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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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On the choice of the regularization parameter in ill-posed problems with approximately given noise level of data

U. Hämarik / T. Raus

Institute of Applied Mathematics, University of Tartu, Estonia. E-mails: ,

Citation Information: Journal of Inverse and Ill-posed Problems jiip. Volume 14, Issue 3, Pages 251–266, ISSN (Online) 1569-3953, ISSN (Print) 0928-0219, DOI: 10.1515/156939406777340928,

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We consider regularization of linear ill-posed problems Au = ƒ in Hilbert spaces. Approximations u r to the solution u * can be constructed by the Tikhonov method or by the Lavrentiev method, by iterative or by other methods. We assume that instead of ƒ ∈ R(A) noisy data are available with the approximately given noise level δ: in process δ → 0 it holds || − ƒ||/δc with unknown constant c. We propose a new a-posteriori rule for the choice of the regularization parameter r = r(δ) guaranteeing u r(δ)u * for δ → 0. Note that such convergence is not guaranteed for the parameter choice given by the L-curve rule, by the GCV-rule, by the quasioptimality criterion and also for discrepancy principle ||Au r|| = with b < c. The error estimates are given, which in case || − ƒ|| ≤ δ are quasioptimal and order-optimal.

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Attila Gábor and Julio R. Banga
BMC Systems Biology, 2015, Volume 9, Number 1
[2]
Uno Hämarik, Urve Kangro, Reimo Palm, Toomas Raus, and Ulrich Tautenhahn
Inverse Problems in Science and Engineering, 2014, Volume 22, Number 1, Page 10
[3]
Uno Hämarik, Reimo Palm, and Toomas Raus
Journal of Computational and Applied Mathematics, 2012, Volume 236, Number 8, Page 2146
[4]
Hilbeth P. Azikri de Deus, Claudio R. Ávila S. Jr., Ivan Moura Belo, and André T. Beck
Applied Mathematical Modelling, 2012, Volume 36, Number 10, Page 4687
[5]
Frank Bauer and Mark A. Lukas
Mathematics and Computers in Simulation, 2011, Volume 81, Number 9, Page 1795
[6]
U. Hämarik and R. Palm
Mathematical Modelling and Analysis, 2007, Volume 12, Number 1, Page 61
[7]
T. Raus and U. Hämarik
Mathematical Modelling and Analysis, 2009, Volume 14, Number 2, Page 187
[8]
T Raus and U Hämarik
Journal of Physics: Conference Series, 2008, Volume 135, Page 012087
[9]
B. Tomas Johansson and Roman Chapko
Inverse Problems and Imaging, 2008, Volume 2, Number 3, Page 317
[10]
[11]
U. Hämarik, R. Palm, and T. Raus
Journal of Inverse and Ill-posed Problems, 2007, Volume 15, Number 3, Page 277
[12]
Fermín S. Viloche Bazán and Leonardo S. Borges
BIT Numerical Mathematics, 2010, Volume 50, Number 3, Page 481
[13]
Fermín S Viloche Bazán and Juliano B Francisco
Inverse Problems, 2009, Volume 25, Number 4, Page 045007
[14]
Fermín S Viloche Bazán
Inverse Problems, 2008, Volume 24, Number 3, Page 035001

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