Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error (Engl et al. in Regularization of inverse problems. Mathematics and its applications, vol. 375, Kluwer Academic, Dordrecht, 1996; Mathé in SIAM J. Numer. Anal. 42(3):968–973, 2004; Mathé and Pereverzev in Inverse Probl. 19(3):789–803, 2003; Vainikko in USSR Comput. Math. Math. Phys. 22(3): 1–19, 1982). 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 qualification extends the definition introduced by Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003), mainly in the sense that the functions associated with 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 in Mathé and Pereverzev (Inverse Probl. 19(3):789–803, 2003) 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.