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Open AccessArticle

The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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Mathematics 2020, 8(3), 331; https://doi.org/10.3390/math8030331
Received: 7 February 2020 / Revised: 20 February 2020 / Accepted: 22 February 2020 / Published: 3 March 2020
(This article belongs to the Special Issue Numerical Analysis: Inverse Problems – Theory and Applications)
This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93). View Full-Text
Keywords: Tikhonov regularization; oversmoothing penalty; discrepancy principle; nonlinear ill-posed problems; Hilbert scales; convergence; rates Tikhonov regularization; oversmoothing penalty; discrepancy principle; nonlinear ill-posed problems; Hilbert scales; convergence; rates
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Hofmann, B.; Hofmann, C. The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties. Mathematics 2020, 8, 331.

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