### Generalized Qualification and Qualification Levels for Spectral Regularization Methods

#### Abstract

The concept of qualifcation for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of

qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualifcation extends the definition introduced

by Mathé and Pereverzev in 2003 (Mathé, P. and Pereverzev, S. V.: "Geometry of linear ill-posed problems in variable Hilbert scales", Inverse Problems, 19(3), 789-803 (2003)), mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also

have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a 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 three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathé and Pereverzev are shown. 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 this theory in the

context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.

qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualifcation extends the definition introduced

by Mathé and Pereverzev in 2003 (Mathé, P. and Pereverzev, S. V.: "Geometry of linear ill-posed problems in variable Hilbert scales", Inverse Problems, 19(3), 789-803 (2003)), mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also

have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a 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 three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathé and Pereverzev are shown. 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 this theory in the

context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.