A variational approach for the solution of the electromagnetic interior transmission problem for anisotropic media
Fioralba Cakoni Houssem Haddar
Inverse Problems & Imaging 2007, 1(3): 443-456 doi: 10.3934/ipi.2007.1.443
The interior transmission problem plays a basic role in the study of inverse scattering problems for inhomogeneous medium. In this paper we study the interior transmission problem for the Maxwell equations in the electromagnetic scattering problem for an anisotropic inhomogeneous object. We use a variational approach which extends the method developed in [15] to the case when the index of refraction is less than one as well as for partially coated scatterers. In addition, we also describe the structure of the transmission eigenvalues.
keywords: interior transmission problem; anisotropic media.
Integral equations for inverse problems in corrosion detection from partial Cauchy data
Fioralba Cakoni Rainer Kress
Inverse Problems & Imaging 2007, 1(2): 229-245 doi: 10.3934/ipi.2007.1.229
We consider the inverse problem to recover a part $\Gamma_c$ of the boundary of a simply connected planar domain $D$ from a pair of Cauchy data of a harmonic function $u$ in $D$ on the remaining part $\partial D\setminus \Gamma_c$ when $u$ satisfies a homogeneous impedance boundary condition on $\Gamma_c$. Our approach extends a method that has been suggested by Kress and Rundell [17] for recovering the interior boundary curve of a doubly connected planar domain from a pair of Cauchy data on the exterior boundary curve and is based on a system of nonlinear integral equations. As a byproduct, these integral equations can also be used for the problem to extend incomplete Cauchy data and to solve the inverse problem to recover an impedance profile on a known boundary curve. We present the mathematical foundation of the method and illustrate its feasibility by numerical examples.
keywords: Inverse boundary value problem integral equations partial boundary measurements impedance boundary condition.
New results on transmission eigenvalues
Fioralba Cakoni Drossos Gintides
Inverse Problems & Imaging 2010, 4(1): 39-48 doi: 10.3934/ipi.2010.4.39
We consider the interior transmission eigenvalue problem corresponding to the inverse scattering problem for an isotropic inhomogeneous medium. We first prove that transmission eigenvalues exist for media with index of refraction greater or less than one without assuming that the contrast is sufficiently large. Then we show that for an arbitrary Lipshitz domain with constant index of refraction there exists an infinite discrete set of transmission eigenvalues that accumulate at infinity. Finally, for the general case of non constant index of refraction we provide a lower and an upper bound for the first transmission eigenvalue in terms of the first transmission eigenvalue for appropriate balls with constant index of refraction.
keywords: inverse scattering problem. Interior transmission problem transmission eigenvalues inhomogeneous medium
Transmission eigenvalues for inhomogeneous media containing obstacles
Fioralba Cakoni Anne Cossonnière Houssem Haddar
Inverse Problems & Imaging 2012, 6(3): 373-398 doi: 10.3934/ipi.2012.6.373
We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.
keywords: Interior transmission problem inhomogeneous medium transmission eigenvalues inverse scattering problem.
Fioralba Cakoni Houssem Haddar Michele Piana
Inverse Problems & Imaging 2009, 3(2): i-i doi: 10.3934/ipi.2009.3.2i
This special issue is dedicated to Professors David Colton and Rainer Kress in honor of their contribution and leadership in the area of direct and inverse scattering theory for more then 30 years. The papers in this special issue were solicited from the invited speakers at the International Conference on Inverse Scattering Problems organized in honor of the 65th birthdays of David Colton and Rainer Kress held in the seaside resort of Sestry Levante, Italy, May 8-10, 2008.
    As organizers of this conference and close collaborators of Professors Colton and Kress, we are very honored to have had the opportunity to facilitate this special scientific and social event. It was a particular occasion that gathered together long term colleagues, collaborators, former students and friends of Professors Colton and Kress. And now it gives us particular pleasure to be guest editors of this special issue of Inverse Problems and Imaging which is a collection of original research papers in the area of scattering theory and inverse problems. Much of the work presented here has been directly or indirectly influenced by these two scientists, offering the reader a glimpse of their significant impact in this research area.
   We would like to thank all of those who have contributed a paper for this special issue. A special thanks goes to the Editor in Chief of Inverse Problems and Imaging, Lassi Päivärinta, for supporting and facilitating this publication. We would also like to thank all the participants of the Sestri Levante Conference who made such a successful, stimulating and pleasant event possible. Last (but definitely not least!) we would like to thank the sponsors of the conference: the European Office of Aerospace Research and Development of the United States Air Force Office of Scientific Research, the University of Genova, the University of Verona, the Istituto Nazionale di Alta Matematica - Gruppo Nazionale di Calcolo Scientifico, the University of Göttingen, the University of Delaware and INRIA Center of Saclay Ile de France.
Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast
Fioralba Cakoni Shari Moskow Scott Rome
Inverse Problems & Imaging 2018, 12(4): 971-992 doi: 10.3934/ipi.2018041

In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [8], which requires that the contrast of the inhomogeneity is of one-sign only near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. In addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.

keywords: Interior transmission problem transmission eigenvalues periodic inhomogeneous medium inverse scattering problem homogenization

Year of publication

Related Authors

Related Keywords

[Back to Top]