Direction des Relations Internationales (DRI)

Programme INRIA "Equipes Associées"
(Demande de prolongation)

I. DEFINITION

Rapport scientifique de l'année 2008

Description de l'activité scientifique de l'équipe associée et des résultats obtenus : publications, communications, organisation de colloques, formation, soutenances de thèse, valorisation économique, sociale, industrielle, enregistrement de logiciels, dépôt de brevets ... (1 à 2 pages)

• One of the main scientific activities related to the associated team was the organization of the "International Conference on Inverse Scattering Problems", honoring David Colton and Rainer Kress. This conference was held at Sestri Levante (Italy), 8th-10th May, 2008. The organization of this conference was jointly done with Michele Piana from the University of Genova. The associated team provided financial support for the invited speakers (20 invited speakers). The conference also benefited from financial support provided by the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, United States Air Force Research Laboratory (http://www.london.af.mil), the University of Genova, the University of Goettingen and the University of Delaware.
  
  For more details, visit the web page of the ICISP conference
  
  
• Related to the conference organization, F. Cakoni, H. Haddar and M. Piana are editing a special issue of the journal Inverse Problems and Imaging. The deadline for authors is November 30th and we expect around 18 submitted papers. Submitted articles will undergo the usual reviewing process (as required by the journal policy). We expect the special issue being ready by mid-2009.
  
  
• F. Cakoni, D. Colton and H. Haddar have written two articles related to some of the objectives announced in the program of 2008:
  
  
  1. The Computation of Lower Bounds for the Norm of the Index of Refraction, accepted in 2008 for publication in Journal of Integral Equations and Applications (JIEA).
     In this article we considered the scattering of time harmonic electromagnetic plane waves by a bounded, inhomogeneous, anisotropic dielectric medium and showed that under certain assumptions a lower bound on the norm of the (matrix) index of refraction can be obtained from a knowledge of the smallest transmission eigenvalue corresponding to the medium. We also provided numerical examples showing the efficaciousness of our estimates.
     
     
  2. The inverse scattering problem from inhomogeneous medium with cavities. This article is almost finalized and will be sumbitted to SIAM Journal of Math. Anal. We consider in this work the inverse scattering problem from inclusions that contain voids and focus on the mathematical analysis of the so-called interior transmission problem related to this problem. The presence of voids introduces a serious difficulty in the analysis of this problem as the usual techniques cannot be applied. The proposed technique makes use of the Calderon projection associated with voids. We also analyse the dependence of the so-called transmission eigenvalues with respect to size of the void.
  
  
  
• F. Cakoni and H. haddar have written an article entitled: On existence on transmission eigenvalues for anisotropic Maxwell equations, that has been submitted to the Journal of Applicable Analysis. In this work we partially answer the difficult question related to the existence of transmission eigenvalues for Maxwell equations. This work is inspired by the ideas of Sylvester and Paivairinta for  the scalar case. Our approach is mainly based on the special splitting of the problem into coecive and compact form that has been introduced and employed in a previous work dealing with the solution of the interior transmission problem.
  
  
• During the three months training of Q. Chen within the DEFI team, we started investigating the extension of the linear sampling method to inverse scattering problems in the time domain. We established a mathematical framework that extends the fixed-frequency analysis to the time domain problem for near field data setting. The first numerical results are still not completed, but suggest that 1) a special care has to be made in the choice of the regularization parameter (a time dependent regularization seems to be needed) and 2) the inversion of the time domain matrix is time consuming and would greatly penalize the efficiency of the algorithm. This work will certainly be pursued in 2008 and 2009 in collaboration with P. Monk (the PhD advisor of Q. Chen) and A. Lechleiter (a Post-Doc at the DeFI team).

Rapport financier 2008

Bilan des échanges effectués en 2008

1. Chercheurs Seniors

(1) DR / CR / professeur
(2) colloque, thèse, stage, visite....
(3) précisez l'unité (mois, semaine..)

2. Juniors

(1) post-doc / doctorant / stagiaire
(2) colloque, thèse, stage, visite....
(3) précisez l'unité (mois, semaine..)

III. PREVISIONS 2009

Programme de travail

• Transmission eigenvalues and their use in inverse scattering problems: the available results on the existence of eigenvalues are not totally satisfactory since one is only able to prove their existence for large contrasts. The treatment of the general case and questions related to the number of these eigenvalues remain open. We shall pursue our investigations in an effort to answer these questions. It would also be of great interest to improve the lower bounds on the index of refraction of the scattering object in terms of these eigenvalues (numerical results have shown that available estimates are far from being optimal). Let us recall that this type of result is very important for target identification problems where no a priori knowledge is available on the  physical nature of the scatterers. This is the case for instance in most of the urban infrastructure imaging problems since the objects lying beneath the surface of an urban environment range from abandoned facilities, rock formations and unmarked burial sites to corroded chemical waste deposits. In particular,  a priori assumptions on the material or topological properties of such objects would be totally unrealistic. A related problem with practical interests that we plan to investigate, is the analysis of the interior transmission problem and the corresponding eigenvalues in the case when the contrast of the scattering object changes sign.
  We also continue with our goal to improve the numerical algorithm that has been designed to compute  transmission eigenvalues. More investigations are needed to explain the observed instability in the cases of limited aperture data. As a first step, one needs a rigorous justification of why the norm of the solution must blow up at resonant frequencies (at the moment only heuristic considerations are provided). Furthermore, the results for the scalar case need to be extended to the full electromagnetic case. We plan to propose a  training program for Master Degree students related to the numerical aspects of this problem.
  
  
• Consider the problem of the imaging of a perfectly conducting object coated by a thin anisotropic dielectric layer as well as the analogous problem for a dielectric object coated by a thin highly conducting anisotropic layer. Such problems arise for instance in corrosion detection and identification problems where one needs to distinguish between real targets from coated decoys. We shall study both the forward and inverse problems.
  For the forward problem, the first step consists in extending the asymptotic models developed within the DeFI team to the anisotropic case (which fits more to realistic cases). The second step will be the incorporation of these models into a forward solver which is either based on integral equation methods (which will be conducted in collaboration with Cerfacs) or on a volumic discretization approach such as  the ultra weak formulation developed by P. Monk.
  For the inverse problem, the methods previously developed at Delaware and INRIA on the identification of a constant surface impedance needs to be extended both to the case of an anisotropic surface impedance as well as  the case of generalized impedance boundary conditions where the impedance operator is a second order differential boundary operator. The latter case will be studied in the framework of a training program for Master Degree students. The boundary coeficient determination is based on the derivation of an integral equation for the surface impedance or surface conductivity. More analysis of the corresponding nonstandard integral operator is needed in order to have a better understanding of the numerical implementation of the method.
  
  
• Extend the gap reciprocity method for detecting buried objects in a piecewise homogeneous background to the case of a fully nonhomogeneous background (which is the case in most practical applications). We will begin by investigating theoretical questions linked with the justification of the method in this case, in particular the construction of a dense  set of test functions. On the numerical side, the main problem  will be to efficiently compute the Green tensor. Some special asymptotic cases, such as highly oscillating media (which may constitute a very good approximation for relatively large wavelengths), offer the possibility to use an effective Green tensor (or homogenized tensor). The incorporation of these ideas into the numerical method  offers a valuable alternative to the full numerical computation of the Green tensor. These considerations will be addressed over a period of several years.
  
  
• Investigate the possibility of establishing qualitative methods for studying imaging problems in the time domain (at the present time all qualitative methods are essentially restricted to the frequency domain). This is an ambitious  and open ended project  and only a common framework like the Equipe associe would enable a significant progress in this direction.

Programme d'échanges avec budget prévisionnel

1. Echanges

• F. Cakoni and H. Haddar are organizing a minisymposium in the Conference on Applied Inverse Problems  (one of the major conferences dedicated to inverse problems) that will be held in Vienna, July 20-24, 2009. Some of the money will be used to cover the expenses of the invited speakers.
• The Department of Mathematical Sciences of the University of Delaware is organizing a 3 weeks summer program for graduate students on the mathematics of inverse problems during June 15 - July 3, 2009.  D. Colton is one of the key speakers for one week, whereas F. Cakoni and P. Monk are in the organizing committee and are giving supplementary lectures. Part of the EA money would be used to cover the participation of the DeFI members in this summer school as speaker or participant.
  
• We expect short time visit for senior researchers on a basis similar to 2008.
• A. Lechleiter, who is a post-doc at the DeFI team plans to visit the UDEL team for one month. Q. Chen, PhD student of P. Monk, plan also to make a long visit to the DeFI team to continue working on time domain linear sampling method.
  
• We hope to be successuful in hiring at least two Master Degree students for training at INRIA on subjects related to the program proposed for 2009 with the possibility of continuing into PhD studies that will be co-directed by H. Haddar and F. Cakoni.

2. Cofinancement

- The members of the  Delaware team have a grant from the U.S. Air Force Office of Scientific Research. Part of this funding will be used to pay for one trip per year for a person at INRIA to the University of Delaware.

- The Department of Mathematical Sciences will provide supplementary money for each senior person of the team (Cakoni. Colton, Monk) toward travel expences to visit INRIA.

3. Demande budgétaire

Indiquez, dans le tableau ci-dessous, le coût global estimé de la proposition et le budget demandé à la DRI dans le cadre de cette Equipe Associée.
(maximum 20 K€ pour une prolongation en 2e année et 10 K€ pour une 3e année).

Remarques ou observations :