Skip to main content
SpringerLink
Log in

Two approaches to the solution of coefficient inverse problems for wave equations

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

Two approaches to solving coefficient inverse problems for wave equations are compared. One approach is based on integral representations obtained with the help of the Green’s function for the wave equation. In the other approach, the gradient of the error functional is directly computed in terms of the solution of the adjoint problem for a partial differential equation. The methods developed are intended for finding inhomogeneities in homogeneous media and can be applied in medicine diagnostics, acoustic and seismic near surface exploration, engineering seismics, etc.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. N. Hounsfield, “Computed Medical Imaging,” Nobel Lectures: Physiology or Medicine 1971–1980 (World Scientific, Singapore, 1992).

    Google Scholar 

  2. B. M. Glinskii, A. L. Sobisevich, and M. S. Khairetdinov, “Experience of Vibroseismic Sounding of Complex Geological Structures (for the Shugo Mud Volcano),” Dokl. Earth Sci. 413, 397–401 (2007).

    Article  Google Scholar 

  3. E. V. Kuchunova and V. M. Sadovskii, “Numerical Study of Seismic Wave Propagation in Block Media on Multiprocessor Computers,” Vychisl. Metody Program. Novye Vychisl. Tekhnol. 9(1), 66–76 (2008).

    Google Scholar 

  4. M. M. Lavrent’ev, “On a Class of Inverse Problems for Differential Equations,” Dokl. Akad. Nauk SSSR 160, 32–35 (1965).

    MathSciNet  Google Scholar 

  5. M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics (Nauka, Moscow, 1980; Am. Math. Soc., Providence, R.I., 1986).

    MATH  Google Scholar 

  6. G. Chavent, “Deux resultats sur le probléme inverse dans les équations aux deriveées partielles du deuxieme ordre an t et sur l’unicité de la solution du probléme inverse de la diffusion,” C.R. Acad. Sci. Paris 270, 25–28 (1970).

    MathSciNet  MATH  Google Scholar 

  7. A. G. Ramm, Multidimensional Inverse Scattering Problems (Longman Group, London, 1992).

    MATH  Google Scholar 

  8. V. G. Romanov and S. I. Kabanikhin, Inverse Problems for Maxwell’s Equations (VSP, Utrecht, 1994).

    MATH  Google Scholar 

  9. S. I. Kabanikhin, A. D. Satybaev, and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbole Problems (VSP, Utrecht, 2004).

    Google Scholar 

  10. M. Yu. Kokurin, “On the Reduction of a Nonlinear Inverse Problem for a Two-Dimensional Hyperbolic Equation to a Linear Integral Equation,” Vychisl. Metody Program. Novye Vychisl. Tekhnol. 10(1), 300–305 (2009).

    Google Scholar 

  11. L. Beilina and M. V. Klibanov, “A Globally Convergent Numerical Method for a Coefficient Inverse Problem,” SIAM J. Sci. Comput. 31, 478–509 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  12. L. Beilina, M. V. Klibanov, and M. Yu. Kokurin, Preprint No. 2009:47, Chalmers Preprint Series,.

  13. A. B. Bakushinsky and A. V. Goncharsky, Ill-posed problems. Theory and applications. Dordrect: Kluwer Acad. Publs.. B (1994).

    Book  Google Scholar 

  14. A. V. Goncharskii and S. Yu. Romanov, “On a Three-Dimensional Diagnostics Problem in the Wave Approximation,” Comput. Math. Math. Phys. 40, 1308–1311 (2000).

    MathSciNet  MATH  Google Scholar 

  15. A. V. Goncharskii, S. L. Ovchinnikov, and S. Yu. Romanov, “On One Problem of Wave Diagnostics,” Vestn. Mosk. Gos. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 1, 7–13 (2010).

  16. A. V. Goncharskii and S. Yu. Romanov, “On a Problem of Wave Diagnostics,” Vychisl. Metody Program. Novye Vychisl. Tekhnol. 7(1), 36–40 (2006).

    Google Scholar 

  17. A. V. Goncharskii, S. Yu. Romanov, and S. A. Kharchenko, “Inverse Problem of Acoustic Diagnostics of here-Dimensional Media,” Vychisl. Metody Program. Novye Vychisl. Tekhnol. 7(1), 113–121 (2006).

    Google Scholar 

  18. S. L. Ovchinnikov and S. Yu. Romanov, “Organization of Parallel Computations in Solving an Inverse Problem of Wave Diagnostics,” Vychisl. Metody Program. Novye Vychisl. Tekhnol. 9(2), 338–345 (2008).

    Google Scholar 

  19. E. E. Tyrtyshnikov, Methods of Numerical Analysis (Akademiya, Moscow, 2007) [in Russian].

    Google Scholar 

  20. F. Natterer and F. Wubbeling, “A Propagation-Backpropagation Method for Ultrasound Tomography: Inverse Problems,” 11, 1225–1232 (1995).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Computing Center, Moscow State University, Moscow, 119992, Russia

    A. V. Goncharskii & S. Yu. Romanov

Authors
  1. A. V. Goncharskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Yu. Romanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Goncharskii.

Additional information

Original Russian Text © A.V. Goncharskii, S.Yu. Romanov, 2012, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2012, Vol. 52, No. 2, pp. 263–269.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goncharskii, A.V., Romanov, S.Y. Two approaches to the solution of coefficient inverse problems for wave equations. Comput. Math. and Math. Phys. 52, 245–251 (2012). https://doi.org/10.1134/S0965542512020078

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542512020078

Keywords

Search

Navigation