Newton methods for nonlinear inverse problems with random noise |
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| Authors: | Frank Bauer Thorsten Hohage Axel Munk |
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| Affiliation: | 1. Department of Knowledge-Based Mathematics, Johannes Kepler University of Linz, Softwarepark 21, 4232 Hagenberg, Austria;2. Institute of Numerical and Applied Mathematics, Univ. Göttingen, Lotzestr. 16-18, 37083 Göttingen, Germany;3. Institute of Mathematical Stochastics, Univ. Göttingen, Maschmühlenweg 8-10, 37073 Göttingen, Germany |
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| Abstract: | We study the convergence of regularized Newton methods applied to nonlinear operator equations in Hilbert spaces if the data are perturbed by random noise. We show that under certain conditions it is possible to achieve the minimax rates of the corresponding linearized problem if the smoothness of the solution is known. If the smoothness is unknown and the stopping index is determined by Lepskij's balancing principle, we show that the rates remain the same up to a logarithmic factor due to adaptation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) |
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