Identification of Current Sources in 3D Electrostatics

  • Aron SommerEmail author
  • Andreas Helfrich-Schkarbanenko
  • Vincent Heuveline
Reference work entry


Motivated by passive airborne geoexploration we consider a source identification problem. This problem setting arises in electrostatics and it turns out to be a linear, ill-posed inverse problem. After developing a theoretical framework for corresponding elliptic forward problem, an approach for reconstructing current sources from local electric potential data is illustrated. A pseudo-solution is achieved by means of Tikhonov regularization. The performance of the method is shown by three-dimensional synthetic and real-life numerical examples. For numerical modeling, we choose Method of Finite Elements provided by COMSOL Multiphysics and apply MATLAB for developing a reconstruction algorithm.


Inverse Problem Direct Problem Tikhonov Regularization Unique Solvability Forward Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Jörg Bäuerle for his fruitful comments.


  1. Adams RA, Fournier JJF (2003) Sobolev spaces. Second edition. Elsevier, AmsterdamzbMATHGoogle Scholar
  2. Braess D (2003) Finite Elemente. Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer, Berlin Heidelberg New YorkGoogle Scholar
  3. Brenner SC, Scott LR (1994) The mathematical theory of finite element methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. Colton D, Kress R (1998) Inverse acoustic and electromagnetic scattering theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. Devaney AJ, Sherman G (1982) Nonuniqueness in inverse source and scattering problems. IEEE Trans Antennas Propag 30(5):1034–1037MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dobrowolski M (2006) Angewandte Funktionalanalysis. Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Springer, Berlin Heidelberg New YorkGoogle Scholar
  7. Evans LC (2008) Partial differential equations. Graduate studies in mathematics, vol 19. American Mathematical Society, Providence, Rhode IslandGoogle Scholar
  8. Gilbarg D, Trudinger NS (2001) Elliptic partial differential equations of second order, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  9. Groetsch HW (1993) Inverse problems in the mathematical sciences. Vieweg, BraunschweigCrossRefzbMATHGoogle Scholar
  10. Hadamard J (1915) Four lectures on mathematics. Columbia University Press, New YorkzbMATHGoogle Scholar
  11. Hanke M, Rundell W (2011) On rational approximation methods for inverse source problems. Inverse Probl Imaging (IPI) 5(1):185–202MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hansen P (1998) Rank-deficient and discrete ill-podes problems. Numerical aspects of linear inversion. SIAM, PhiladelphiaCrossRefGoogle Scholar
  13. Helfrich Schkarbanenko A (2011) Elektrische Impedanztomografie in der Geoelektrik. Dissertation, Karlsruher Institut für TechnologieGoogle Scholar
  14. Kirsch A (1996) An introduction to the mathematical theory of inverse problems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  15. Korovkin NV, Chechurin VL, Hayakawa M (2007) Inverse problems in electric circuits and electromagnetics. Springer, New YorkzbMATHGoogle Scholar
  16. Königsberger K (2004) Analysis 2, 5. korrigierte Auflage. Springer, Berlin Heidelberg New YorkCrossRefzbMATHGoogle Scholar
  17. Kress R, Kühn L, Potthast R (2002) Reconstruction of a current distribution from its magnetic field. Inverse Probl 18:1127–1146CrossRefMathSciNetzbMATHGoogle Scholar
  18. Marengo EA, Devaney AJ (1999) The inverse source problem of electromagnetics: linear inversion formulation and minimum energy solution. IEEE Trans Antennas Propag 47(2):410–412MathSciNetCrossRefzbMATHGoogle Scholar
  19. Phillips DL (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:84–97MathSciNetCrossRefzbMATHGoogle Scholar
  20. Rieder A (2003) Keine Probleme mit Inversen Problemen. Eine Einführung in ihre stabile Lösung. Vieweg Verlag, WiesbadenCrossRefGoogle Scholar
  21. Salsa S (2008) Partial differential equations in action. From modeling to theory. Springer, MilanzbMATHGoogle Scholar
  22. Sommer A (2012) Passive Erdölexploration aus der Luft – Theorie und Numerik eines linearen inversen Problems. Diploma thesis, Karlsruher Institut für TechnologieGoogle Scholar
  23. Steinbach O (2008) Numerical approximation methods for elliptic boundary value problems. Finite and boundary elements. Springer, New YorkCrossRefzbMATHGoogle Scholar
  24. Tikhonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Sov Math Dokl 4:1035–1038zbMATHGoogle Scholar
  25. Toponogov VA (2006) Differential geometry of curves and surfaces. A concise guide. Brikhäuser, BostonGoogle Scholar
  26. Wirgin A (2008) The inverse crime. arXiv:math-ph/0401050v1Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Aron Sommer
    • 1
    Email author
  • Andreas Helfrich-Schkarbanenko
    • 2
  • Vincent Heuveline
    • 3
    • 4
    • 5
  1. 1.Institut für Informationsverarbeitung (TNT)Leibniz Universität HannoverHannoverGermany
  2. 2.Institute for Applied and Numerical MathematicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.Engineering Mathematics and Computing Lab (EMCL)Karlsruhe Institute of TechnologyKarlsruheGermany
  4. 4.Institute for Applied and Numerical MathematicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  5. 5.University of Heidelberg, Interdisciplinary Center for Scientific ComputingEngineering Mathematics and Computing LabHeidelbergGermany

Personalised recommendations