Recent Advances in the Theory of Regularization Methods for Ill-Posed Problems and Applications

RUBEN D. SPIES

Arlington

Encuentro; Computational Mathematics Program Review Meeting; 2012

AFOSR

First, sufficient conditions on the penalizers in generalized Tikhonov-Phillips functionals guaranteeing existence, uniqueness and stability of the minimizers are found. The particular cases in  which the penalizers are given by the bounded variation norm, by powers of seminorms and by linear combinations of powers of seminorms associated to closed operators, are studied. For the case of penalizers given by linear combinations of seminorms induced by closed operators, convergence of the regularized solutions is proved when the vector regularizationrule approaches the origin through appropriate radial and differentiable paths. Characterizations of the limiting solutions are given. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, regularization methods onthese scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases convergence is proved and orders and optimal orders of convergence are shown. The definition of qualification for spectral regularization methods is extended and three different levels are introduced: weak, strong and optimal.  It is shown that the weak qualification extends the definition introduced by P. Mathé and S. Pereverzev. It is shown that certain methods possessing infinite classical qualification, e.g. TSVD, Landweber's and Showalter's methods, also have generalized qualification leading to an optimal order ofconvergence of the regularization error. Sufficient conditions for a spectral regularization method (SRM) to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all threequalification levels are provided and the relationships between them as well as with the classical concept of qualification  and the qualification introduced by P. Mathe and S. Pereverzev areshown. In particular, SRMs having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of thistheory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.  The concept of saturation of an arbitrary regularization method is formalized. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the totalerror must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error.Finally the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order, for methodswith maximal qualification and for methods with optimal qualification. Several examples and applications to image processing are shown.