Section: New Results

Qualitative methods for inverse scattering problems

Identifying defects in an unknown background using differential measurements

Participants : Lorenzo Audibert, Houssem Haddar.

With Alexandre Girard, we developed a new qualitative imaging method capable of selecting defects in complex and unknown background from differential measurements of farfield operators: i.e. far measurements of scattered waves in the cases with and without defects. Indeed, the main difficulty is that the background physical properties are unknown. Our approach is based on a new exact characterization of a scatterer domain in terms of the far field operator range and the link with solutions to so-called interior transmission problems. We present the theoretical foundations of the method and some validating numerical experiments in a two dimensional setting [10] . This work is based on the generalized formulation of the Linear Sampling Method with exact characterization of targets in terms of farfield measurements that has been introduced in [1] .

The Factorization Method for a Cavity in an Inhomogeneous Medium

Participants : Houssem Haddar, Shixu Meng.

With F. Cakoni we considered the inverse scattering problem for a cavity that is bounded by a penetrable anisotropic inhomogeneous medium of compact support where on is interested in determining the shape of the cavity from internal measurements on a curve or surface inside the cavity. We derived a factorization method which provides a rigorous characterization of the support of the cavity in terms of the range of an operator which is computable from the measured data. The support of the cavity is determined without a-priori knowledge of the constitutive parameters of the surrounding anisotropic medium provided they satisfy appropriate physical as well as mathematical assumptions imposed by our analysis. Numerical examples were given showing the viability of our method [7] .

Asymptotic analysis of the transmission eigenvalue problem for a Dirichlet obstacle coated by a thin layer of non-absorbing media

Participant : Houssem Haddar.

With F. Cakoni and N. Chaulet we considered the transmission eigenvalue problem for an impenetrable obstacle with Dirichlet boundary condition surrounded by a thin layer of non-absorbing inhomogeneous material. We derived a rigorous asymptotic expansion for the first transmission eigenvalue with respect to the thickness of the thin layer. Our convergence analysis is based on a Max–Min principle and an iterative approach which involves estimates on the corresponding eigenfunctions. We provided explicit expressions for the terms in the asymptotic expansion up to order 3 [3] .

Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell's Equations

Participants : Houssem Haddar, Shixu Meng.

In this work, we considered the transmission eigenvalue problem for Maxwell's equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. Following the approach developed by Cossonnière-Haddar in the scalar case, we formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equation and prove that assuming that the contrast is constant near the boundary of the support of the inhomogeneity, the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point. This is a joint work with F. Cakoni.

Invisibility in scattering theory

Participant : Lucas Chesnel.

We investigated a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We considered cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we showed that the non-scattering wavenumbers form a (possibly empty) discrete set. This result is important in the justification of certain reconstruction techniques like the Linear Sampling Method in practical applications.

In a second step, for a given real wavenumber and a given domain D, we developed a constructive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which allows to provide examples of (approximated) invisible inclusions. This is a joint work with A.-S. Bonnet-Ben Dhia and S.A. Nazarov [11] .

Invisibility in electrical impedance tomography

Participant : Lucas Chesnel.

We adapted the technique to construct invisible isotropic conductivities in for the point electrode model in electrical impedance tomography. Again, the theoretical approach, based on solving a fixed point problem, is constructive and allows the implementation of an algorithm for approximating the invisible perturbations. We demonstrated the functionality of the method via numerical examples. This a joint work with N. Hyvönen and S. Staboulis [13] .

A quasi-backscattering problem for inverse acoustic scattering in the Born regime

Participants : Houssem Haddar, Jacob Rezac.

In this work we propose a data collection geometry in which to frame the inverse scattering problem of locating unknown obstacles from far-field measurements of time-harmonic scattering data. The measurement geometry, which we call the quasi-backscattering set-up, is configurated such that one device acts as a transmitter and a line of receivers extends in one-dimension a small distance from the transmitter. We demonstrate that the data collected can be used to locate inhomogeneities whose physical properties are such that the Born approximation applies. In particular, we are able to image a two-dimensional projection of the location of an obstacle by checking if a test function which corresponds to a point in 2 belongs to the range of a measurable operator. The reconstruction algorithm is based on the MUSIC (Multiple SIgnal Classification) algorithm.