BIT Numerical Mathematics

, Volume 52, Issue 1, pp 45–63 | Cite as

Convex source support in three dimensions

  • Martin Hanke
  • Lauri Harhanen
  • Nuutti Hyvönen
  • Eva Schweickert


This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. The original algorithm for reconstructing the convex source support is inherently two-dimensional as it utilizes Möbius transformations. However, replacing the Möbius transformations by inversions with respect to suitable spheres and introducing the corresponding Kelvin transforms, the basic ideas of the algorithm carry over to three spatial dimensions. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for electrical impedance tomography with a single pair of boundary current and potential as the measurement data.


Electrical impedance tomography Convex source support Obstacle problem Inverse elliptic boundary value problem 

Mathematics Subject Classification (2000)

65N21 35R30 


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  1. 1.
    Arfken, G.: Mathematical Methods for Physicists. Academic Press, New York (1970) Google Scholar
  2. 2.
    Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer, London (2001) MATHCrossRefGoogle Scholar
  3. 3.
    Borcea, L.: Electrical impedance tomography. Inverse Probl. 18, R99–R136 (2002) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bourgeois, L., Chambeyron, C., Kusiak, S.: Locating an obstacle in a 3D finite depth ocean using the convex scattering support. J. Comput. Appl. Math. 204, 387–399 (2007) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brühl, M., Hanke, M.: Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Probl. 16, 1029–1042 (2000) MATHCrossRefGoogle Scholar
  6. 6.
    Cheney, M., Isaacson, D., Newell, J.C.: Electrical impedance tomography,. SIAM Rev. 41, 85–101 (1999) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Springer, Berlin (1988) CrossRefGoogle Scholar
  8. 8.
    El Badia, A., Ha Duong, T.: Some remarks on the problem of source identification from boundary measurements,. Inverse Probl. 14, 883–891 (1998) MATHCrossRefGoogle Scholar
  9. 9.
    El Badia, A., Ha Duong, T.: An inverse source problem in potential analysis. Inverse Probl. 16, 651–663 (2000) MATHCrossRefGoogle Scholar
  10. 10.
    El Badia, A.: Inverse source problem in an anisotropic medium by boundary measurements. Inverse Probl. 21, 1487–1506 (2005) MATHCrossRefGoogle Scholar
  11. 11.
    Haddar, H., Kusiak, S., Sylvester, J.: The convex back-scattering support. SIAM J. Appl. Math. 66, 591–615 (2005) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hakula, H., Hyvönen, N.: Two noniterative algorithms for locating inclusions using one electrode measurement of electric impedance tomography. Inverse Probl. 24, 055018 (2008) CrossRefGoogle Scholar
  13. 13.
    Hämäläinen, M., Hari, R., Ilmoniemi, R.J., Knuutila, J., Lounasmaa, O.V.: Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of signal processing in the human brain. Rev. Mod. Phys. 65, 413–497 (1993) CrossRefGoogle Scholar
  14. 14.
    Hanke, M., Harrach, B., Hyvönen, N.: Justification of point electrode models in electrical impedance tomography. Math. Models Methods Appl. Sci. doi: 10.1142/s0218202511005362
  15. 15.
    Hanke, M., Hyvönen, N., Lehn, M., Reusswig, S.: Source supports in electrostatics. BIT 48, 245–264 (2008) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Hanke, M., Hyvönen, N., Reusswig, S.: Convex source support and its application to electric impedance tomography. SIAM J. Imag. Sci. 1, 364–378 (2008) MATHCrossRefGoogle Scholar
  17. 17.
    Hanke, M., Hyvönen, N., Reusswig, S.: Convex backscattering support in electric impedance tomography. Numer. Math. 117, 373–396 (2011) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Harhanen, L., Hyvönen, N.: Convex source support in half-plane. Inverse Probl. Imag. 4, 429–448 (2010) MATHCrossRefGoogle Scholar
  19. 19.
    Isakov, V.: Inverse Source Problems. Am. Math. Soc., Providence (1990) MATHGoogle Scholar
  20. 20.
    Isakov, V.: Inverse Problems for Partial Differential Equations, 2nd edn. Springer, New York (2006) MATHGoogle Scholar
  21. 21.
    Kang, H., Lee, H.: Identification of simple poles via boundary measurements and an application of EIT. Inverse Probl. 20, 1853–1863 (2004) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kusiak, S., Sylvester, J.: The scattering support. Commun. Pure Appl. Math. 56, 1525–1548 (2003) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Kusiak, S., Sylvester, J.: The convex scattering support in a background medium. SIAM J. Math. Anal. 36, 1142–1158 (2005) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer, Berlin (1972) Google Scholar
  25. 25.
    Potthast, R., Sylvester, J., Kusiak, S.: A ‘range test’ for determining scatterers with unknown physical properties. Inverse Probl. 19, 533–547 (2003) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Schweickert, E.: Characterisierung des diskoidalen Quellträgers der Poissongleichung in der Kugel. Master’s Thesis, Johannes Gutenberg–Universität Mainz, Mainz (2008) (in German) Google Scholar
  27. 27.
    Sylvester, J., Kelly, J.: A scattering support for broadband sparse far field measurements. Inverse Probl. 21, 759–771 (2005) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Sylvester, J.: Notions of support for far fields. Inverse Probl. 22, 1273–1288 (2006) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Uhlmann, G.: Electrical impedance tomography and Calderón’s problem. Inverse Probl. 25, 123011 (2009) CrossRefGoogle Scholar
  30. 30.
    Vessella, S.: Locations and strengths of point sources: stability estimates. Inverse Probl. 8, 911–917 (1992) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987) MATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  • Martin Hanke
    • 1
  • Lauri Harhanen
    • 2
  • Nuutti Hyvönen
    • 2
  • Eva Schweickert
    • 1
  1. 1.Institut für MathematikJohannes Gutenberg-Universität MainzMainzGermany
  2. 2.Department of Mathematics and Systems AnalysisAalto UniversityAaltoFinland

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