Inverse problems for reconstructive imaging of inhomogeneities inside the body : mathematical theory, numerical algorithms, and biomedical applications

Abstract

This thesis considers various types of inverse problems such as inverse boundary value problems for the heat equation, a free boundary problem arising in plasma physics, the electrical impedance tomography (EIT), and the magnetic resonance electrical impedance tomography (MREIT). The theoretical foundations such as the uniqueness, the existence, and the stability are investigated in each case. We present a unified framework of the Newton-type iterative reconstruction algorithm based on the domain derivative.

First, we present a geometrical approach to the domain derivative, which provides the straightforward differentiation method of a boundary value problem. Using some differential geometric notions we mathematically justify this method. The obtained closed formula of the domain derivative is congruent to the known results on some kinds of partial differential equations by the standard weak formulation method, and moreover our formula is applicable to other types of partial differential equations without modifications.

Second, the inverse heat problems formulated by the aid of initial-boundary value problems with Dirichlet and Neumann boundary condition are considered. We show the uniqueness of these inverse problems and develop a regularized Newton-type reconstruction algorithm. In order to reduce the computational time we suggest a collocation method using boundary integral formulation. On the other hand, we show that the null space of the linearized map induced by the domain derivative consists of tangential vector fields, the counterpart of which on the inverse obstacle problem governed by Helmholtz equation is still unresolved.

Third, initial guess finding is important in the Newton-type reconstruction algorithm. This thesis presents a real time and stable algorithm for giving a good initial guess in some types of inverse problems. We present a rigorous proof and numerical simulations for the algorithm giving a good ...