Generalized Qualification and Qualification Levels for Spectral Regularization Methods

T. HERDMAN, R. D. SPIES AND K. G. TEMPERINI

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS

Springer

Año: 2007

0022-3239

Abstract. The concept of quali¯cation for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error ([2], [5], [6], [11]). In this article, the de¯nition of quali¯cation is extended and three di®erent levels are introduced: weak, strong and optimal. It is shown that the weak quali¯cation extends the de¯nition introduced by Math¶e and Pereverzev ([6]), 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 in¯nite classical quali¯cation, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized quali¯cation leading to an optimal order of convergence of the regularization error. Su±cient conditions for a SRM to have weak quali¯cation are provided and necessary and su±cient conditions for a given order of convergence to be strong or optimal quali¯cation are found. Examples of all three quali¯cation levels are provided and the relationships between them as well as with the classical concept of quali¯cation and the quali¯cation introduced in [6] are shown. In particular, SRMs having extended quali¯cation in each one of the three levels and having zero or in¯nite classical quali¯cation 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.The concept of quali¯cation for spectral regularization methods (SRM) for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error ([2], [5], [6], [11]). In this article, the de¯nition of quali¯cation is extended and three di®erent levels are introduced: weak, strong and optimal. It is shown that the weak quali¯cation extends the de¯nition introduced by Math¶e and Pereverzev ([6]), 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 in¯nite classical quali¯cation, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized quali¯cation leading to an optimal order of convergence of the regularization error. Su±cient conditions for a SRM to have weak quali¯cation are provided and necessary and su±cient conditions for a given order of convergence to be strong or optimal quali¯cation are found. Examples of all three quali¯cation levels are provided and the relationships between them as well as with the classical concept of quali¯cation and the quali¯cation introduced in [6] are shown. In particular, SRMs having extended quali¯cation in each one of the three levels and having zero or in¯nite classical quali¯cation 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.