A globally convergent convexification algorithm for the inverse problem of electromagnetic frequency sounding in one dimension

Authors

  • M.V. Klibanov
  • A. Timonov

Keywords:

электромагнитное частотное зондирование
минимизация выпуклых функций
сходимость
аппроксимация
метод наименьших квадратов
итерационные алгоритмы
математическое моделирование

Abstract

A globally convergent convexification algorithm for the numerical solution of the inverse problem of electromagnetic frequency sounding in one dimension is presented. This algorithm is based on the concept of convexification of a multiextremal objective function proposed recently by the authors. A key point in the proposed algorithm is that unlike conventional layer-stripping algorithms, it provides the stable approximate solution via minimization of a finite sequence of strictly convex objective functions resulted from applying the nonlinear weighted least squares method with Carleman’s weight functions. The other advantage of the proposed algorithm is that its convergence to the «exact» solution does not depend on a starting vector. Thus, the uncertainty inherent to the local methods, such as the gradient or Newton-like methods, is eliminated. The 1-D inverse model of magnetotelluric sounding is selected to exemplify the convexification approach. Based on the localizing property of Carleman’s weight functions, it is proven that the distance between the approximate and «exact» solutions is small if the approximation error is small. The results of computational experiments with several realistic and synthetic marine shallow water configurations are presented to demonstrate the computational feasibility of the proposed algorithm.


Downloads

Published

2003-02-10

Issue

Section

Section 1. Numerical methods and applications

Author Biographies

M.V. Klibanov

University of North Carolina at Charlotte,
Department of Mathematics and Statistics
Fretwell 376, 9201 University City Blvd., Charlotte, NC 28223, USA

A. Timonov

University of North Carolina at Charlotte,
Department of Mathematics and Statistics
Fretwell 376, 9201 University City Blvd., Charlotte, NC 28223, USA


References

  1. Klibanov M.V. and Timonov A. «A new slant on the inverse problems of electromagnetic frequency sounding: «convexification» of a multiextremal objective function via the Carleman weight functions», Inverse Problems, T. 17: 1865-1887, 2001.
  2. Tikhonov A.N. Ön determining the electrical characteristics of deep layers of the Earth’s crust», Sov. Math. Dokl., 2: 295-297, 1950.
  3. Tikhonov A.N. «Mathematical basis of the theory of electromagnetic soundings», U.S.S.R. Comput. Math. Mathemat. Phys., T. 5: 207-211, 1965.
  4. Cagniard L. «Basic theory of the magnetotelluric method of geophysical prospecting», Geophysics, T. 37: 605-635, 1953.
  5. Berdichevsky M.N. and Zhdanov M.S. Advanced Theory of Deep Geomagnetic Sounding. New York: Elsevier Science Publishing Inc., 1984.
  6. Constable S.C. «Marine electromagnetic induction studies», Surv. Geophys., T. 11: 303-327, 1990.
  7. Palshin N.A. Öceanic electromagnetic studies: a review», Surv. Geophys., T. 17: 455-491, 1996.
  8. MARELEC 96, 99, 01. Proceedings of the 1st, 2nd, and 3rd International Conferences on Marine Electromagnetics. London, UK, June 1997; Brest, France, July 1999; Stockholm, Sweden, July 2001.
  9. Haber E., Asher U.M., and Oldenburg D. Ön optimization techniques for solving nonlinear inverse problems», Inverse Problems, T. 16: 1263-1280, 2000.
  10. Newman G.A. and Harvesten G.M. «Solution strategies for two- and three-dimensional electromagnetic inverse problems», Inverse Problems, T. 16: 1357-1375, 2000.
  11. Dmitriev V.I. and Alekseeva N.V. «An algorithm for the numerical solution of the inverse problem of MT sounding», in: Software Library for Geophysics (in Russian), Moscow: Moscow State University, 1984.
  12. Alexander J.C. and Yorke J.A. «The homotopy continuation method: numerically implementable topological procedures», Trans. Am. Math. Soc., T. 242: 271-284, 1978.
  13. Ramlau R. «A steepest descent algorithm for the global minimization of the Tikhonov functional», Inverse Problems, T. 18: 381-403, 2002.
  14. Bakushinsky A.B. and Goncharsky A.V. Ill-posed Problems: Theory and Applications. Dodrecht: Kluwer Academic Publishers, 1994.
  15. Himmelblau D.M. Applied Nonlinear Programming. New York: McGraw-Hill, 1972.
  16. Sylvester J. Layer Stripping. Surveys on Solution Methods for Inverse Problems. Ed. D. Colton, H. Engl, W. Rundell, etc. New York: Springer-Verlag, 2000.
  17. Chen Y., Rokhlin V. Ön the inverse scattering problem for the Helmholtz equation in one dimension», Inverse Problems, T. 8: 365-391, 1992.
  18. Somersalo E. «Layer stripping for time-harmonic Maxwell’s equations with high frequency», Inverse Problems, T. 10: 449-466, 1994.
  19. Klibanov M.V. and Timonov A. «A sequential minimization algorithm based on the convexification approach», Inverse Problems, 2003 (to appear).
  20. Levitan B.M. Inverse Sturm-Liouville Problems. Zeist: VSP, 1987.
  21. Khruslov E.Ya. and Shepelsky D.G. «Inverse scattering method in electromagnetic sounding theory», Inverse Problems, T. 10: 1-37, 1994.
  22. Rundell W. and Sacks P.E. «Reconstruction techniques for classical inverse Sturm-Liouville problems», Math Comput., T. 58: 161-183, 1992.
  23. Brown B.M., Samko V.S., Knowles I W., etc. «Inverse spectral problem for the Sturm-Liouville equation», Inverse Problems, T. 19: 235-252, 2003.
  24. Isakov V. Inverse Problems for Partial Differential Equations. New York: Springer, 1998.
  25. Klibanov M.V. «Global convexity in a three-dimensional inverse acoustic problem», SIAM J. Math. Anal., T. 28: 1371-1388, 1997.
  26. Tikhonov A.N. Ön stability of inverse problems», Sov. Math. Dokl., T. 39: 195-198, 1943.
  27. Lavrentiev M.M., Romanov V.G., and Shishatskii S.P. Ill-Posed Problems of Mathematical Physics and Analysis. Providence R.I.: AMS, 1986.
  28. Arestov V.V. «Approximation of unbounded operators by bounded operators and related extremal problems», Russian Math. Surv., T. 51: 1093-1126, 1996.
  29. Lasdon L.S., Waren A.D., Jain A., and Ratner M. «Design and testing of a Generalized Reduced Gradient Code for nonlinear programming», ACM Trans. on Math. Software, T. 4: 34-50, 1978.
  30. Krylstedt P., Mattsson J., and Timonov A. 愦灭;quot愦灭;graveNumerical modeling of electromagnetic frequency sounding in marine environments: a comparison of local optimization techniques», in: Proceedings of the 3rd International Conference on Marine Electromagnetics, Stockholm, Sweden, July 2001.
  31. Romanov V.G. Inverse Problems of Mathematical Physics. Utrecht: VNU, 1987.