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Title of Journal: J Optim Theory Appl

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Abbravation: Journal of Optimization Theory and Applications

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Springer US

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DOI

10.1007/s00381-012-1963-x

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1573-2878

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Generalized Qualification and Qualification Levels

Authors: T Herdman R D Spies K G Temperini
Publish Date: 2009/01/21
Volume: 141, Issue: 3, Pages: 547-567
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Abstract

The concept of qualification for spectral regularization methods SRM for inverse illposed problems is strongly associated to the optimal order of convergence of the regularization error Engl et al in Regularization of inverse problems Mathematics and its applications vol 375 Kluwer Academic Dordrecht 1996 Mathé in SIAM J Numer Anal 423968–973 2004 Mathé and Pereverzev in Inverse Probl 193789–803 2003 Vainikko in USSR Comput Math Math Phys 223 1–19 1982 In this article the definition of qualification is extended and three different levels are introduced weak strong and optimal It is shown that the weak qualification extends the definition introduced by Mathé and Pereverzev Inverse Probl 193789–803 2003 mainly in the sense that the functions associated with orders of convergence and source sets need not be the same It is shown that certain methods possessing infinite classical qualification eg truncated singular value decomposition TSVD Landweber’s method and Showalter’s method also have generalized qualification leading to an optimal order of convergence of the regularization error Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced in Mathé and Pereverzev Inverse Probl 193789–803 2003 are shown In particular SRMs having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented Finally several implications of this theory in the context of orders of convergence converse results and maximal source sets for inverse illposed problems are shownThis work was supported by DARPA/SPO NASA LaRC and the National Institute of Aerospace under Grant VT031 2535 by AFOSR Grants F496200310243 and FA95500710273 by Consejo Nacional de Investigaciones Científicas y Técnicas CONICET and by Universidad Nacional del Litoral UNL Argentina through Project CAI+D 2006 PE 236

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