Section: New Results

Sampling methods for inverse scattering problems

Sampling methods with time dependent data

Participants : Houssem Haddar, Armin Lechleiter, Simon Marmorat.

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 .

We also extended the near-field version of the linear sampling method to causal time-dependent wave data for smooth, band-limited incident pulses, considering different boundary conditions as for instance Dirichlet, Neumann or Robin conditions [27] .

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), 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.

Inverse problems for Stokes-Brinkmann flows

Participants : Armin Lechleiter, Tobias Rienmüller.

Geometric inverse problems for flows arise for instance when conrolling pipelines and oil reservoirs. In this project, we considered the Stokes-Brinkmann equations that model, for instance, porous penetrable inclusions in a free background. The factorization method is able to characterize the inclusions from the relative Dirchlet-to-Neumann operator. Numerical examples show the feasibility of the method.

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 the so-called factorization method to this setting. We also analyzed a data completion algorithm based on integral equation method for the Helmholtz equation. This algorithm is then coupled to the so-called RG-LSM algorithm to retrieve cracks inside a locally homogeneous background. This work is conducted in collaboration with F. Ben Hassen.

Transmission Eigenvalues and their application to the identification problem

Participants : Anne Cossonnière, Houssem Haddar, Giovanni Giorgi.

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. A review article on the state of art concerning the transmission eigenvalue problem has been written in collaboration with F. Cakoni [24] .

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 [14] . This work is then extended to the case of medium with perfectly conducting inclusions. Only the scalar case has been studied [35] . In collaboration with M. Fares and F. Collino from CERFACS we investigated the use of a surface 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 also analyzed the transmission eigenvalue problem using this surface integral equation approach. This technique allowed us to generalize discretness results on the spectrum to cases where the contrast can change sign [2] .

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 [26] .

The factorization method for EIT with inhomogeneous background

Participants : Giovanni Migliorati, Houssem Haddar.

We developed a numerical inversion scheme based on the Factorization Method to solve the (continuous model of) Electrical Impedance Tomography problem with inhomogeneous background. The numerical scheme relies on the well chosen approximation by the finite element method of the solution to the dipole-like Neumann boundary-value problem. Two regularization techniques are tested, i.e. the Tikhonov regularization embedding Morozov principle, and the classical Picard Criterion. The numerical analysis of the method and the results obtained are presented in the INRIA report [28] .