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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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1569-3945
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Volume 21, Issue 2

Issues

The Levenberg–Marquardt iteration for numerical inversion of the power density operator

Guillaume Bal / Wolf Naetar / Otmar Scherzer
  • Computational Science Center, University of Vienna, Nordbergstr. 15, A-1090 Vienna, Austria; and Radon Institute of Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria
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/ John Schotland
Published Online: 2013-04-03 | DOI: https://doi.org/10.1515/jip-2012-0091

Abstract.

In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg–Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity σ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg–Marquardt algorithm and tested it on simulated data.

Keywords: Inverse problems; nonlinear ill-posed problems; iterative regularization; elliptic equations; hybrid imaging

About the article

Received: 2012-11-26

Published Online: 2013-04-03

Published in Print: 2013-04-01


Citation Information: Journal of Inverse and Ill-Posed Problems, Volume 21, Issue 2, Pages 265–280, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jip-2012-0091.

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Citing Articles

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[1]
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[2]
Simon Hubmer, Ekaterina Sherina, Andreas Neubauer, and Otmar Scherzer
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[3]
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Inverse Problems, 2018, Volume 34, Number 7, Page 075005
[4]
Xizi Song, Yanbin Xu, Feng Dong, and Russell S. Witte
IEEE Sensors Journal, 2017, Volume 17, Number 24, Page 8206
[5]
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SIAM Journal on Mathematical Analysis, 2017, Volume 49, Number 6, Page 4704
[6]
Xizi Song, Yanbin Xu, and Feng Dong
Measurement Science and Technology, 2017, Volume 28, Number 4, Page 045404
[7]
Xizi Song, Yanbin Xu, and Feng Dong
Measurement Science and Technology, 2016, Volume 27, Number 11, Page 114003
[8]
Guillaume Bal, Francis J. Chung, and John C. Schotland
SIAM Journal on Mathematical Analysis, 2016, Volume 48, Number 2, Page 1332
[9]
Bastian Harrach, Eunjung Lee, and Marcel Ullrich
Inverse Problems, 2015, Volume 31, Number 9, Page 095003
[10]
Kristoffer Hoffmann and Kim Knudsen
Sensing and Imaging, 2014, Volume 15, Number 1
[11]
Guillaume Bal and John C. Schotland
Physical Review E, 2014, Volume 89, Number 3
[12]
Guillaume Bal, Chenxi Guo, and Francçois Monard
Inverse Problems and Imaging, 2014, Volume 8, Number 1, Page 1

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