Conformal mapping and inverse conductivity problem with one measurement

Marc Dambrine; Djalil Kateb

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 1, page 163-177
  • ISSN: 1292-8119

Abstract

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This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

How to cite

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Dambrine, Marc, and Kateb, Djalil. "Conformal mapping and inverse conductivity problem with one measurement." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 163-177. <http://eudml.org/doc/249929>.

@article{Dambrine2007,
abstract = { This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method. },
author = {Dambrine, Marc, Kateb, Djalil},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Inverse conductivity problem; conformal mapping.; inverse conductivity problem; conformal mapping},
language = {eng},
month = {2},
number = {1},
pages = {163-177},
publisher = {EDP Sciences},
title = {Conformal mapping and inverse conductivity problem with one measurement},
url = {http://eudml.org/doc/249929},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Dambrine, Marc
AU - Kateb, Djalil
TI - Conformal mapping and inverse conductivity problem with one measurement
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/2//
PB - EDP Sciences
VL - 13
IS - 1
SP - 163
EP - 177
AB - This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.
LA - eng
KW - Inverse conductivity problem; conformal mapping.; inverse conductivity problem; conformal mapping
UR - http://eudml.org/doc/249929
ER -

References

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  1. I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems18 (2002) 1659–1672.  Zbl1069.35090
  2. M. Dambrine and D. Kateb, Work in progress.  
  3. E. Fabes, H. Kang and J.K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks. SIAM J. Math. Anal.30 (1999) 699–720.  Zbl0930.35194
  4. G.M. Golutsin, Geometrische Funktionentheorie. Deutscher Verlag der Wissenschaften, Berlin (1957).  
  5. H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems. Inverse Problems21 (2005) 935–953.  Zbl1071.35123
  6. P. Henrici, Applied and computational complex analysis, Vol 1,3. John Wiley & Sons (1986).  Zbl0578.30001
  7. N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Noordhoff, Groniningen (1953).  Zbl0052.41402
  8. M. Taylor, Partial Differential Equations, Vol. 1: Basic Theory. Applied Math. Sciences115, Springer-Verlag, New York (1996).  

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