THE 2014 NSF-CBMS CONFERENCE ON
INVERSE SCATTERING AND TRANSMISSION EIGENVALUES
May 27-May 31, 2014

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The two basic references for Prof. Colton's lectures are the following:
D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, 3rd edition, Springer Series in Applied Mathematical Sciences, Vol. 93, Springer, New York, 2013.
F. Cakoni and D. Colton, A qualitative approach to inverse scattering theory, Springer Series in Applied Mathematical Sciences, Volume 188, Springer, New York, 2014.

The prerequisite for these CMBS lectures is a familiarity with the elementary theory of functional analysis and Sobolev spaces, for example, Chapter 1 of the book by Cakoni and Colton (26 pages) listed above. It is recommended that the participants read that chapter (or its equivalent) before attending the conference.

The lectures of Professor Fioralba Cakoni will show how transmission eigenvalues can be determined from the far field data and used to determine the material properties of the scattering object:
Lecture 1 by Prof. Cakoni
Lecture 2 by Prof. Cakoni
Notes for Lecture 2 by Prof. Cakoni
Lecture 3 by Prof. Cakoni

The lecture by Prof. Vassilis Papanicolaou will be on the inverse special transmission eigenvalue problem:
Lecture by Prof. Papanicolaou

The contents of the ten lectures to be delivered by Prof. Colton will be as follows:

Lecture 1: The Helmholtz Equation
This lecture is designed to provide the basic ingredients needed to understand the subsequent lectures. In particular, we introduce spherical harmonics, spherical Bessel functions, the Sommerfeld radiation condition, Green's formula, the far field pattern and Rellich's lemma.

Lecture 2: Scattering by an Inhomogeneous Medium
This lecture is designed to provide a basic introduction to the direct scattering problem for time harmonic acoustic waves. In particular, it is assumed that a time harmonic incident plane wave is scattered by a non-absorbing isotropic inhomogeneous medium of compact support and a mathematical formulation is given for this problem. We will then show that there exists a unique solution to this problem through the use of the unique continuation principle and the method of integral equations.

Lecture 3: The Far Field Operator
The far field operator will play a central role in all that follows. In this lecture we define the far field operator and show that it is normal. Herglotz wave functions are defined and the transmission eigenvalue problem is introduced. It is shown that if the wave number is not a transmission eigenvalue then the far field operator is injective with dense range.

Lecture 4: Inverse Scattering
This lecture will introduce the inverse scattering problem for time harmonic incident waves and show that the index of refraction is uniquely determined by the far field pattern of the scattered waves corresponding to a fixed frequency and plane waves incident from all directions. We will then explain that the inverse scattering problem is both nonlinear and ill-posed and discuss the numerical problems that arise due to these two facts.

Lecture 5: Ill-Posed Problems
In order to numerical solve the inverse scattering problem a knowledge of linear ill-posed problems is essential. This lecture is designed to be a basic introduction to linear ill-posed problems including the concept of a regularization method, Morozov's discrepancy principle, Picard's theorem and Tikhonov regularization.

Lecture 6: The Factorization Method
In many applications it is only desired to reconstruct the support of the inhomogeneous medium from inexact measurements of the far field pattern. In this case a linear method can be derived which makes no "weak scattering" assumptions (i.e. respects the nonlinear nature of the inverse scattering problem) and obtains the support of the inhomogeneous medium by solving a linear operator equation of the first kind. There are two variations of this approach to the inverse scattering problem called the linear sampling method and the factorization method respectively. In this lecture we will discuss the factorization method.

Lecture 7: The Linear Sampling Method
This is a continuation of Lecture 6 where we focus on the linear sampling method.

Lecture 8: Anisotropic Media-The Direct Problem
We are now concerned with the scattering problem for anisotropic media which presents problems that are not present in the anisotropic case. To this end, we introduce the Dirichlet-to-Neumann map and then reduce the scattering problem to a variational problem over a bounded domain. This variational problem is then solved by using the Lax-Milgram Lemma. We conclude by showing that the far field operator in this case is again normal.

Lecture 9: Anisotropic Media-The Inverse Problem
We introduce the transmission eigenvalue problem for anisotropic media and show that the far field operator is injective with dense range provided that the wave number is not a transmission eigenvalue. After stating a result that the support of an anisotropic medium is uniquely determined from the far field pattern, we derive the factorization method for determining the support of the scatterer from noisy far field data.

Lecture 10: An Inverse Spectral Problem
In this lecture we will consider a simple inverse spectral problem for transmission eigenvalues in which the inhomogeneous media is spherically stratified and only the eigenvalues associated with spherically symmetric eigenfunctions are considered. In this case it will be shown that complex eigenvalues exist, and, under appropriate assumptions, that the (real and complex) transmission eigenvalues uniquely determine the index of refraction.




Tuncay Aktosun aktosun@uta.edu
Last modified: May 31, 2014