Abstract
In this paper we consider a particular variant of finite-dimensional Tikhonov regularization for ill-posed operator equations. Convergence rates are established and an a-posteriori parameter choice method is derived that leads to optimal convergence rates with respect to data errors and with respect to the finite-dimensional subspace, without using any information about the exact solution. Finally, using linear splines we present several numerical examples that confirm the theoretical results.
Zusammenfassung
In dieser Arbeit betrachten wir eine besondere Variante endlich-dimensionaler Tikhonov-Regularisierung schlechtgestellter Operatorgleichungen. Konvergenzraten werden nachgewiesen und eine a-posteriori Parameterwahl wird hergeleitet, die optimale Konvergenzraten bezüglich des Datenfehlers und der endlich-dimensionalen Approximation liefert, ohne Information über die exakte Lösung zu benötigen. Schließlich präsentieren wir auch einige numerische Beispiele, bei denen lineare Splinefunktionen verwendet werden, die die theoretischen Ergebnisse bestätigen.
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This author's research was partially supported by the Austrian Fonds zu Förderung der wissenschaftlichen Forschung (project S 32/03). This author is on leave from Universität Linz, Austria; the travel support from the Fulbright Commission is gratefully acknowledged.
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King, J.T., Neubauer, A. A variant of finite-dimensional Tikhonov regularization with a-posteriori parameter choice. Computing 40, 91–109 (1988). https://doi.org/10.1007/BF02247939
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DOI: https://doi.org/10.1007/BF02247939