|
|
|
| e-Pub |
Previous | 
Home
Section: New Results
Qualitative methods for inverse scattering problems
A generalized formulation of the Linear Sampling Method
Participants : Lorenzo Audibert, Houssem Haddar.
We proposed and analyzed a new formulation of the Linear Sampling Method that uses an exact characterization of the targets shape in terms of the so-called farfield operator (at a fixed frequency). This characterization is based on constructing nearby solutions of the farfield equation using minimizing sequences of a least squares cost functional with an appropriate penalty term. We first provided a general framework for the theoretical foundation of the method in the case of noise-free and noisy measurements operator. We then explicited applications for the case of inhomogeneous inclusions and indicate possible straightforward generalizations. We finally validated the method through some numerical tests and compare the performances with classical LSM and the factorization methods.
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 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.
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.cmap.polytechnique.fr/~defi/ISIP/isip.html 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. We also edited a spacial issue of the journal Inverse Problems dedicated to the use of these transmission eigenvalues in inverse problems http://iopscience.iop.org/0266-5611/29/10/100201/ . Our recent contributions are the following:
-
Together with A. Cossonnière we analyzed the Fredholm properties of the interior transmission problem for the cases where the index contrast changes sign outside the boundary by using a surface integral equation approach.
-
With F. Cakoni and N. Chaulet we investigated the asymptotic behaviour of the first transmission eigenvalue of a thin coating with respect to the coating thickness.
The factorization method for inverse scattering problems
The factorization method for cracks with impedance boundary conditions
Participant : Houssem Haddar.
With Y. Boukari we used 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 [5] .
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 [15] .
The factorization method for GIBC
Participants : Mathieu Chamaillard, Houssem Haddar.
With N. Chaulet, we studied 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.