Inverse boundary value problems for time-harmonic waves: Conditional stability and iterative reconstruction - Qiu, Lingyun (ProQuest Dissertations & Theses, 2013)
Many inverse problems arising in different disciplines including exploration geophysics, medical imaging and nondestructive evaluation can be formulated as a nonlinear operator equation, F(x) = y, where F models the corresponding forward problem. Usually, the inverse problem is an ill-posed problem in the sense that a small perturbation in the data can lead to a significant impact in the reconstruction. In the first part of this dissertation, we focus on the analysis of iterative methods in Banach spaces. We assume certain conditional Hölder or Lipschitz type stability of the inverse problem and prove a linear or sublinear convergence rate for the Landweber iteration and a projected steepest descent iteration. This is a novel view point for the convergence analysis of the iterative methods. The second part of this dissertation is concerned with the conditional Lipschitz stability estimate for the inverse boundary value problem for time-harmonic waves. Assuming that the wavespeed (density) is piece-wise constant with discontinuities on a finite number of known interfaces, we provide a Lipschitz stability estimate for the inverse problems of acoustic (elastic) waves. In the third part, we study the inverse boundary value problem for the acoustic time-harmonic waves. It is to determine the property of the medium inside a domain from the measurements of the displacement and normal stress on its boundary. The governing equation is the Helmholtz equation. A hierarchy algorithm is proposed and analysed for the iterative reconstruction with multi-frequency data. The algorithm is based on a projected steepest descent iteration with stability constraints.