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Optimized Algorithms for Solving Structural Inverse Gravimetry and Magnetometry Problems on GPUs

  • Elena N. AkimovaEmail author
  • Vladimir E. Misilov
  • Andrey I. Tretyakov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 753)

Abstract

In this article, we construct new variants of iteratively regularized linearized gradient-type methods for solving structural inverse gravimetry and magnetometry problems, namely the regularized conjugate gradient method, the modified regularized conjugate gradient method, and the hybrid regularized conjugate gradient method.

The main idea of the modification is to calculate the Jacobian matrix of the integral operator at a fixed point, without updating it during the entire iteration process.

We also developed memory-optimized and time-efficient parallel algorithms and programs on the basis of the constructed modified methods. The memory optimization uses the block-Toeplitz structure of the Jacobian matrix. The algorithms were implemented on GPUs using the NVIDIA CUDA technology. We performed an efficiency and speedup analysis, and solved a model problem with synthetic disturbed data.

Keywords

Nonlinear gradient-type methods Parallel algorithms Gravimetry and magnetometry problems Toeplitz matrix GPU 

References

  1. 1.
    Akimova, E.N., Martyshko, P.S., Misilov, V.E.: Algorithms for solving the structural gravity problem in a multilayer medium. Dokl. Earth Sci. 453(2), 1278–1281 (2013). doi: 10.1134/S1028334X13120180 CrossRefzbMATHGoogle Scholar
  2. 2.
    Akimova, E.N., Martyshko, P.S., Misilov, V.E.: Parallel algorithms for solving structural inverse magnetometry problem on multucore and graphics processors. In: Proceedings of 14th International multidisciplinary scientific GeoConference SGEM 2014, vol. 1(2), pp. 713–720 (2014)Google Scholar
  3. 3.
    Akimova, E.N., Martyshko, P.S., Misilov, V.E.: A fast parallel gradient algorithm for solving structural inverse gravity problem. In: AIP Conference Proceedings, vol. 1648, 850063 (2015). doi: 10.1063/1.4913118
  4. 4.
    Akimova, E.N., Misilov, V.E.: A fast componentwise gradient method for solving structural inverse gravity problem. In: Proceedings of 15th International Multidisciplinary Scientific GeoConference SGEM 2015, vol. 3(1), pp. 775–782 (2015)Google Scholar
  5. 5.
    Bakushinskiy, A., Goncharsky, A.: Ill-Posed Problems: Theory and Applications. Springer Science & Business Media, Netherlands (1994). doi: 10.1007/978-94-011-1026-6 CrossRefGoogle Scholar
  6. 6.
    Malkin, N.R.: On solution of inverse magnetic problem for one contact surface (the case of layered masses). DAN SSSR, Ser. A, vol. 9, pp. 232–235 (1931)Google Scholar
  7. 7.
    Martyshko, P.S., Akimova, E.N., Misilov, V.E.: Solving the structural inverse gravity problem by the modified gradient methods. Izv. Phys. Solid Earth 52(5), 704–708 (2016). doi: 10.1134/S1069351316050098 CrossRefzbMATHGoogle Scholar
  8. 8.
    Martyshko, P.S., Fedorova, N.V., Akimova, E.N., Gemaidinov, D.V.: Studying the structural features of the lithospheric magnetic and gravity fields with the use of parallel algorithms. Izv. Phys. Solid Earth 50(4), 508–513 (2014). doi: 10.1134/S1069351314040090 CrossRefGoogle Scholar
  9. 9.
    Martyshko, P.S., Prutkin, I.L.: Technology of depth distribution of gravity field sources. Geophys. J. 25(3), 159–168 (2003)Google Scholar
  10. 10.
    Numerov, B.V.: Interpretation of gravitational observations in the case of one contact surface. Doklady Akad. Nauk SSSR, pp. 569–574 (1930)Google Scholar
  11. 11.
    Vasin, V.V.: Irregular nonlinear operator equations: Tikhonov’s regularization and iterative approximation. Inverse Ill-Posed Prob. 21(1), 109–123 (2013). doi: 10.1515/jip-2012-0084 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Vasin, V.V.: Modified steepest descent method for nonlinear irregular operator equations. Dokl. Math. 91(3), 300–303 (2015). doi: 10.1134/S1064562415030187 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of RASYekaterinburgRussia
  2. 2.Ural Federal UniversityYekaterinburgRussia

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