Visibility and Invisibility in Inverse Problems

Abstract

In this thesis, we describe two projects dealing with two different aspects of inverse problems. The first project concerns \emph{visibility}, i.e., the problem of reconstructing the internal properties of a medium from external measurements. Specifically, we study the inverse problem for the conductivity equation \[ \divg (\gamma \nabla u) = 0 \] in a bounded Lipschitz domain $\Omega \subset \R^n$. The goal is to reconstruct the positive scalar function $\gamma$ from its Dirichlet-to-Neumann map $\Lambda_\gamma$, which maps the Dirichlet data of solutions to the conductivity equation to their Neumann data. For dimensions $n \geq 3$, injectivity of the map $\gamma \mapsto \Lambda_\gamma$ was proved for $\gamma \in C^2$ by Sylvester and Uhlmann in 1987. Later in 1988, Nachman provided a constructive procedure for computing $\gamma$ from $\Lambda_\gamma$. In the same year, Alessandrini proved log-type stability estimates for $\|\gamma_1-\gamma_2\|_{L^\infty}$ in terms of the operator norm of $\Lambda_{\gamma_1}-\Lambda_{\gamma_2}$. Since then, the problems of injectivity, stability and reconstruction for less regular conductivities has seen considerable interest. Here, we show the validity of Nachman's reconstruction procedure for $\gamma \in W^{3/2,2n}(\Omega)$ such that $\gamma \equiv 1$ near $\pa \Omega$, and derive log-type stability estimates under the slightly stronger assumption that $\gamma_1,\gamma_2 \in W^{2-s, n/s}(\Omega)$ for some $0<s<1/2$. In the second part of the thesis, we consider the problem of making an object \emph{invisible} with respect to electromagnetic measurements made on its boundary. The idea is to enclose a region $D \subset \R^3$ by an ``invisibility cloak" with carefully designed electromagnetic parameters so that electromagnetic measurements on the boundary of the cloak are indistinguishable from measurements that would be obtained in empty space, regardless of the contents of $D$. For the conductivity equation, such a cloak was designed by Greenleaf, Lassas and Uhlmann in 2003, based on the transformation properties of $\gamma$ under change of coordinates. The cloaking parameters obtained in this way are anisotropic and degenerate at the boundary between the cloak and the object. This presents a serious difficulty for theoretical analysis as well as practical implementation, and was later addressed by Greenleaf, Kurylev, Lassas and Uhlmann in 2008. The problem of degeneracy was addressed by using regular approximations to the coordinate transformation involved in the construction of the degenerate cloak, so that one obtains an \textit{approximate cloak} of arbitrary accuracy. Using inverse homogenization techniques, these regularized cloaks were further approximated by cloaks whose parameters are both non-degenerate and isotropic. In this thesis, we construct a similar non-degenerate and isotropic approximate cloak for the time-harmonic Maxwell's equations. This was a joint work with Dr. Tuhin Ghosh.