Skip to main content
SpringerLink
Log in
SpringerLink
Log in

A Posteriori Error Estimate in the Lagrangian Setting for an Inverse Problem Based on a New Formulation of Maxwell’s System

  • Conference paper
  • First Online:
Inverse Problems and Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 120))

  • 1619 Accesses

Abstract

In this paper we consider an inverse problem of determination of a dielectric permittivity function from a backscattered electromagnetic wave. The inverse problem is formulated as an optimal control problem for a certain partial differential equation derived from Maxwell’s system. We study a solution method based on finite element approximation and provide a posteriori error estimate for the use in an adaptive algorithm.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Notes

  1. 1.

    Throughout the remaining part of the text we will use ‘≲’ to indicate approximate estimation in the following sense: \(a\lesssim b\) if and only if there exists a constant \(C>0\) and some \(b^*\approx b\) such that \(a\leq Cb^*\).

References

  1. Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39 (2000)

    Google Scholar 

  2. Beilina, L.: Adaptive hybrid finite element/difference methods: applications to inverse elastic scattering. J. Inverse Ill-Posed Probl. 11 (2003)

    Google Scholar 

  3. Beilina, L.: Energy estimates and numerical verification of the stabilized domain decomposition finite element/finite difference approach for time-dependent Maxwell’s system. Cent. Eur. J. Math. 11 (2013)

    Google Scholar 

  4. Beilina, L., Johnson, C.: A posteriori error estimation in computational inverse scattering. Math. Model. Method. Appl. Sci. 15 (2005)

    Google Scholar 

  5. Beilina, L., Klibanov, M.V.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York (2012)

    Book  MATH  Google Scholar 

  6. Beilina, L., Klibanov, M.V., Kokurin, M. Yu.: Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem. J. Math. Sci. 3 (2010)

    Google Scholar 

  7. Beilina, L., Thành, N.T., Klibanov, M.V., Fiddy, M.A.: Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation. Inverse Probl. 30 (2014)

    Google Scholar 

  8. Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Studentlitteratur, Lund (1996)

    Google Scholar 

  9. Hammond, P., Sykulski, J.K.: Engineering Electromagnetism: Physical Processes and Computation. Oxford University Press, Oxford (1994)

    Google Scholar 

  10. Jin, J.: The Finite Element Method in Electromagnetics. Wiley, New York (2002)

    Google Scholar 

  11. Johnson, C., Szepessy, A.: Adaptive finite element methods for conservation laws based on a posteriori error estimates. Comm. Pure Appl. Math. 48 (1995)

    Google Scholar 

  12. Klibanov, M.V., Fiddy, M.A., Beilina, L., Pantong, N., Schenk, J.: Picosecond scale experimental verification of a globally convergent numerical method for a coefficient inverse problem. Inv. Probl. 26, 04500–3 (2010)

    Article  MathSciNet  Google Scholar 

  13. Kuzhuget, A.V., Beilina, L., Klibanov, M.V., Sullivan, A., Nguyen, L., Fiddy, M.A.: Quantitative image recovery from measured blind backscattered data using a globally convergent inverse method. IEEE Trans. Geosci. Remote Sens. 51 (2013)

    Google Scholar 

  14. Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon, Oxford (2003)

    Book  MATH  Google Scholar 

  15. Poulsen, K.D., Lynch, D.R.: Elimination of vector parasites in finite element maxwell solutions. IEEE Trans. Microw. Theory Tech. 39 (1991)

    Google Scholar 

  16. Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V., Yagola, A.G.: Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publishers, Dordrecht. (1995)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his gratitude to his supervisor Larisa Beilina for many good suggestions and advice, and to Mohammad Asadzadeh for many improved formulations.

This research was supported by the Swedish Research Council and by the Swedish Institute, Visby Program.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology and Gothenburg University, SE-42196, Gothenburg, Sweden

    John Bondestam Malmberg

Authors
  1. John Bondestam Malmberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Bondestam Malmberg .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Chalmers University of Technology Gothenburg University, Gothenburg, Sweden

    Larisa Beilina

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Malmberg, J. (2015). A Posteriori Error Estimate in the Lagrangian Setting for an Inverse Problem Based on a New Formulation of Maxwell’s System. In: Beilina, L. (eds) Inverse Problems and Applications. Springer Proceedings in Mathematics & Statistics, vol 120. Springer, Cham. https://doi.org/10.1007/978-3-319-12499-5_3

Download citation

Publish with us

Policies and ethics

Search

Navigation