DOI: https://doi.org/10.26089/NumMet.v24r426

Efficient algorithms for solving inverse gravimetry and magnetometry problem on graphics processors

Authors

  • Elena N. Akimova
  • Vladimir E. Misilov
  • Andrey I. Tretyakov

Keywords:

inverse gravimetry problem
inverse magnetometry problem
gradient methods
Toeplitz matrices
GPU
CUDA

Abstract

The work is devoted to algorithms for solving the inverse gravimetry problem of finding an interface between media from gravity data and the magnetometry problem for the case of an arbitrarily directed magnetization from magnetic data and their implementation on graphics processors. Based on the conjugate gradient method using the Toeplitz-block-Toeplitz structure of the matrix of integral operator derivatives, we construct the efficient modified algorithms for solving inverse gravimetry and magnetometry problems. A new componentwise method is elaborated for solving the inverse magnetometry problem for the case of an arbitrarily directed magnetization vector. Numerical experiments are carried out on the GPU to study the applicability and performance of the developed algorithms.

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Published

2023-11-08

Issue

Vol. 24 (2023): Issue 4.

Section

Methods and algorithms of computational mathematics and their applications

Author Biographies

Elena N. Akimova

N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS (IMM UB RAS)
• Leading Researcher
Ural Federal University (UrFU),
• Professor

Vladimir E. Misilov

N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS (IMM UB RAS)
• Researcher
Ural Federal University (UrFU),
• Associate Professor

Andrey I. Tretyakov

N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS (IMM UB RAS)
• Leading Programmer
Ural Federal University (UrFU),
• Senior Lecturer

References

  1. B. V. Numerov, “Interpretation of Gravity Observations in the Case of a Contact Surface,” Dokl. Akad. Nauk SSSR. Ser. A, No. 21, 569-574 (1930).
  2. N. R. Malkin, “On the Solution of the Inverse Magnetic Problem for the Case of a Contact Surface (the Case of Layered Masses),” Dokl. Akad. Nauk SSSR. Ser. A, No. 9, 232-235 (1931).
  3. V. N. Strakhov, “Inverse Problem of the Logarithmic Potential for a Contact Surface,” Dokl. Akad. Nauk SSSR 200 (4), 817-820 (1971).
  4. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Springer, Dordrecht, 1994).
    doi 10.1007/978-94-011-1026-6
  5. P. S. Martyshko and I. L. Prutkin, “Techniques for Separating Gravity Field Sources in Depth,” Geofyzicheskii Zhurnal 25 (3), 159-168 (2003).
  6. P. S. Martyshko, I. V. Ladovskii, and A. G. Tsidaev, “Construction of Regional Geophysical Models Based on the Joint Interpretation of Gravity and Seismic Data,” Fiz. Zemli, No. 11, 23-35 (2010) [Izv., Phys. Solid Earth 46 (11), 931-942 (2010)].
    doi 10.1134/S1069351310110030
  7. P. S. Martyshko, I. V. Ladovskii, N. V. Fedorova, et al., Theory and Methods for Complex Interpretation of Geophysical Data (Ural Branch Russ. Acad. Sci., Ekaterinburg, 2016).
  8. A. I. Kobrunov, E. N. Motryuk, and D. O. Lominskiy, “Method of Geological Environment Interpolation in a Fragmented Set of Data,” Fundamentalnye Issledovania, No. 2 (Part 24), 5340-5345 (2015).
  9. E. N. Akimova and V. V. Vasin, “Stable Parallel Algorithms for Solving the Inverse Gravimetry and Magnetometry Problems,” Int. J. Eng. Model. 17 (1-2), 13-19 (2004).
    http://gradst.unist.hr/Portals/9/docs/EM/EM-2004-NO_1-2/ENGMOD_2004_Vol-17_No_1-2_AKIMOVA.pdf . Cited October 19, 2023.
  10. E. N. Akimova, V. E. Misilov, A. F. Skurydina, and A. I. Tretyakov, “Gradient Methods for Solving Inverse Gravimetry and Magnetometry Problems on the Uran Supercomputer,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 16 (1), 155-164 (2015).
    doi 10.26089/NumMet.v16r116
  11. V. V. Vasin and A. F. Skurydina, “Two-Stage Method of Construction of Regularizing Algorithms for Nonlinear Ill-Posed Problems,” Tr. Inst. Mat. i Mech. UrO RAN 23 (1), 57-74 (2017) [Proc. Steklov Inst. Math. 301 (Suppl. 1), 173-190 (2018)].
    doi 10.1134/S0081543818050152
  12. A. I. Tretyakov, “Development of the Parallel Programs Complex for Solving the Inverse Gravimetric and Magnetometry Problems for Large Grids,” Vestn. Yuzhn. Ural. Gos. Univ. Ser. Vychisl. Mat. Inf. 11 (1), 57-78 (2022).
    doi 10.14529/cmse220104
  13. I. L. Prutkin, “The Solution of Three-Dimensional Inverse Gravimetric Problem in the Class of Contact Surfaces by the Method of Local Corrections,” Izv. Akad. Nauk SSSR, Fiz. Zemli, No. 1, 67-77 (1986) [Izv., Phys. Solid Earth 22 (1), 49-60 (1986)].
  14. 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)].
    doi 10.1134/S1064562415030187
  15. E. N. Akimova, V. E. Misilov, and A. I. Tretyakov, “Optimized Algorithms for Solving Structural Inverse Gravimetry and Magnetometry Problems on GPUs,” in Communications in Computer and Information Science (Springer, Cham, 2017), Vol. 753, pp. 144-155.
    doi 10.1007/978-3-319-67035-5_11
  16. E. Akimova and V. Misilov, “A Fast Componentwise Gradient Method for Solving Structural Inverse Gravity Problem,” in Proc. 15th Int. Multidisciplinary Scientific GeoConference SGEM, Albena, Bulgaria, June 18-24, 2015 (Bulgarian Acad. Sci., Albena, 2015), Vol. 3, pp. 775-782.
  17. E. N. Akimova, V. E. Misilov, and A. I. Tretyakov, “Modified Componentwise Gradient Method for Solving Structural Magnetic Inverse Problem,” in Communications in Computer and Information Science (Springer, Cham, 2018), Vol. 910, pp. 162-173.
    doi 10.1007/978-3-319-99673-8_12
  18. V. V. Vasin, Fundamentals of the Theory of Ill-Posed Problems (Siberian Branch Russ. Acad. Sci., Novosibirsk, 2020) [in Russian].
  19. J. Nocedal and S. J. Wright, Numerical Optimization (Springer, New York, 2006).
  20. V. E. Misilov, “On Solving the Structural Inverse Magnetic Problem of Finding a Contact Surface in the Case of Arbitrary Directed Magnetization,”
    https://www.earthdoc.org/content/papers/10.3997/2214-4609.201600473 . Cited October 19, 2023.
  21. E. N. Akimova and V. E. Misilov, “Conjugate Gradient Method for Solving the Inverse Gravimetry Problem in Multilayered Medium: Parallel Implementation,” AIP Conf. Proc. 2164 (1), Article Number 120001 (2019).
    doi 10.1063/1.5130861. Cited October 19, 2023

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Copyright (c) 2023 Е. Н. Акимова, В. Е. Мисилов, А. И. Третьяков

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