Conformal mapping and inverse conductivity problem with one measurement
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 163-177.

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.

DOI : 10.1051/cocv:2007006
Classification : 49N45, 49Q10, 30C30
Mots-clés : inverse conductivity problem, conformal mapping 
@article{COCV_2007__13_1_163_0,
     author = {Dambrine, Marc and Kateb, Djalil},
     title = {Conformal mapping and inverse conductivity problem with one measurement},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {163--177},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {1},
     year = {2007},
     doi = {10.1051/cocv:2007006},
     mrnumber = {2282107},
     zbl = {1136.49026},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2007006/}
}
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
PY  - 2007
SP  - 163
EP  - 177
VL  - 13
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2007006/
DO  - 10.1051/cocv:2007006
LA  - en
ID  - COCV_2007__13_1_163_0
ER  - 
%0 Journal Article
%A Dambrine, Marc
%A Kateb, Djalil
%T Conformal mapping and inverse conductivity problem with one measurement
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2007
%P 163-177
%V 13
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2007006/
%R 10.1051/cocv:2007006
%G en
%F COCV_2007__13_1_163_0
Dambrine, Marc; Kateb, Djalil. Conformal mapping and inverse conductivity problem with one measurement. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 163-177. doi : 10.1051/cocv:2007006. http://archive.numdam.org/articles/10.1051/cocv:2007006/

[1] I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems 18 (2002) 1659-1672. | Zbl

[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. | Zbl

[4] G.M. Golutsin, Geometrische Funktionentheorie. Deutscher Verlag der Wissenschaften, Berlin (1957). | MR | Zbl

[5] H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems. Inverse Problems 21 (2005) 935-953. | Zbl

[6] P. Henrici, Applied and computational complex analysis, Vol 1,3. John Wiley & Sons (1986). | MR | Zbl

[7] N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Noordhoff, Groniningen (1953). | MR | Zbl

[8] M. Taylor, Partial Differential Equations, Vol. 1: Basic Theory. Applied Math. Sciences 115, Springer-Verlag, New York (1996). | MR | Zbl

Cité par Sources :