Volume 53, pp. 313-328, 2020.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

Bernd Hofmann and Robert Plato


For Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Hölder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence for regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low-order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

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Key words

ill-posed problem, inverse problem, Tikhonov regularization, oversmoothing penalty, a priori parameter choice strategy, discrepancy principle, logarithmic source condition

AMS subject classifications

65J20, 65J15, 65J22, 47J06, 47J05

ETNA articles which cite this article

Vol. 57 (2022), pp. 101-126 Philip Miller and Thorsten Hohage: Convergence rates for oversmoothing Banach space regularization

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