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Automatic Optimal Regularization for Ill‐Posed Problems with Stochastical Noise without Imposing Smoothness Assumptions
Author(s) -
Bauer Frank
Publication year - 2005
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200510297
Subject(s) - regularization (linguistics) , smoothness , separable space , operator (biology) , mathematics , noise (video) , gaussian noise , hilbert space , gaussian , algorithm , mathematical optimization , computer science , mathematical analysis , artificial intelligence , physics , image (mathematics) , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene
We consider the compact operator A : → for the separable Hilbert spaces and . The problem Ax = y is called ill‐posed when the singular values s k , k = 1, 2, … of the operator A tend to zero. Classically one assumes that y is biased with “deterministic noise”; we will also consider “stochastic noise” where the noise element is a weak Gaussian random variable. There classical stopping rules (e.g. Morozov) do not work. We will show that both for the “deterministic noise” case as well for the “stochastical noise” case we can regularize in an (asymptotically almost) optimal way without knowledge of the smoothness of the solution using Lepskij's method. Furthermore the method also works for estimated error levels and error behavior. So we can assure regularization which is just dependent on measurements obtainable in reality, e.g. satellite problems. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)