Recovering Coefficients of Second-Order Hyperbolic and Plate Equations via Finite Measurements on the Boundary

Abstract In this dissertation, we consider the inverse problem for a second-order hyperbolic equation of recovering n + 3 unknown coefficients defined on an open bounded domain with a smooth enough boundary. We also consider the inverse problem of recovering an unknown coefficient on the Euler- Bernoulli plate equation on a lower-order term again defined on an open bounded domain with a smooth enough boundary. For the second-order hyperbolic equation, we show that we can uniquely and (Lipschitz) stably recover all these coefficients from only using half of the corresponding boundary measurements of their solutions, and for the plate equation, we show that we can uniquely and stably recover the coefficient by using two measurements on the boundary. The proofs for solving both inverse problems are based on a post-Carleman estimate strategy developed by Isakov in [19], continuous observability inequalities, and regularity theory

Image Segmentation Applied to Electrical Impedance Tomography

Electrical impedance tomography (EIT) has many significant applications and has gained popularity due to its ease of use and its non-invasiveness. While it has notable applications, EIT is severely ill-posed. Since EIT is ill-posed, various techniques have been considered to solve the inverse problem. While the inverse problem has been well-studied, a common question is to ask how we can improve in solving the inverse problem of the electrical impedance tomography? In this thesis, we consider using image segmentation alongside with iteratively-regularized Gauss-Newton Method (IRGN) to solve the inverse problem of EIT in various geometries. A comparison between the reconstruction of the image with IRGN and with IRGN alongside image segmentation is presented. Alongside the comparison, we analyze the parameter and residual error between IRGN and image segmentation to show the efficiency of image segmentation in solving the inverse problem of EIT. In the end, we discuss future work that could be done to extend the results of this thesis