Inexact Newton regularizations with uniformly convex stability terms: a unified convergence analysis

We present a unified convergence analysis of inexact Newton regularizations for nonlinear ill-posed problems in Banach spaces. These schemes consist of an outer (Newton) iteration and an inner iteration which provides the update of the current outer iterate. To this end the nonlinear problem is linearized about the current iterate and the resulting linear system is approximately (inexactly) solved by an inner regularization method. In our analysis we only rely on generic assumptions of the inner methods and we show that a variety of regularization techniques satisfies these assumptions. For instance, gradient-type and iterated-Tikhonov methods are covered. Not only the technique of proof is novel, but also the results obtained, because for the first time uniformly convex penalty terms stabilize the inner scheme

Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data

We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the well-known iteratively regularized Gauss-Newton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the Kullback-Leibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the far-field pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples

Sparsity Regularization in Diffuse Optical Tomography

The purpose of this dissertation is to improve image reconstruction in Diffuse Optical Tomography (DOT), a high contrast imaging modality that uses a near infrared light source. Because the scattering and absorption of a tumor varies significantly from healthy tissue, a reconstructed spatial representation of these parameters serves as tomographic image of a medium. However, the high scatter and absorption of the optical source also causes the inverse problem to be severely ill posed, and currently only low resolution reconstructions are possible, particularly when using an unmodulated direct current (DC) source. In this work, the well posedness of the forward problem and possible function space choices are evaluated, and the ill posed nature of the inverse problem is investigated along with the uniqueness issues stemming from using a DC source. Then, to combat the ill posed nature of the problem, a physically motivated additional assumption is made that the target reconstructions have sparse solutions away from simple backgrounds. Because of this, and success with a similar implementation in Electrical Impedance Tomography, a sparsity regularization framework is applied to the DOT inverse problem. The well posedness of this set up is rigorously proved through new regularization theory results and the application of a Hilbert space framework similar to recent work. With the sparsity framework justified in the DOT setting, the inverse problem is solved through a novel smoothed gradient and soft shrinkage algorithm. The effectiveness of the algorithm, and the sparsity regularization of DOT, is evaluated through several numerical simulations using a DC source with comparison to a Levenberg Marquardt implementation and published error results