Nonlinear methods for inverse problems
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The general inverse problem is formulated as a nonlinear operator equation. The solution of this via the Newton-Kantorovich method is outlined. Fréchet differentiability of the operator is given by the implicit function theorem. We also consider questions such as uniqueness, stability and regularization of the inverse problem. This general theory is then applied to a number of different inverse problems. The Newton-Kantorovich method is derived for each example and Fréchet differentiability examined. In some cases numerical results are provided, for others our work provides a theoretical basis for results obtained by different authors. The problems considered include an interior measurement inverse problem from steady-state diffusion, and a boundary measurement problem for electrical conductivity imaging. We also examine the determination of refractive indices and scattering boundaries for the Helmholtz equation from measurements of the farfield. In addition an inverse problem from geometric optics is investigated.