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Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration

  • The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.

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Author:Gilles BlanchardGND, Peter Mathé
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Potsdam (1 (2012) 7)
Document Type:Preprint
Year of Completion:2012
Publishing Institution:Universität Potsdam
Release Date:2012/01/06
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC Classification:62-XX STATISTICS / 62Gxx Nonparametric inference / 62G05 Estimation
62-XX STATISTICS / 62Lxx Sequential methods / 62L15 Optimal stopping [See also 60G40, 91A60]
65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J20 Improperly posed problems; regularization
Collections:Universität Potsdam / Schriftenreihen / Preprints des Instituts für Mathematik der Universität Potsdam, ISSN 2193-6943 / 2012
Licence (German):License LogoKeine Nutzungslizenz vergeben - es gilt das deutsche Urheberrecht
Notes extern:RVK-Klassifikation: SI 990 , SK 830