Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

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

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS

SPRINGER/PLENUM PUBLISHERS

Lugar: New York; Año: 2009

0022-3239

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by Neubauer [6]. Necessary and su±cient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error 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, two converse results are proved and the theory is applied to ¯nd su±cient conditions which ensure the existence of global saturation for spectral methods with classical quali¯cation of ¯nite positive order and for methods with maximal quali¯cation. Finally, several examples of regularization methods possessing global saturation are shown.