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On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs

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Barbara Kaltenbacher, Andrej Klassen

Numerical Analysis · Submitted 30 Jan 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions.

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arXiv ID1801.10181 (version 1)
Pith IDHZ2X5QLT, Pith's permanent record of this paper and its review
CitationAndrej Klassen, Barbara Kaltenbacher. "On convergence and convergence rates for Ivanov and Morozov regularization and application to some parameter identification problems in elliptic PDEs." arXiv:1801.10181 (2018). https://pith.science/paper/HZ2X5QLT
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arXiv listing abstract v1. Results and the main claim are underlined; .

In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy principle for choosing the regularization parameter, and Morozov regularization (also called method of the residuals). After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions. Finally, we apply these results to some linear and nonlinear parameter identification problems in elliptic boundary value problems.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy principle for choosing the regularization parameter, and Morozov regularization (also called method of the residuals). contribution 0.98 0.80 0.10 asserted 0.99
s1 After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions. contribution 0.63 0.95 0.60 asserted 1.00
s2 Finally, we apply these results to some linear and nonlinear parameter identification problems in elliptic boundary value problems. contribution 0.96 0.91 0.14 asserted 0.98

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Original abstract

In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy principle for choosing the regularization parameter, and Morozov regularization (also called method of the residuals). After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions. Finally, we apply these results to some linear and nonlinear parameter identification problems in elliptic boundary value problems.

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