Fioralba Cakoni

The Linear Sampling Method in Inverse Electromagnetic Scattering

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in and . Included in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions.

The existence of an infinite discrete set of transmission eigenvalues

We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwells equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e., subregions with contrast zero.

The linear sampling method for cracks

We consider the inverse scattering problem of determining the shape of an infinite cylinder having an open arc as cross section from a knowledge of the TM-polarized scattered electromagnetic field corresponding to time-harmonic incident plane waves propagating from arbitrary directions. We assume that the arc is a (possibly) partially coated perfect conductor and develop the linear sampling method, which was originally developed for solving the inverse scattering problem for obstacles with nonempty interior, to include the above case of obstacles with empty interior.

The linear sampling method for anisotropic media

We consider the inverse scattering problem of determining the support of an anisotropic inhomogeneous medium from a knowledge of the incident and scattered time harmonic acoustic wave at fixed frequency. To this end, we extend the linear sampling method from the isotropic case to the case of anisotropic medium. In the case when the coefficients are real we also show that the set of transmission eigenvalues forms a discrete set.

The Interior Transmission Eigenvalue Problem

We consider the inverse problem of determining the spherically symmetric index of refraction

The inverse electromagnetic scattering problem for anisotropic media

n(r)

On the existence of transmission eigenvalues in an inhomogeneous medium

from a knowledge of the corresponding transmission eigenvalues (which can be determined from field pattern of the scattered wave). We also show that for constant index of refraction

The Direct and Inverse Scattering Problems for Partially Coated Obstacles

n(r)=n

A qualitative approach to inverse scattering theory

, the smallest transmission eigenvalue suffices to determine n, complex eigenvalues exist for n sufficiently small and, for homogeneous media of general shape, determine a region in the complex plane where complex eigenvalues must lie.

THE DETERMINATION OF THE SURFACE IMPEDANCE OF A PARTIALLY COATED OBSTACLE FROM FAR FIELD DATA

The inverse electromagnetic scattering problem for anisotropic media plays a special role in inverse scattering theory due to the fact that the (matrix) index of refraction is not uniquely determined from the far field pattern of the scattered field even if multi-frequency data are available. In this paper, we describe how transmission eigenvalues can be determined from the far field pattern and be used to obtain upper and lower bounds on the norm of the index of refraction. Numerical examples will be given for the case when the scattering object is an infinite cylinder and the inhomogeneous medium is orthotropic.