Modular Regularization Algorithms 
Michael Jacobsen

Abstract  The class of linear illposed problems is introduced along with a range of standard numerical tools and basic concepts from linear algebra, statistics and optimization. Known algorithms for solving linear inverse illposed problems are analyzed to determine how they can be decomposed into independent modules. These modules are then combined to form new regularization algorithms with other properties than those we started out with. Several variations are tested using the Matlab toolbox MOORe Tools created in connection with this thesis.
Object oriented programming techniques are explained and used to set up the illposed problems in the toolbox. Hereby, we are able to write regularization algorithms that automatically exploit structure in the illposed problem without being rewritten explicitly. We explain how to implement a stopping criteria for a parameter choice method based upon an iterative method. The parameter choice method is also used to demonstrate the implementation of the standardform transformation. We have implemented a simple preconditioner aimed at the preconditioning of the generalform Tikhonov problem and demonstrate its simplicity and effciency.
The steps taken with MOORe Tools to produce several of the figures are demonstrated in the toolbox tutorial. We have included the article "Subspace Preconditioned LSQR for IllPosed Problems" that discusses an algorithm that is not easily implemented with MOORe Tools. 
Type  Ph.D. thesis [Academic thesis] 
Year  2004 
Publisher  Informatics and Mathematical Modelling, Technical University of Denmark, DTU 
Address  Richard Petersens Plads, Building 321, DK2800 Kgs. Lyngby 
Series  IMMPHD2004140 
Note  Supervised by Prof. Per Christian Hansen 
Electronic version(s)  [pdf] [ps] 
BibTeX data  [bibtex] 
IMM Group(s)  Scientific Computing 