Skip to main content
SpringerLink
Log in

A Two Dimensional Inverse Scattering Problem for Shape and Conductive Function for a Dielectic Cylinder

  • Conference paper
  • First Online:
Intelligent Mathematics II: Applied Mathematics and Approximation Theory

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 441))

  • 873 Accesses

Abstract

The inverse problem under consideration is to simultaneously reconstruct the conductive function and shape of a coated homogeneous dielectric infinite cylinder from the far-field pattern for scattering of a time-harmonic E-polarized electromagnetic plane wave. We propose an inverse algorithm that combine the approaches suggested by Ivanyshyn et al. [1–3], and extend the approaches from the case of impenetrable scatterer to the case of penetrable scatterer. It is based on a system of non-linear boundary integral equation associated with a single-layer potential approach to solve the forward scattering problem. We present the mathematical foundations of the method and exhibit its feasibility by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ivanyshyn, O., Kress, R.: Nonlinear integral equations in inverse obstacle scattering. In: Fotiatis, D., Massalas, C. (eds.) Mathematical Methods in Scattering Theory and Biomedical Engineering. World Scientific, Singapore, pp. 39–50 (2006)

    Google Scholar 

  2. Kress, R., Rundell, W.: Inverse scattering problem for shape and impedance. Inverse Prob. 17, 1075–1085 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kress, R., Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Prob. 21, 1207–1223 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  5. Gerlach, T., Kress, R.: Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Prob. 12, 619–625 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hassen, M.F.B., Ivanyshyn, O., Sini, M.: The 3D acoustic scattering by complex obstacles. Inverse Prob. 26, 105–118 (2010)

    Google Scholar 

  7. Cakoni, F., Colton, D.: The determination of the surface impedance of a partially coated obstacle from the far field data. SIAM J. Appl. Math. 64, 709–723 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cakoni, F., Colton, D., Monk, P.: The determination of boundary coefficients from far field measurements. J. Integr. Eqn. Appl. 22, 167–191 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eckel, H., Kress, R.: Nonlinear integral equations for the inverse electrical impedance problem. Inverse Prob. 23, 475–491 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fang, W., Zeng, S.: A direct solution of the Robin inverse problem. J. Integr. Eqn. Appl. 21, 545–557 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ivanyshyn, O., Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comput. Phys. 230, 3443–3452 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jakubik, P., Potthast, R.: Testing the integrity of some cavity-the Cauchy problem and the range test. Appl. Numer. Math. 58, 899–914 (2008)

    Google Scholar 

  13. Liu, J., Nakamura, G., Sini, M.: Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder. SIAM J. Appl. Math. 67, 1124–1146 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nakamura, G., Sini, M.: Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder. SIAM J. Anal. 39, 819–837 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Altundag, A., Kress, R.: On a two dimensional inverse scattering problem for a dielectric. Appl. Anal. 91, 757–771 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Altundag, A., Kress, R.: An iterative method for a two-dimensional inverse scattering problem for a dielectric. J. Inverse Ill-Posed Prob. 20, 575–590 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Altundag, A.: On a two-dimensional inverse scattering problem for a dielectric, Dissertation, Göttingen, Feb 2012

    Google Scholar 

  18. Altundag, A.: A hybrid method for inverse scattering problem for a dielectric. In: Advances in Applied Mathematics and Approximation Theory, vol. 41, pp. 185–203. Springer (2013)

    Google Scholar 

  19. Altundag, A.: Inverse obstacle scattering with conductive boundary condition for a coated dielectric cylinder. J. Concr. Appl. Math. 13, 11–22 (2015)

    MathSciNet  MATH  Google Scholar 

  20. Altundag, A.: A second degree Newton method for an inverse scattering problem for a dielectric. Hittite J. Sci. Eng. 2, 115–125 (2015)

    Article  Google Scholar 

  21. Akduman, I., Kress, R., Yaman, F., Yapar, A.: Inverse scattering for an impedance cylinder buried in a dielectric cylinder. Inverse Prob. Sci. Eng. 17, 473–488 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yaman, F.: Location and shape reconstructions of sound-soft obstacle buried in penetrable cylinders. Inverse Prob. 25, 1–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kress, R.: Integral Equation, 2nd edn. Springer, Berlin (1998)

    Google Scholar 

  24. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000)

    Google Scholar 

  25. Kress, R., Sloan, I.H.: On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numerische Mathematik 66, 199–214 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Prob. 22, 663–680 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ivanyshyn, O., Johansson, T.: Boundary integral equations for acoustical inverse sound-soft scattering. J. Inverse Ill-Posed Prob. 15, 1–14 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Colton, D., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley-Interscience Publications, New York (1983)

    MATH  Google Scholar 

  29. Kress, R., Roach, G.F.: Transmission problems for the Helmholtz equation. J. Math. Phys. 19, 1433–1437 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kress, R.: On the numerical solution of a hypersingular integral equation in scattering theory. J. Comput. Appl. Math. 61, 345–360 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Burger, M., Kaltenbacher, B., Neubauer, A.: Iterative solution methods. In: Scherzer (ed.) Handbook of Mathematical Methods in Imaging. Springer, Berlin, pp. 345–384 (2011)

    Google Scholar 

Download references

Acknowledgments

The author would like to thank Professor Rainer Kress for the helpful discussions and suggestions on the topic of this paper.

Author information

Authors and Affiliations

  1. Istanbul Sabahattin Zaim University, Istanbul, Turkey

    Ahmet Altundag

Authors
  1. Ahmet Altundag
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ahmet Altundag .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, USA

    George A. Anastassiou

  2. Department of Mathematics, TOBB Economics and Technology University, Ankara, Turkey

    Oktay Duman

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Altundag, A. (2016). A Two Dimensional Inverse Scattering Problem for Shape and Conductive Function for a Dielectic Cylinder. In: Anastassiou, G., Duman, O. (eds) Intelligent Mathematics II: Applied Mathematics and Approximation Theory. Advances in Intelligent Systems and Computing, vol 441. Springer, Cham. https://doi.org/10.1007/978-3-319-30322-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-30322-2_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30320-8

  • Online ISBN: 978-3-319-30322-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation