The focus of the project is on stabilization of large-scale inverse problems where structured models and iterative algorithms are necessary for computing approximate solutions. For this purpose, we study various iterative Krylov methods and their abilities to produce regularized solutions. Some of the Krylov methods have previously been studied and identified as iterative regularization methods, whereas others have been proposed in the literature, but only sparsely studied in practise. This thesis considerably improves the understanding of these methods.
Image deblurring problems constitute a nice class of large-scale problems for which the various methods can be tested. Therefore, this present work includes a separate study of the matrix structures that appear in this connection – not least to create a common basis for discussions.
Another important part of the thesis is regularization matrices for the formulation of inverse problems on general form. Special classes of regularization matrices for large-scale problems (among these also two-dimensional problems) have been analyzed. Moreover, the above mentioned Krylov methods have also been analyzed in connection with the solution of problems on general form, and a new extension to the methods has been developed for this purpose.
The L-curve method is one among several parameter choice methods that can be used in connection with the solution of inverse problems. A part of the work has resulted in a new heuristic for the localization of the corner of a discrete L-curve. This heuristic is implemented as a part of a larger algorithm which is developed in collaboration with G. Rodriguez and P. C. Hansen.
Last, but not least, a large part of the project has, in different ways, revolved around the object-oriented Matlab toolbox MOORe Tools developed by PhD Michael Jacobsen. New implementations have been added, and several bugs and shortcomings have been fixed.
The work has resulted in three papers that are all included in an appendix for convenience.