Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T02:09:22.102Z Has data issue: false hasContentIssue false
  • English
  • Français

Conformal mapping and inverse conductivity problem withone measurement

Published online by Cambridge University Press:  14 February 2007

Marc Dambrine  and
Djalil Kateb
Marc Dambrine
Affiliation:
Laboratoire de Mathématiques Appliquées de Compiègne. Université de Technologie de Compiègne. Centre de Recherche de Royalieu 60200 Compiègne, France; [email protected]
Djalil Kateb
Affiliation:
Laboratoire de Mathématiques Appliquées de Compiègne. Université de Technologie de Compiègne. Centre de Recherche de Royalieu 60200 Compiègne, France; [email protected]
Get access

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.

Keywords

Inverse conductivity problemconformal mapping.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 1 , January 2007 , pp. 163 - 177
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems 18 (2002) 1659–1672.
M. Dambrine and D. Kateb, Work in progress.
Fabes, E., Kang, H. and Seo, J.K., Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks. SIAM J. Math. Anal. 30 (1999) 699–720. CrossRef
G.M. Golutsin, Geometrische Funktionentheorie. Deutscher Verlag der Wissenschaften, Berlin (1957).
Haddar, H. and Kress, R., Conformal mappings and inverse boundary value problems. Inverse Problems 21 (2005) 935–953. CrossRef
P. Henrici, Applied and computational complex analysis, Vol 1,3. John Wiley & Sons (1986).
N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Noordhoff, Groniningen (1953).
M. Taylor, Partial Differential Equations, Vol. 1: Basic Theory. Applied Math. Sciences 115, Springer-Verlag, New York (1996).