PhD Defence: Fast Gradient-Based Iterative Regularization Methods for Nonlinear Ill-Posed Problems – Theory and Applications

Fast Gradient-Based Iterative Regularization Methods for Nonlinear Ill-Posed Problems – Theory and Applications

In this thesis, we investigate a class of fast gradient-based iterative regularization methods for nonlinear Inverse and Ill-Posed Problems based on Landweber iteration and Nesterov's acceleration scheme. These so-called Two Point Gradient (TPG) methods have been found to be very useful in practical applications, since they are both easy to implement and lead to a great speedup compared to standard gradient-based methods. While methods utilizing second-order information, which are known for their fast convergence, often become infeasible when dealing with large datasets, gradient-based methods are usually more flexible and able to deal with large datasets, at the disadvantage of requiring a large number of iterations. TPG methods have the potential to bridge the apparent gap between these two classes of methods, being both fast, flexible, and able to deal with large datasets, which are important requirements for any iterative regularization method used for solving inverse problems.

This thesis provides a convergence analysis of TPG methods under the common assumption of a tangential cone condition, and covers some well-known choices of commonly used stepsizes. Furthermore, a convergence analysis with the tangential cone condition being replaced by a local convexity assumption more natural to Nesterov's original acceleration idea is performed. These results provide the first successful convergence analysis for TPG methods for the solution of nonlinear ill-posed problems.

Apart from these theoretical results, this thesis presents a number of numerical examples showing the usefulness of TPG methods in practical applications. In a number of academic examples, the assumptions required for convergence are considered in detail, and precise comparisons between several TPG methods and standard gradient-based methods are performed. Afterwards, TPG methods are applied to two problems arising in Medical Imaging. The first of these two problems is the imaging technique of Single Photon Emission Computed Tomography (SPECT), where TPG methods are shown to lead to a large speedup in the reconstruction process. The second problem concerns Magnetic Resonance Advection Imaging (MRAI), which is a novel imaging technique for mapping the pulse wave velocity in brain vessel from Magnetic Resonance Imaging (MRI) measurements. For this problem, for which a precise modelling is performed, TPG methods are essential due to the large datasets involved.