| Popis: |
In this thesis, we cover the so-called heuristic (aka error-free or data-driven) parameter choice rules for the regularisation of ill-posed problems (which just so happen to be prominent in the treatment of inverse problems). We consider the linear theory associated with both continuous regularisation methods, such as that of Tikhonov, and also iterative procedures, such as Landweber's method. We provide background material associated with each of the aforementioned regularisation methods as well as the standard results found in the literature. In particular, the convergence theory for heuristic rules is typically based on a noise-restricted analysis. We also introduce some more recent developments in the linear theory for certain instances: in case of operator perturbations or weakly bounded noise for linear Tikhonov regularisation. In both the aforementioned cases, novel parameter choice rules were derived; for the case of weakly bounded noise, through necessity and in the case of operator perturbations, an entirely new class of parameter choice rules are discussed (so-called semi-heuristic rules which could be said to be the ``middle ground" between heuristic rules and a-posteriori rules). We then delve further into the abyss of the relatively unknown; namely the nonlinear theory (by which we mean that the regularisation is nonlinear) for which the development and analysis of heuristic rules are still in their infancy. Most notably in this thesis, we present a recent study of the convergence theory for heuristic Tikhonov regularisation with convex penalty terms which attempts to generalise, to some extent, the restricted noise analysis of the linear theory. As the error in this setting is measured in terms of the Bregman distance, this naturally lends itself to the introduction of some novel parameter choice rules. Finally, we illustrate and supplement most of the preceding by including a numerics section which displays the effectiveness of heuristic parameter choice rules and conclude with a discussion of the results as well as a speculation of the potential future scope of research in this exciting area of applied mathematics. submitted by Kemal Raik, MA MSc. Universität Linz, Dissertation, 2020 (VLID)5343227 |
