Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Abhishake Rastogi

doi:10.3934/cpaa.2020183

Article Contents

2020, Volume 19, Issue 8: 4111-4126. Doi: 10.3934/cpaa.2020183

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Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems

Abhishake Rastogi

Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Straße 24-25, 14476 Potsdam, Germany

Received: July 31, 2019

Revised: December 31, 2019

Published: May 2020

This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG)-SFB1294/1-318763901

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Abstract

In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.

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Mathematics Subject Classification: Primary: 62G20; Secondary: 62G08, 65J15, 65J20, 65J22.

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