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Journal of Optimization Theory and Applications

, Volume 148, Issue 1, pp 164–196 | Cite as

Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

  • Terry Herdman
  • Ruben D. Spies
  • Karina G. Temperini
Article

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 (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.

Keywords

Ill-posed Inverse problem Qualification Saturation 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Terry Herdman
    • 1
  • Ruben D. Spies
    • 2
    • 3
  • Karina G. Temperini
    • 2
    • 4
  1. 1.Interdisciplinary Center for Applied Mathematics, ICAMVirginia TechBlacksburgUSA
  2. 2.Instituto de Matemática Aplicada del LitoralIMAL, CONICET-UNLSanta FeArgentina
  3. 3.Departamento de Matemática, Facultad de Ingeniería QuímicaUniversidad Nacional del LitoralSanta FeArgentina
  4. 4.Departamento de Matemática, Facultad de Humanidades y CienciasUniversidad Nacional del LitoralSanta FeArgentina

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