Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

TERRY HERDMAN; RUBEN D. SPIES; KARINA G. TEMPERINI

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS

SPRINGER/PLENUM PUBLISHERS

Lugar: New York; Año: 2011 vol. 148 p. 164 - 196

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 (Beiträge zur angewandten Analysis und Informatik, pp. 262–270, 1994). 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 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 find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.