On the method of calculating the modulus of continuity of the inverse operator and its modifications with application to non-linear problems of geoelectrics

Authors

DOI:

https://doi.org/10.26089/NumMet.v21r430

Keywords:

inverse problem, modulus of continuity of an operator, a priori and a posteriori estimates, Monte Carlo, geoelectrics

Abstract

The article considers a priori estimates of the ambiguity (error) of approximate solutions of conditionally correct nonlinear inverse problems based on the modulus of continuity of the inverse operator and its modifications. It is shown that in the class of piecewise constant solutions defined on a given parametrization grid, the modulus of continuity of the inverse operator and its modifications monotonously increase with increasing mesh dimension. A method is proposed for constructing an optimal parameterization grid that has a maximum dimension provided that the modulus of continuity of the inverse operator does not exceed a given value. A numerical algorithm for calculating the modulus of continuity of the inverse operator and its modifications using Monte Carlo algorithms is presented; questions of convergence of the algorithm are investigated. The proposed method is also applicable for calculating classical posterior error estimates. Numerical examples are given for nonlinear inverse problems of geoelectrics.

Author Biography

M.I. Shimelevich

References

  1. M. M. Lavrent’ev, On Certain Ill-Posed Problems of Mathematical Physics (Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 1962) [in Russian].
  2. V. K. Ivanov, “On Ill-Posed Problems,” Mat. Sb. 61 (2), 211-223 (1963).
  3. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978; VSP Press, Utrecht, 2002).
  4. A. B. Bakushinsky and A. V. Goncharsky, Ill-Posed Problems. Numerical Methods and Applications. (Mosk. Gos. Univ., Moscow, 1989) [in Russian].
  5. M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980; AMS Press, Providence, 1986).
  6. O. V. Novik, Mathematical Problems in the Reduction of the Number of Geophysical Data Necessary for Oil and Gas Exploration , available from VIEMS, No. 485-MG (VIEMS Ross. Akad. Nauk, Moscow, 1987).
  7. M. I. Shimelevich, Optimization Algorithms for Solving the Inverse Problems of electromagnetic sounding , available from VIEMS, No. 796—MG-89 (VIEMS Ross. Akad. Nauk, 1989).
  8. M. I. Shimelevich, “Methods for Increasing the Stability of the Geoelectrics Inverse Problems on the Base of Neural Network Modeling,” Geofizika, No. 4, 49-56 (2013).
  9. M. I. Shimelevich, E. A. Obornev, I. E. Obornev, and E. A. Rodionov, “Numerical Methods for Estimating the Degree of Practical Stability of Inverse Problems in Geoelectrics,” Izv. Ross. Akad. Nauk, Fiz. Zemli, No. 3, 58-64 (2013) [Izv., Phys. Solid Earth 49 (3), 356-362 (2013)].
  10. M. I. Shimelevich, E. A. Obornev, I. E. Obornev, and E. A. Rodionov, “The Neural Network Approximation Method for Solving Multidimensional Nonlinear Inverse Problems of Geophysics,” Izv. Ross. Akad. Nauk, Fiz. Zemli, No. 4, 100-109 (2017) [Izv., Phys. Solid Earth 53 (4), 588-597 (2017)].
  11. A. V. Goncharskii and A. G. Yagola, “Uniform Approximation of Monotone Solutions of Ill-Posed Problems,” Dokl. Akad. Nauk SSSR, 184 (4), 771-773 (1969).
  12. A. V. Goncharskii, A. M. Cherepashchuk, and A. G. Yagola, Numerical Methods for Solving Inverse Problems of Astrophysics (Nauka, Moscow, 1978) [in Russian].
  13. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Nauka, Moscow 1990) [in Russian].
  14. A. G. Yagola and K. Yu. Dorofeev, “Sourcewise Representation and a Posteriori Error Estimates for Ill-Posed Problems,” in Fields Institute Communications: Operator Theory and Its Applications (AMS Press, Providence, 2000), Vol. 25, pp. 543-550.
  15. K. Yu. Dorofeev and A. G. Yagola, “The Method of Extending Compacts and a Posteriori Error Estimates for Nonlinear Ill-Posed Problems,” J. Inverse Ill Posed Probl. 12 (6), 627—636 (2004).
  16. A. S. Leonov, “Which of Inverse Problems Can Have a Priori Approximate Solution Accuracy Estimates Comparable in Order with the Data Accuracy,” Sib. Zh. Vych. Mat. 17 (4), 339-348 (2014) [Numer. Anal. Appl. 7 (4), 284-292 (2014)].
  17. A. S. Leonov, “A Posteriori Accuracy Estimations of Solutions to Ill-Posed Inverse Problems and Extra-Optimal Regularizing Algorithms for Their Solution,” Sib. Zh. Vych. Mat. 15 (1), 85-102 (2012) [Numer. Anal. Appl. 5 (1), 68-83 (2012)].
  18. A. B. Bakushinskii and A. S. Leonov, “New a Posteriori Error Estimates for Approximate Solutions to Irregular Operator Equations,” Vychisl. Metody Programm. 15, 359-369 (2014).
  19. V. I. Dmitriev, Inverse Problems of Geophysics (MAKS Press, Moscow, 2012) [in Russian].
  20. V. B. Glasko and V. I. Starostenko, “Regularizing Algorithm for Solving a System of Nonlinear Equations in Inverse Problems of Geophysics,” Izv. Akad. Nauk SSSR, Fiz. Zemli, No. 3, 44-53 (1976).
  21. V. N. Strakhov, “On the Parameterization in the Inverse Problems of Gravimetry,” Izv. Akad. Nauk SSSR, Fiz. Zemli, No. 6, 39-49 (1978).
  22. M. I. Shimelevich and E. A. Obornev, “Application of the Optimization Approach in the Interpretation of MTS Data,” Izv. Vyssh. Uchebn. Zaved., Geol. Razved., No. 2, 109-115 (1997).
  23. A. S. Leonov and A. G. Yagola, “Optimal Methods for the Solution of Ill-Posed Problems with Sourcewise Represented Solutions,” Fundam. Prikl. Mat. 4 (3), 1029-1046 (1998).
  24. L. A. Lyusternik and V. I. Sobolev, A Short Course on Functional Analysis (Vysshaya Shkola, Moscow, 1982) [in Russian].
  25. E. V. Tabarintseva, “Estimating the Accuracy of a Method of Auxiliary Boundary Conditions in Solving an Inverse Boundary Value Problem for a Nonlinear Equation,” Sib. Zh. Vych. Mat. 21 (3), 291-310 (2018) [Numer. Anal. Appl. 11 (3), 236-255 (2018)].
  26. V. V. Vasin, “Modified Steepest Descent Method for Nonlinear Irregular Operator Equations,” Dokl. Akad. Nauk 462 (3), 264-267 (2015) [Dokl. Math. 91 (3), 300-303 (2015)].
  27. M. I. Shimelevich, “Algorithms for Calculating the Modulus of Continuity of the Inverse Operator and Its Modifications Using Monte Carlo Methods as Applied to Geoelectrics,” in Proc. Int. Conf. on Advanced Mathematics, Computations and Applications, Novosibirsk, Russia, July 1-5, 2019 (Novosib. Gos. Univ., Novosibirsk, 2019).
    https://elibrary.ru/item.asp?id=41893341 . Cited October 14, 2020.
  28. V. A. Zorich, Mathematical Analysis I (Fazis, Moscow, 1997; Springer, Berlin, 2004).
  29. I. M. Sobol’, Monte Carlo Numerical Methods (Nauka, Moscow, 1973) [in Russian].
  30. A. N. Shiryaev, Probability (MCCME, Moscow, 2007; Springer, New York, 2008).
  31. S. M. Ermakov, Monte Carlo Method and Related Topics (Nauka, Moscow, 1975) [in Russian].
  32. I. M. Varentsov, V. A. Kulikov, A. G. Yakovlev, and D. V. Yakovlev, “Possibilities of Magnetotelluric Methods in Geophysical Exploration for Ore Minerals,” Izv. Ross. Akad. Nauk, Fiz. Zemli, No. 3, 9-29 (2013) [Izv., Phys. Solid Earth 49 (3), 309-328 (2013)].
  33. A. N. Tikhonov, “Determining the Electrical Characteristics of Deep-Lying Layers in the Earth’s Crust,” Dokl. Akad. Nauk SSSR 73 (2), 295-297 (1950).
  34. H. Wiese, Geomagnetische Tiefentellurik (Geomagnet.Inst. Deutsch. Akad. Wiss., Berlin, 1965).
  35. U. Schmucker, Anomalies of Geomagnetic Variations in the Southwestern United States (Univ. California Press, Berkley, 1970).
  36. P. Weidelt, “The Inverse Problem of Geomagnetic Induction,” Zeitschrift für Geophysik 38, 257-289 (1972).
  37. M. N. Berdichevskii and M. S. Zhdanov, Interpretation of Earth’s Electromagnetic Field Anomalies (Nedra, Moscow, 1981) [in Russian].
  38. M. N. Berdichevskii and V. I. Dmitriev, Models and Methods of Magnetotellurics (Nauch. Mir, Moscow, 2009) [in Russian].
  39. M. S. Zhdanov, Theory of Inverse Problems and Regularization in Geophysics (Nauch. Mir, Moscow, 2007) [in Russian].
  40. G. W. Hohmann, “Three-Dimensional EM Modeling,” Geophys. Surv. 6, 27-53 (1983).
  41. V. I. Dmitriev, “Direct and Inverse Problems in Electromagnetic Sounding of Three-Dimensional Heterogeneous Medium,” Izv. Ross. Akad. Nauk, Fiz. Zemli, No. 3, 46-51 (2013) [Izv., Phys. Solid Earth 49 (3), 344-349 (2013)].
  42. V. I. Dmitriev, Electromagnetic Fields in Nonhomogeneous Media (Mosk. Gos. Univ., Moscow, 1969) [in Russian].
  43. V. I. Dmitriev and E. V. Zakharov, Integral Equation Method in Computational Electrodynamics (MAKS Press, Moscow, 2008) [in Russian].
  44. D. B. Avdeev, “The Integral Equation Method for the Solution of Direct Problems in Geoelectrics,” in Electromagnetic Studies of the Earth’s Interior (Nauch. Mir, Moscow, 2005), pp. 11—32.
  45. M. N. Berdichevskii and V. I. Dmitriev, “Inverse Problems of Magnetotellurics: A Modern Formulation,” Izv. Ross. Akad. Nauk, Fiz. Zemli, No. 4, 12-29 (2004) [Izv., Phys. Solid Earth 40 (4), 276-292 (2004)].
  46. V. V. Spichak, “Magnetotelluric Fields in Three-Dimensional Geoelectric Models,” (Nauch. Mir, Moscow, 1999) [in Russian].
  47. M. N. Yudin, “Calculation of the Magnetotelluric Field by the Grid Method in Three-Dimensional Inhomogeneous Media” in Marine Electromagnetic Studies (IZMIR AN SSSR, Moscow, 1980), pp. 96-101.
  48. M. N. Yudin, “Alternating Method for the Numerical Solution of Direct Geoelectric Problems,” in Mathematical Methods in Geoelectrics (IZMIR AN SSSR, Moscow, 1982), pp. 47-52.
  49. V. I. Dmitriev, “Two-Dimensional Inverse Problem of Magnetotelluric Sounding in a Nonhomogeneous Medium,” in Applied Mathematics and Informatics (MAKS Press, Moscow, 2017), Vol. 56, pp. 5-17.
  50. V. I. Dmitriev, “Unique Solvability of the Three-Dimensional Inverse Problem of Electromagnetic Sounding,” in Applied Mathematics and Informatics (MAKS Press, Moscow, 2018), Vol. 57, pp. 5-20.
  51. M. V. Dmitrieva, Numerical Modeling of Physical Processes in the Plasma of Tokamak Installations under the Influence of Electromagnetic Waves in the Alfvén Frequency Range , Candidate’s Dissertation in Mathematics and Physics (Keldysh Institute of Applied Mathematics, Moscow, 1985).
  52. S. C. Eisenstat, M. H. Schultz, and A. H. Sherman, “Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination,” SIAM J. Sci. Stat. Comput. 2 (2), 225-237.
  53. M. V. Dmitrieva, A. G. Elfimov, A. A. Ivanov, et al., Numerical Simulation of Alfvén Plasma Heating in a Torus , Preprint No. 51 (Keldysh Institute of Applied Mathematics, Moscow, 1988).

Published

03-11-2020

How to Cite

Шимелевич М. On the Method of Calculating the Modulus of Continuity of the Inverse Operator and Its Modifications With Application to Non-Linear Problems of Geoelectrics // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2020. 21. 350-372. doi 10.26089/NumMet.v21r430

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Section

Methods and algorithms of computational mathematics and their applications