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.

Keywords: 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://www.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://www.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://www.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, Volume 13 (2007) no. 1, pp. 163-177. doi : 10.1051/cocv:2007006. http://www.numdam.org/articles/10.1051/cocv:2007006/

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

and ,[2] Work in progress.

and ,[3] Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks. SIAM J. Math. Anal. 30 (1999) 699-720. | Zbl

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

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

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

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

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

,*Cited by Sources: *