1 Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark3 Department of Applied Mathematics and Computer Science, Technical University of Denmark
The class of linear ill-posed 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 ill-posed 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 ill-posed 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 standard-form transformation. We have implemented a simple preconditioner aimed at the preconditioning of the general-form 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 Ill-Posed Problems" that discusses an algorithm that is not easily implemented with MOORe Tools.