Elliptic Inverse Problems

Abstract

Inverse problems arise in various areas of science and engineering including medical imaging, computer vision, geophysics, solid mechanics, astronomy, and so forth. A wide range of these problems involve elliptic operators. We call them elliptic inverse problems. In this thesis we discuss three elliptic inverse problems. The first one is photo-acoustic tomography problem. Photo-acoustic tomography is a hybrid medical imaging modality, it combines a high-resolution modality and a high-contrast modality. The underlying mathematical problem is a typical coupled physics inverse problem which involves two types of waves. We introduce the physical mechanism of photo-acoustic tomography, and focus on the recovery of diffusion coefficient and absorption coefficient from internal measurements in the partial data case. The second one is electro-seismic conversion problem. Electro-seismic conversion is a phenomenon where electromagnetic waves and seismic waves are coupled. It has been successfully applied in the modern oil prospection. The mathematical model is another coupled physics inverse problem. We concentrate on the reconstruction of the coupling coefficient and the conductivity in the presence of internal measurements. The third one contains a few identification results for the first order perturbation of the bi-harmonic operator in an infinite slab. We demonstrate the unique determination of a magnetic potential and an electrical potential from knowing the partial Cauchy data set.