method that would provide the Cauchy data from the scattered field. We showed that the obtained scheme has the same convergence properties as RG-LSM with exact data and remains competitive with respect to classical approaches. Preliminary numerical results in a 2-D configuration confirmed these conclusions and also gave further insight on the sampling resolution: Due to the ill-posedness of the first step, only the propagative part of the wave is well reconstructed, which may results in poor approximations of the field. However, the second step (RG-LSM) seems not being affected by this error and therefore is the reconstruction of the target. In a joint work with O. Ozdemir we first extended this approach to the case of rough interfaces [25] . Motivated by microwave imaging experiments, we are currently investigating the cases where the inclusions are buried under thin rough layers for which the use of approximate interface conditions (atc) would be appropriate. We first investigated the accuracy of a continuation method using atc for multilayered interfaces [26] . The second step would be to incorporate this procedure as a pre-processing of the reciprocity gap sampling method. A long time prospective of this work is to tackle the 3-D electromagnetic case.

#### Inverse scattering in 3D waveguides

Participant : Armin Lechleiter.

Time-harmonic acoustic waves in an ocean of finite height can be modeled by the Helmholtz equation inside a layer with suitable boundary conditions. Scattering in this geometry features phenomena unknown in free space: resonances might occur at special frequencies and wave fields consist of partly evanescent modes. Inverse scattering in waveguides hence needs to cope with energy loss and limited aperture data due to the planar geometry. In this project, we analyzed direct wave scattering in a 3D planar waveguide and showed that resonance frequencies do not exist for a certain class of bounded penetrable scatterers. More important, we proposed the Factorization method for solving inverse scattering problems in the 3D waveguide. This fast inversion method requires near-field data for special incident fields and we rigorously proved a method to generate this data from standard point sources. This is a joint project with Tilo Arens and Drossos Gintides (see [10] ).

#### Inverse scattering from screens with impedance boundary conditions

Participants : Yosra Boukari, Houssem Haddar.

We are interested in solving the inverse problem of determining a screen (or a crack) from multi-static measurements of electromagnetic (or acoustic) scattered field at a given frequency. An impedance boundary condition is assumed to be verified at both faces of the screen. We extended in a first step the use of the linear sampling method and the reciprocity-gap sampling method to retrieve the shape of the screen [32] and we are currently finalizing the theoretical justification of the so-called factorization method. We pursue the analysis of the accuracy of these methods with respect to the impedances values as well as using this analysis to derive a priori estimates on the impedances values. This work is conducted in collaboration with F. Ben Hassen.

#### Transmission Eigenvalues and their application to the identification problem

Participants : Anne Cossonnière, 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 equivalent 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.

The main topic of the PhD thesis of A. Cossonnière is to extend some of the results obtained above (for the scalar problem) to the Maxwell's problem. In this perspective, theoretical results related to solutions of the interior transmission problem for medium with cavities and existence of transmission eigenvalues have been obtained [37] . In collaboration with M. Fares and F. Collino from CERFACS we investigated the use of an integral equation approach to find the 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. The obtained procedure has been validated numerically in 2D and 3D cases. We are currently extending this work to the case of medium with perfectly conducting inclusions.