Abstract
We construct memory-optimized and time-efficient parallel algorithms (and the corresponding programs) taking advantage of regularized modified \(\alpha \)-processes, namely the modified steepest descent method and the modified minimal residual method, for solving the nonlinear equation of the structural inverse gravimetry problem. Memory optimization relies on the block-Toeplitz structure of the Jacobian matrix. The algorithms are implemented on multicore CPUs and GPUs through the use of, respectively, OpenMP and NVIDIA CUDA technologies. We analyze the efficiency and speedup of the algorithms. In addition, we solve a model problem of gravimetry and conduct a comparative study regarding the number of iterations and computation time against algorithms based on conjugate gradient-type methods and the componentwise gradient method. The comparison demonstrates that the algorithms based on \(\alpha \)-processes perform better, reducing the number of iterations and the computation time by as much as 50%.
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References
Akimova, E.N., Martyshko, P.S., Misilov, V.E.: Algorithms for solving the structural gravity problem in a multilayer medium. Doklady Earth Sci. 453(2), 1278–1281 (2013). https://doi.org/10.1134/S1028334X13120180
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, no. 2, pp. 713–720 (2014)
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, p. 850063 (2015). https://doi.org/10.1063/1.4913118
Akimova, E.N., Martyshko, P.S., Misilov, V.E., Kosivets, R.A.: An efficient numerical technique for solving the inverse gravity problem of finding a lateral density. Appl. Math. Inf. Sci. 10(5), 1681–1688 (2016). https://doi.org/10.18576/amis/100506
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, no. 1, pp. 775–782 (2015)
Akimova, E.N., Misilov, V.E. Arguchinsky, M.S.: Memory efficient algorithm for solving the inverse problem of finding a density in a curvilinear layer. In: CEUR Workshop Proceedings, vol. 2076, pp. 1–8 (2018)
Akimova, E.N., Misilov, V.E., Tretyakov, A.I.: Regularized methods for solving nonlinear inverse gravity problem. In: EAGE Geoinformatics (2016). https://doi.org/10.3997/2214-4609.201600458
Akimova, E.N., Misilov, V.E., Tretyakov, A.I.: Optimized algorithms for solving the structural inverse gravimetry and magnetometry problems on GPUs. Commun. Comput. Inf. Sci. 753, 144–155 (2017). https://doi.org/10.1007/978-3-319-67035-5_11
Akimova, E.N., Vasin, V.V.: Stable parallel algorithms for solving the inverse gravimetry and magnetometry problems. Int. J. Eng. Model. - Univ. Split Croatia 17(1–2), 13–19 (2004)
Akimova, E.N., Vasin, V.V., Misilov, V.E.: Iterative Newton type algorithms and its applications to inverse gravimetry problem. Bull. South Ural State Univ. Ser. Math. Modell. Programm. Comput. Softw. 6(3), 26–37 (2013)
Bakushinskiy, A., Goncharsky, A.: Ill-Posed Problems: Theory and Applications. Springer, Heidelberg (1994). https://doi.org/10.1007/978-94-011-1026-6
Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces, International Series of Monographs in Pure and Applied Mathematics. The Macmillan Co., New York (1964)
Malkin, N.R.: On solution of inverse magnetic problem for one contact surface (the case of layered masses). In: DAN SSSR, Ser. A, no. 9, pp. 232–235 (1931)
Martyshko, P.S., Akimova, E.N., Misilov, V.E.: Solving the structural inverse gravity problem by the modified gradient methods. Izvestiya, Physics of the Solid Earth 52(5), 704–708 (2016)
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. Izvestiya Phys. Solid Earth 50(4), 508–513 (2014). https://doi.org/10.1134/S1069351314040090
Martyshko, P.S., Prutkin, I.L.: Technology of depth distribution of gravity field sources. Geophys. J 25(3), 159–168 (2003)
Numerov, B.V.: Interpretation of gravitational observations in the case of one contact surface. In: Doklady Akad. Nauk SSSR, pp. 569–574 (1930)
Vasin, V.V.: Irregular nonlinear operator equations: Tikhonov’s regularization and iterative approximation. J. Inverse Ill-Posed Probl. 21(1), 109–123 (2013). https://doi.org/10.1515/jip-2012-0084
Vasin, V.V.: Modified steepest descent method for nonlinear irregular operator equations. Doklady Math. 91(3), 300–303 (2015). https://doi.org/10.1134/S1064562415030187
Vasin, V.V.: Regularized modified \(\alpha \)-processes for nonlinear equations with monotone operators. Dokl. Math. 94(1), 361–364 (2016). https://doi.org/10.1134/S1064562416040062
Vasin, V.V., Skurydina, A.F.: Two-stage method of construction of regularizing algorithms for nonlinear Ill-posed problems. Proc. Steklov Inst. Math. 301, 173–190 (2018). https://doi.org/10.1134/S0081543818050152
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Akimova, E.N., Misilov, V.E., Tretyakov, A.I. (2019). Using Multicore and Graphics Processors to Solve The Structural Inverse Gravimetry Problem in a Two-Layer Medium by Means of \(\alpha \)-Processes. In: Sokolinsky, L., Zymbler, M. (eds) Parallel Computational Technologies. PCT 2019. Communications in Computer and Information Science, vol 1063. Springer, Cham. https://doi.org/10.1007/978-3-030-28163-2_20
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