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Justification of the Steepest Descent Method for the Integral Statement of an Inverse Problem for a Hyperbolic Equation

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Abstract

Under consideration is the steepest descent method for solving the problem of determination of a coefficient in a hyperbolic equation in integral statement. The properties of solutions to the direct and inverse problems are studied. Estimates for the objective functional and its gradient are obtained. Convergence in the mean is proved for the steepest descent method for minimizing the residual functional.

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References

  1. Tikhonov A. N., “On solution of ill-posed problems and the regularization method,” Dokl. Akad. Nauk SSSR, 151, No. 3, 501–504 (1963).

    Google Scholar 

  2. Alekseev A. S., “Inverse dynamical seismic problems,” in: Some Methods and Algorithms for Interpretation of Geophysical Data [in Russian], Nauka, Moscow, 1967, pp. 9–84.

    Google Scholar 

  3. Bamberger A., Chavent G., and Lailly P., “About the stability of the inverse problem in 1–D wave equations. Application to the interpretation of seismic profiles,” Appl. Math. Optim., 5, 1–47 (1979).

    Google Scholar 

  4. Santosa F. and Symes W., An Analysis of Least-Squares Velocity Inversion. Vol. 4, Soc. Exploration Geophys., Tulsa (USA) (1989).

    Google Scholar 

  5. Dmitriev V. I. and Fëdorova E. A., “On a solution of the inverse problem by the method of partial probing of a layered medium in: Bibliographical Program on Geophysics [in Russian], Moscow Univ., Moscow, 1983, pp. 11–18.

    Google Scholar 

  6. Nyambaa Sh. and Cheverda V. A., An Optimizational Method for Solving the Electroexploration Inverse Problem on Continuous Current for Vertically-Inhomogeneous Media [Preprint, No. 794], Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk (1988).

    Google Scholar 

  7. Iskakov K. T. and Kabanikhin S. I., “The solution of one-dimensional inverse problem of geoelectrics by the method of conjugate gradients,” Russian J. Theoret. Appl. Mech., 2, No. 3, 197–222 (1992).

    Google Scholar 

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

    Google Scholar 

  9. Kabanikhin S. I., “Numerical analysis of inverse problems,” J. Inverse Ill-Posed Probl., 3, No. 4, 278–304 (1995).

    Google Scholar 

  10. Karchevsky A. L., “Properties of the misfit functional for a nonlinear one-dimensional coefficient hyperbolic inverse problem,” J. Inverse Ill-Posed Probl., 5, No. 2, 139–165 (1997).

    Google Scholar 

  11. Romanov V. G., Inverse Problems of Mathematical Physics [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  12. Kabanikhin S. I. and Bakanov G. V., “The optimizational method for solving the discrete inverse problem for hyperbolic equation,” J. Inverse Ill-Posed Probl., 6, No. 4, 513–530 (1996).

    Google Scholar 

  13. Azamatov J. S. and Kabanikhin S. I., “Nonlinear operator equations. L 2-theory,” J. Inverse Ill-Posed Probl., 7, No. 6, 497–529 (1999).

    Google Scholar 

  14. Vasil' ev F. P., Numerical Methods for Solving Extremal Problems [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  15. Karmanov V. G., Mathematical Programming [in Russian], Nauka, Moscow (1988).

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk

    S. I. Kabanikhin

  2. Karaganda State University, Karaganda

    K. T. Iskakov

Authors
  1. S. I. Kabanikhin
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  2. K. T. Iskakov
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Kabanikhin, S.I., Iskakov, K.T. Justification of the Steepest Descent Method for the Integral Statement of an Inverse Problem for a Hyperbolic Equation. Siberian Mathematical Journal 42, 478–494 (2001). https://doi.org/10.1023/A:1010471125870

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