Associate Team ISIP

Associate Team ISIP

Problèmes de diffraction inverse et d'identification
(Inverse Scattering and Identification Problems)

INRIA project-team : DeFI	Partner : Department of Mathematical Sciences University of Delaware
Centre de recherche INRIA : INRIA Saclay Ile de France Thème INRIA : NumD	Pays : France

	French Coordinator	Partner Coordinator étranger
Nom, prénom	HADDAR Houssem	CAKONI Fioralba
Grade/statut	DR2 (Habilitation)	Associate Professor
Organisme d'appartenance	Equipe DeFI (INRIA Saclay Ile de France/Ecole Polytechnique)	Department of Mathematical Sciences, University of Delaware
Adresse postale	CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex France	Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-2553 USA
URL	http://www-rocq.inria.fr/~haddar/	http://www.math.udel.edu/~cakoni/
Téléphone	+33 1 69 33 46 41	+1 302 831 0592
Télécopie	+33 1 69 33 30 11	+1 302 831 4511
Courriel	Housem.Haddar@inria.fr	cakoni@math.udel.edu

Team Members

David Colton (UDel)
Armin Lechleiter (DeFI)
Peter Monk (Udel)

Brief Description of the Research theme

The associated team will concentrate on the use of qualitative methods in electromagnetic inverse scattering theory with applications to the imaging of urban infrastructure, the nondestructive evaluation of coated materials and medical imaging. Most of the effort will be focused in the solution of the inverse problems using time harmonic waves, in particular for frequencies in the resonance regime.
The aim of research in this field is to not only detect but also to identify unknown objects in real time. Mathematically, such problems lead to nonlinear and severely ill-posed equations. Until a few years ago, essentially all existing algorithms for target identification were based on either a weak scattering approximation or on the use of nonlinear optimization techniques. In recent years alternative methods for imaging, known as qualitative methods, have been developed which avoid incorrect model assumptions inherent in weak scattering approximations and, as opposed to nonlinear optimization techniques, do not require a priori information. In addition, these methods are non iterative and are based on finding an indicator function which is usually a solution of a linear ill-posed integral equation. This leads to an easily implementable and fast imaging technique. The best known qualitative method is the linear sampling method and it's close relative the reciprocitygap functional method.
We will use the linear sampling method and the reciprocity gap functional method to investigate a number of complex imagining problems in the areas listed above in which there is practically no a priori information on the geometry and physical properties of the scatterer and the aim is to reconstruct the shape and/or estimate the constitutive physical parameters of the object.

Scientific Report

Proposal initial document

Transmission Eigenvalues in Inverse Scattering and their Application to Nondestructive Testing

In the past 3 years the associate team members have made significant progress in the investigation of the interior transmission problem and its related eigenvalues, known as transmission eigenvalues. Transmission eigenvalues are a new class of eigenvalues arising in inverse scattering theory for an inhomogeneous medium in the frequency regime. They provide information about the index of refraction of the inhomogeneous medium. Since transmission eigenvalues can be computed from measured multistatic data, a fact that was first proven in [II], they can be used in electromagnetic or acoustic interrogation of composite materials to detect flaws in the material such as anisotropy structural changes or the presence of cavities [IV]. A typical application of this situation is e.g. testing of airplane canopies using electromagnetic waves, but applicability of such techniques can potentially be extended to many other problems in nondestuctive testing and medical imaging. Mathematically, the study of transmission eigenvalues is challenging due to fact that it is a non-selfadjoint, and nonlinear eigenvalue problem. As a result of the associate team collaborative efforts, the existence of infinitely many real transmission eigenvalues for non-absorbing media, a long standing open problem, has been proven [III], [V], the mathematical framework for the study of interior transmission problem for media with cavities has been developed [III], and bounds for the index of refraction of dielectric media in terms of the first transmission eigenvalue has been provided [I], [V]. These joint publications are listed below with a short description of their contents. They also inspired other works conducted by the associate team members ([1], [2], [4] and [6]) but which are not joint publications.

I.: The Computation of Lower Bounds for the Norm of the Index of Refraction, J. Integral Equations and Applications, 21 203-227 (2009).
Authors: F. Cakoni, H. Haddar, D. Colton.

Abstract: 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.

II.: On the Determination of Dirichlet or Transmission Eigenvalues from Far Field Data, Comptes Rendus Mathematique, 348, No 7-8, 379-383 (2010).
Authors: F. Cakoni, H. Haddar, D. Colton.

Abstract: We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber

belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for $-\Delta$ . When the scatterer is a non absorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues.

III.: On the existence of transmission eigenvalues in an inhomogenous medium, Applicable Analysis, 88 no 4, 475-493 (2009).
Authors: F. Cakoni, H. Haddar

Abstract: 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 Païvaïrinta for the scalar case. Our approach is mainly based on a 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.

IV.: The Interior Transmission Problem For regions with Cavities, SIAM Journal of Mathematical Analysis , 42, No 1, 145-162 (2010).
Authors: F. Cakoni, H. Haddar, D. Colton.

Abstract: We consider the interior transmission problem in the case when the inhomogeneous medium has cavities, i.e. regions in which the index of refraction is the same as the host medium. In this case we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. We also derive Faber-Krahn type inequalities for the transmission eigenvalues.

V.: The existence of an infinite discrete set of transmission eigenvalues, SIAM Journal of Mathematical Analysis, 42, No 1, 237-255 (2010).
Authors: F. Cakoni, D. Gintides, H. Haddar.

Abstract: We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic as well as anisotropic inhomogeneous media for the Helmholtz and Maxwell's equations. Our discussion also includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e. subregions with contrast zero.

Time Domain and Multi-frequency Linear Sampling and Factorization Methods

A draw back of sampling methods is that they need a large aperture of multistatic data in order to provide satisfactory results. In many applications of the linear sampling method that we have developed during associate team collaboration in underground imaging and target identification, it is possible to collect multistatic data but only over a small aperture. We have conducted numerous numerical examples and all show that the resolution of the image becomes worse with the decrease of the aperture. Our idea is to use time-domain or multifrequency data to compensate for the lack of spatial measurements. We first investigated a time domain linear sampling as described in [VI]. These investigations have been complemented with time domain factorization method for far field measurements [5], as well as multifrequency linear sampling methods [3], all for the scalar case. These initial results are promising and provide better limited aperture reconstructions. The interest for time domain sampling method is also motivated by imaging of inclusions or non destructive testing of coatings with non linear behaviour (ferromagnetic, piezo-electric, etc.). A first investigation of sampling methods for weakly non linear materials in the frequency regime has been initiated in [7].

VI.: A sampling method for inverse scattering in the time domain, Inverse Problems, 26, 8, paper 085001 (2010).
Authors: Q. Chen, H. Haddar, A. Lechleiter and P. Monk,

Abstract: We consider a near field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features: First, the data for the method consists of measurement of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground penetrating radar imaging. The second feature allows to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it induces the potential to improve the quality of the reconstruction compared to frequency domain methods working with one single-frequency. We demonstrate this potential by several numerical examples.

Scientific community animation

F. Cakoni, H. Haddar, and M. Piana organized the "International Conference on Inverse Scattering Problems", honoring David Colton and Rainer Kress, in Sestri Levante (Italy), 8th-10th May, 2008. The conference was sponsored by the European Office of Aerospace Research and Development, Air Force Office of Scientific Research, United States Air Force Research Laboratory, the University of Genova, the University of Goettingen, the University of Delaware and INRIA. For more details, visit the web page of the ICISP conference http://www.cmap.polytechnique.fr/%7Edefi/ckconf/
Related to the conference "International Conference on Inverse Scattering Problems", F. Cakoni, H. Haddar and M. Piana edited a special issue of the journal Inverse Problems and Imaging. The special issue (Volume 3, Number 2) appeared in 2009. The content of this issue is available at
http://aimsciences.org/journals/displayPapers1.jsp?pubID=304
Active participation to the international conference on Applied Inverse Problems (AIP'09) (one of the major conferences dedicated to inverse problems) that was held in Vienna, 19-24 July 2009 (http://www.ricam.oeaw.ac.at/conferences/aip2009/)
- D. Colton gave a plenary conference as an invited speaker on "Faber-Krahn Type Inequalities in Inverse Scattering Theory".
- F. Cakoni and H. Haddar organized a minisymposium on "The determination of boundary coefficients in inverse boundary value problems and scattering theory"..
Active participation in the 9th International Conference on Mathematical and Numerical Aspects of Waves Propagation https://waves-2009.bordeaux.inria.fr/ (one of the major conferences dedicated to wave propagation), that was held in Pau, 15th-19th June 2009. H. Haddar gave a plenary conference as an invited speaker on "Transmission Eigenfrequencies for Dielectrics and Their Use in The Identification Problem" and F. Cakoni contributed by a joint work with D. Colton and H. Haddar on "Transmission Eigenvalues and Nondestructive Testing".
Active participation in 2009 IMA PI Summer Program for Graduate Students: The Mathematics of Inverse Problems, organized at the university of Delaware on June 15-July 3, 2009 (http://www.ima.umn.edu/2008-2009/PISG6.15-7.3.09/) F. Cakoni, D. Colton and P. Monk were among the organizers and key lecturers of this event. Two Phd students, from the DEFI group attented this summer school with partial support provided from the University of Delaware.
F. Cakoni, H. Haddar, and A. Lechleiter organized the MMNS Workshop on Inverse Problems for Waves: Methods and Applications on March 29-30 at Ecole Polytechnique, Palaiseau, France.
http://www.cmap.polytechnique.fr/~defi/mmsn2010/welcome.html. This event was sponsored by the MMNS chair http://www.cmap.polytechnique.fr/mmnschair/
Active participation in the IV European Conference on Computational Mechanics that was held in Paris, May 16-21, 2010. F. Cakoni gave an invited keynote talk in the minisymposium on Inverse Problems titled "Transmission Eigenvalues and their Application to Inverse Scattering Problems" and A. Lechleiter gave an invited talk in the minisymposium on Inverse Problems titled "A linear sampling method for inverse scattering in the time domain" a joint contribution with H. Haddar; for more details see (http://www.eccm2010.org/).

Additional References

The following list are other publications of the associate team members that are closely related to the associate team program but that are not joint publications of the associate team partners.

F. Cakoni, D. Colton, P. Monk and J. Sun, The inverse electromagnetic scattering problem for anisotropic media, Inverse Problems, 26, No 7, paper 074004 (2010).
F. Cakoni and A. Kirsch, On the interior transmission eigenvalue problem, Int. Jour of Comp. Sci. Math, 3, No 1-2, 142-167 (2010).
B. Guzina, F. Cakoni and C. Bellis, On multi-frequency obstacle reconstruction via linear sampling method, (to appear).
F. Cakoni, D. Colton and D. Gintides The inverse transmission eigenvalue problem, (submitted).
H. Haddar, A. Lechleiter A Far Field Inverse Scattering Problem in the Time Domain, (submitted).
H. Haddar, A. Cossonnière The Electromagnetic Interior Transmission Problem for Regions with Cavities, (submitted).
A. Lechleiter, Explicit characterization of the support of non-linear inclusions, (submitted).