Asymptotic methods in inverse scattering

This work consists of two major components. First, we characterize the effect of a small inhomogeneity on the transmission eigenvalues of a media by deriving an asymptotic expansion. We consider the case of a scalar isotropic media and approach the problem through a variational framework in order to apply tools from the theory of compact operators. Then, we present approximations to the scattered field of the full time-harmonic Maxwell equations with a thin high contrast dielectric scatterer present. Using this approximation, we develop an inversion method to recover the location of the scatterer in a two dimensional plane. We end by presenting numerical simulations demonstrating both the asymptotic result and the efficiency of the inversion method.Ph.D., Mathematics -- Drexel University, 201