BIT Numerical Mathematics

, Volume 48, Issue 2, pp 245–264 | Cite as

Source supports in electrostatics

  • Martin Hanke
  • Nuutti Hyvönen
  • Manfred Lehn
  • Stefanie ReusswigEmail author


We investigate the inverse source problem of electrostatics in a bounded and convex domain with compactly supported source. We try to extract all information about the unknown source support from the given Cauchy data of the associated potential, adopting by this previous work of Kusiak and Sylvester to the case of electrostatics. We introduce, and for the unit disk we also compute numerically, what we call the discoidal source support, i.e., the smallest set made up by the intersection of disks within the domain, which carries a source compatible with the given data.

Key words

inverse source problem scattering support source support 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. V. Ahlfors, Complex Analysis, 3rd edn., McGraw-Hill, New York, 1979.zbMATHGoogle Scholar
  2. 2.
    L. Borcea, Electrical impedance tomography, Inverse Probl., 18 (2002), pp. R99–R136, and Inverse Probl., 19 (2003), pp. 997–998.Google Scholar
  3. 3.
    M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography, Inverse Probl., 16 (2000), pp. 1029–1042.zbMATHCrossRefGoogle Scholar
  4. 4.
    R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2, Springer, Berlin, 1988.Google Scholar
  5. 5.
    A. El Badia, Inverse source problem in an anisotropic medium by boundary measurements, Inverse Probl., 21 (2005), pp. 1487–1506.zbMATHCrossRefGoogle Scholar
  6. 6.
    H. Haddar, S. Kusiak, and J. Sylvester, The convex back-scattering support, SIAM J. Appl. Math., 66 (2005), pp. 591–615.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa, Magnetoencephalography – theory, instrumentation, and applications to noninvasive studies of the working human brain, Rev. Mod. Phys., 65 (1993), pp. 413–497.CrossRefGoogle Scholar
  8. 8.
    M. Hanke, N. Hyvönen, and S. Reusswig, Convex source support and its application to electric impedance tomography, submitted.Google Scholar
  9. 9.
    P. Henrici, Applied and Computational Complex Analysis, vol. 1, Wiley, New York, 1974.Google Scholar
  10. 10.
    R. Kress, Linear Integral Equations, 2nd edn., Springer, Berlin, 1999.zbMATHGoogle Scholar
  11. 11.
    S. Kusiak and J. Sylvester, The scattering support, Commun. Pure Appl. Math., 56 (2003), pp. 1525–1548.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math Anal., 36 (2005), pp. 1142–1158.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I, Springer, Berlin, 1972.Google Scholar
  14. 14.
    R. Potthast, J. Sylvester, and S. Kusiak, A ‘range test’ for determining scatterers with unknown physical properties, Inverse Probl., 19 (2003), pp. 533–547.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer, Berlin, 2002.zbMATHGoogle Scholar
  16. 16.
    J. Sylvester, Notions of support for far fields, Inverse Probl., 22 (2006), pp. 1273–1288.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    J. Sylvester and J. Kelly, A scattering support for broadband sparse far field measurements, Inverse Probl., 21 (2005), pp. 759–771.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • Martin Hanke
    • 1
  • Nuutti Hyvönen
    • 2
  • Manfred Lehn
    • 1
  • Stefanie Reusswig
    • 1
    Email author
  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Institute of MathematicsHelsinki University of TechnologyHUTFinland

Personalised recommendations