Premium
Newton methods for nonlinear inverse problems with random noise
Author(s) -
Bauer Frank,
Hohage Thorsten,
Munk Axel
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700144
Subject(s) - smoothness , minimax , logarithm , nonlinear system , mathematics , random noise , noise (video) , convergence (economics) , inverse problem , operator (biology) , hilbert space , inverse , mathematical optimization , mathematical analysis , computer science , algorithm , physics , biochemistry , chemistry , geometry , quantum mechanics , repressor , artificial intelligence , transcription factor , economics , image (mathematics) , gene , economic growth
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)