Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

TL;DR

This paper formalizes the concept of saturation in regularization methods for inverse ill-posed problems, establishing conditions for global saturation and analyzing spectral methods with various qualifications.

Contribution

It provides necessary and sufficient conditions for global saturation in regularization methods and applies the theory to spectral methods with different qualifications.

Findings

Conditions for global saturation are established.

Spectral methods with classical and maximal qualification are analyzed.

Examples of regularization methods with global saturation are provided.

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 A. Neubauer in 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.