Convergence analysis of (statistical) inverse problems under conditional stability estimates

Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems in Hilbert scales satisfying conditional stability estimates...

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Bibliographic Details
Published inInverse problems Vol. 36; no. 1; pp. 15004 - 15026
Main Authors Werner, Frank, Hofmann, Bernd
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.01.2020
Subjects
conditional stability
convergence rates
Hilbert scales
Lepskij principle
statistical inverse problems
Tikhonov regularization
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Summary:Conditional stability estimates require additional regularization for obtaining stable approximate solutions if the validity area of such estimates is not completely known. In this context, we consider ill-posed nonlinear inverse problems in Hilbert scales satisfying conditional stability estimates characterized by general concave index functions. For that case, we exploit Tikhonov regularization and provide convergence and convergence rates of regularized solutions for both deterministic and stochastic noise. We further discuss a priori and a posteriori parameter choice rules and illustrate the validity of our assumptions in different model and real world situations.
Bibliography:IP-102256.R2
ISSN:0266-5611
1361-6420
DOI:10.1088/1361-6420/ab4cd7