Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

Abstract

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 in 1994 (A. Neubauer, "On converse and saturation results for regularization methods", In BeitrÄage zur angewandten Analysis und Informatik, pp. 262-270. Shaker, Aachen, 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 suffcient conditions which ensure the existence of global saturation for spectral methods with classical qualifcation of finite positive order and for methods with maximal qualifcation. Finally, several examples of regularization methods possessing global saturation are shown.