Integral Equation Methods in Inverse Obstacle Scattering with a Generalized Impedance Boundary Condition

  • Rainer KressEmail author


The inverse problem under consideration is to reconstruct the shape of an impenetrable two-dimensional obstacle with a generalized impedance boundary condition from the far field pattern for scattering of time-harmonic acoustic or E-polarized electromagnetic plane waves. We propose an inverse algorithm that extends the approach suggested by Johansson and Sleeman (IMA J. Appl. Math. 72(1):96–112, 2007) for the case of the inverse problem for a sound-soft or perfectly conducting scatterer. It is based on a system of nonlinear boundary integral equations associated with a single-layer potential approach to solve the forward scattering problem which extends the integral equation method proposed by Cakoni and Kress (Inverse Prob. 29(1):015005, 2013) for a related boundary value problem for the Laplace equation. In addition, we also present an algorithm for reconstructing the impedance function when the shape of the scatterer is known. We present the mathematical foundations of the methods and exhibit their feasibility by numerical examples.


  1. 1.
    Bourgeois, L., Haddar, H.: Identification of generalized impedance boundary conditions in inverse scattering problems. Inverse Prob. Imaging 4(1), 19–38 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourgeois, L., Chaulet, N., Haddar, H.: Stable reconstruction of generalized impedance boundary conditions. Inverse Prob. 27(9), 095002 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bourgeois, L., Chaulet, N., Haddar, H.: On simultaneous identification of a scatterer and its generalized impedance boundary condition. SIAM J. Sci. Comput. 34(3), A1824–A1848 (2012)CrossRefGoogle Scholar
  4. 4.
    Cakoni, F., Kress, R.: Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition. Inverse Prob. 29(1), 015005 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cakoni, F., Hu, Y., Kress, R.: Simultaneous reconstruction of shape and generalized impedance functions in electrostatic imaging. Inverse Prob. 30(10), 105009 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol. 93, 3rd edn. Springer, New York (2012)Google Scholar
  7. 7.
    Duruflé, M., Haddar, H., Joly, P.: High order generalized impedance boundary conditions in electromagnetic scattering problems. C. R. Phys. 7(5), 533–542 (2006)CrossRefGoogle Scholar
  8. 8.
    Haddar, H., Joly, P., Nguyen, H.M.: Generalized impedance boundary conditions for scattering by strongly absorbing obstacles: the scalar case. Math. Models Methods Appl. Sci. 15(8), 1273–1300 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ivanyshyn, O., Johansson, T.: Boundary integral equations for acoustical inverse sound-soft scattering. J. Inverse Ill-Posed Probl. 16(1), 65–78 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Johansson, T., Sleeman, B.: Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern. IMA J. Appl. Math. 72(1), 96–112 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kirsch, A.: Surface gradient and continuity properties of some integral operators in classical scattering theory. Math. Meth. Appl. Sci. 11(6), 789–804 (1989)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kress, R.: Linear Integral Equations. Applied Mathematical Sciences, vol. 82, 3rd edn. Springer, New York (2013)Google Scholar
  13. 13.
    Kress, R.: A collocation method for a hypersingular boundary integral equation via trigonometric differentiation. J. Integral Equ. Appl. 26(2), 197–213 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kress, R., Sloan, I.H.: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numer. Math. 66(1), 199–214 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  16. 16.
    Senior, T.B.A., Volakis, J.L.: Approximate Boundary Conditions in Electromagnetics. IEEE Electromagnetic Waves Series, vol. 41. The Institution of Electrical Engineers, London (1995)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Numerische und Angewandte MathematikUniversität GöttingenGöttingenGermany

Personalised recommendations