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Inverse Electromagnetic Obstacle Scattering

  • David Colton
  • Rainer Kress
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 93)

Abstract

This last chapter on obstacle scattering is concerned with the extension of the results from Chap.  5 on inverse acoustic scattering to inverse electromagnetic scattering. In order to avoid repeating ourselves, we keep this chapter short by referring back to the corresponding parts of Chap.  5 when appropriate. In particular, for notations and for the motivation of our analysis we urge the reader to get reacquainted with the corresponding analysis in Chap.  5 on acoustics. We again follow the general guideline of our book and consider only one of the many possible inverse electromagnetic obstacle problems: given the electric far field pattern for one or several incident plane electromagnetic waves and knowing that the scattering obstacle is perfectly conducting, find the shape of the scatterer.

References

  1. 29.
    Ben Hassen, F., Erhard, K., and Potthast, R.: The point source method for 3d reconstructions for the Helmholtz and Maxwell equations. Inverse Problems 22, 331–353 (2006).MathSciNetCrossRefGoogle Scholar
  2. 34.
    Blöhbaum, J.: Optimisation methods for an inverse problem with time-harmonic electromagnetic waves: an inverse problem in electromagnetic scattering. Inverse Problems 5, 463–482 (1989).MathSciNetCrossRefGoogle Scholar
  3. 49.
    Cakoni, F., and Colton, D.: Combined far field operators in electromagnetic inverse scattering theory. Math. Methods Appl. Sci. 26, 293–314 (2003).MathSciNetCrossRefGoogle Scholar
  4. 51.
    Cakoni, F., and Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin 2006.zbMATHGoogle Scholar
  5. 59.
    Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.CrossRefGoogle Scholar
  6. 85.
    Collino, C., Fares, M., and Haddar, H.: Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems. Inverse Problems 19, 1279–1298 (2003).MathSciNetCrossRefGoogle Scholar
  7. 88.
    Colton, D., Haddar, H., and Monk, P.: The linear sampling method for solving the electromagnetic inverse scattering problem. SIAM J. Sci. Comput. 24, 719–731 (2002).MathSciNetCrossRefGoogle Scholar
  8. 98.
    Colton, D., and Kress, R.: Karp’s theorem in electromagnetic scattering theory. Proc. Amer. Math. Soc. 104, 764–769 (1988).MathSciNetzbMATHGoogle Scholar
  9. 102.
    Colton, D., and Kress, R.: On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces. Math. Methods Applied Science 24, 1289–1303 (2001).MathSciNetCrossRefGoogle Scholar
  10. 104.
    Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefGoogle Scholar
  11. 145.
    Ganesh, M., and Hawkins, S. C.: An efficient surface integral equation method for the time-harmonic Maxwell equations. ANZIAM J. 48, C17–C33 (2007).MathSciNetCrossRefGoogle Scholar
  12. 146.
    Ganesh, M., and Hawkins, S. C.: A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces. J. Comput. Phys. 227, 4543–4562 (2008).MathSciNetCrossRefGoogle Scholar
  13. 163.
    Haas, M., Rieger, W., Rucker, W., and Lehner, G.: Inverse 3D acoustic and electromagnetic obstacle scattering by iterative adaption. In: Inverse Problems of Wave Propagation and Diffraction (Chavent and Sabatier, eds). Springer, Berlin 1997.Google Scholar
  14. 169.
    Haddar, H., and Kress. R.: On the Fréchet derivative for obstacle scattering with an impedance boundary condition. SIAM J. Appl. Math. 65, 194–208 (2004).Google Scholar
  15. 171.
    Hähner, P.: An exterior boundary-value problem for the Maxwell equations with boundary data in a Sobolev space. Proc. Roy. Soc. Edinburgh 109A, 213–224 (1988).MathSciNetCrossRefGoogle Scholar
  16. 173.
    Hähner, P.: A uniqueness theorem for the Maxwell equations with L 2 Dirichlet boundary conditions. Meth. Verf. Math. Phys. 37, 85–96 (1991).MathSciNetzbMATHGoogle Scholar
  17. 174.
    Hähner, P.: A uniqueness theorem for a transmission problem in inverse electromagnetic scattering. Inverse Problems 9, 667–678 (1993).MathSciNetCrossRefGoogle Scholar
  18. 217.
    Ivanyshyn Yaman, O., and Le Louër, F.: Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems. Inverse Problems 32 095003 (2016).MathSciNetCrossRefGoogle Scholar
  19. 271.
    Kress, R., and Päivärinta, L.: On the far field in obstacle scattering. SIAM J. Appl. Math. 59, 1413–1426 (1999).MathSciNetCrossRefGoogle Scholar
  20. 297.
    Leis, R.: Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York 1986.CrossRefGoogle Scholar
  21. 298.
    Le Louër, F.: Spectrally accurate numerical solution of hypersingular boundary integral equations for three-dimensional electromagnetic wave scattering problems. J. Comput. Phys. 275, 662–666 (2014).MathSciNetCrossRefGoogle Scholar
  22. 299.
    Le Louër, F.: A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles. ANZIAM 59, E1–E49 (2018).CrossRefGoogle Scholar
  23. 306.
    Liu, H., Yamamoto, M. and Zou, J.: Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering. Inverse Problems 23, 2357–2366 (2007).MathSciNetCrossRefGoogle Scholar
  24. 325.
    Morrey, C.M.: Multiple Integrals in the Calculus of Variations. Springer, Berlin 1966.zbMATHGoogle Scholar
  25. 350.
    Pieper, M.: Nonlinear integral equations for an inverse electromagnetic scattering problem. Journal of Physics: Conference Series 124, 012040 (2008).Google Scholar
  26. 356.
    Potthast, R.: Domain derivatives in electromagnetic scattering. Math. Meth. in the Appl. Sci. 19, 1157–1175 (1996).MathSciNetCrossRefGoogle Scholar
  27. 423.
    Werner, P. : On the exterior boundary value problem of perfect reflection for stationary electromagnetic wave fields. J. Math. Anal. Appl. 7, 348–396 (1963).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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