Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
Cybernetics And Systems Analysis
International Theoretical Science Journal
Volume 59 >>> № 3 MAY — JUNE 2023
-->

UDC 519.6
О.М. Khimich1, O.V. Popov2, O.V. Chistyakov3, V.O. Kokhanovskyi4


1 V.M. Glushkov Institute of Cybernetics,
National Academy of Sciences of Ukraine,
Kyiv, Ukraine

 khimich505@gmail.com

2 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

 alex50popov@gmail.com

3 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

 alexej.chystyakov@gmail.com

4 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute,” Kyiv, Ukraine

 kokhanovstyy@gmail.com

ADAPTIVE ALGORITHMS FOR SOLVING EIGENVALUE PROBLEMS
IN A VARIABLE COMPUTER ENVIRONMENT OF SUPERCOMPUTERS

Abstract. The authors propose software for the analysis and solution of the algebraic eigenvalue problem using a MIMD computer with GPUs, which includes parallel algorithms and programs with the functions of adaptive configuration of the variable computer environment (multilevel parallelism, variable topology of interprocessor communications, mixed word length, caching, etc.) on the mathematical properties of the problem identified in the computer and the architectural features of the computer to ensure the reliability of the solution results with the efficient use of computing resources.

Keywords: algebraic eigenvalue problem (APEV), changeable computing environment, adaptive algorithms, gradient conjugation method, subspace iteration method, multi-core computers of MIMD-architecture with graphic processors.


full text

REFERENCES

  1. Sergienko I.V, Molchanov I.N, Khimich A.N. Intelligent technologies of high-performance computing. Cybernetics and Systems Analysis. 2010. Vol. 46, N 5. P. 833–844. https://doi.org/ 10.1007/s10559-010-9265-3.

  2. Sergienko I.V., Khimich O.M. Mathematical modeling: From Melms to Exaflops. Visn. NAS of Ukraine. 2019. N 8. P. 37–50. https://doi.org/10.15407/visn2019.08.037.

  3. Dongarra J, Beckman P., Moore T. et al. The International Exascale Software Project roadmap. International Journal of High Performance Computing Applications. 2011. Vol. 25, Iss. 1. P. 3–60. https://doi.org/10.1177/1094342010391989.

  4. Khimich A.N., Molchanov I.N., Popov A.V., Chistyakova T.V., Yakovlev M.F. Parallel algorithms for solving problems of computational mathematics [in Russian]. Kyiv: Nauk. dumka, 2008. 247 p.

  5. Popov O.V., Rudych O.V., Chistyakov O.V. A multilevel model of parallel computations for linear algebra problems. Programming problems. 2018. N 2–3. P. 83–92.

  6. Popov A.V., Chistyakov O.V. On the effectiveness of algorithms with multilevel parallelism. Physico-mathematical modeling and information technologies. 2021. Iss. 33. P. 133–137. https://doi.org/10.15407/fmmit2021.33.133.

  7. Chistyakov O.V. On the peculiarities of software development for solving eigenvalue problems with sparse matrices on hybrid computers. Computer mathematics. 2015. N 1. P. 75–84.

  8. Khimich A.N., Decret V.A., Popov A.V., Chistyakov A.V. Numerical study of the stability of composite materials on hybrid architecture computers. Problemy upravleniya i informatiki. 2018. N 4. P. 73–88.

  9. Khimich O.M., Chistyakov O.V., Brusnikin V.M. A hybrid algorithm of the generalized conjugate gradient method for the eigenvalue problem with symmetric sparse matrices. Mathematical machines and systems. 2015. N 3. P. 3–13.

  10. Khimich A.N., Popov A.V., Chistyakov O.V. Hybrid algorithms for solving the algebraic eigenvalue problem with sparse matrices. Cybernetics and Systems Analisys. 2017. Vol. 53, N. 6. P. 937–949. https://doi.org/10.1007/s10559-017-9996-5.

  11. Khimich A.N., Popov A.V., Chistyakov O.V., Sidoruk V.A. A parallel algorithm for solving a partial eigenvalue problem for block-diagonal bordered matrices. Cybernetics and Systems Analisys. 2020. Vol. 56, N. 6. P. 913–923. https://doi.org/10.1007/s10559-020-00311-z .

  12. Molchanov I.N., Popov A.V., Khimich A.N. Algorithm to solve the partial eigenvalue problem for large profile matrices. Cybernetics and Systems Analysis. 1992. Vol. 28, N 2. P. 281–286. https://doi.org/10.1007/BF01126215.

  13. Message Passing Interface Forum. MPI: A Message-passing Interface Standard. International Journal of Supercomputer Applications. 1994. Vol. 8 (3/4). Р. 157–416.

  14. OpenMP. Architecture Review Board. URL: https://developer.nvidia.com/ cuda-downloads .

  15. Sydoruk V.A., Yershov P.S., Bogurskyi D.O., Marochkanych O.R. Intellectualization of calculations for the problems of mathematical modeling of complex processes and objects. Computer mathematics. 2019. N 1. P. 143–150.

  16. Molchanov I.N., Levchenko I.S., Fedonyuk N.N., Khimich A.N., Chistyakova T.V. Numerical simulation of stress concentration in an elastic half-space with a two-layer inclusion. Prikladnaya mekhanika. 2002. 38. N 3. P. 65–71.

  17. Lurie A.I. Theory of elasticity. Springer-Verlag, 2005. 1050 p.

  18. Nikolaevskaja E.A., Chimich A.N, Chistyakova T.V. Programming with multiple precision. Ser. Studies in Computational Intelligence. Vol. 397. Berlin; Heidelberg: Springer, 2012. 234 p.

  19. Khimich О.М., Chistyakova T.V, Sidoruk V.A., Yershov P.S. Adaptive computer technologies for solving problems of computational and applied mathematics. Cybernetics and Systems Analysis. 2021. Vol. 57, N 6. P. 990–997. https://doi.org/10.1007/s10559-021-00424-z .

  20. Buttari A., Langou J., Kurzak J., Dongarra J. A class of parallel tiled linear algebra algorithms for multicore architectures. Parallel Computing. 2009. Vol. 35, Iss. 1. P. 38–53. https://doi.org/10.1016/j.parco.2008.10.002.

  21. Pissanetzky S. Sparse matrix technology. 1st ed. Elsevier, 1984. 312 p.

  22. Saad Y. Iterative methods for sparse linear systems. 2nd edition. SIAM, 2003. 547 p.

  23. Popov A.V. Parallel algorithms for solving linear systems with sparse symmetric matrices. Programming problems. 2008. N 2–3. P. 111–118.

  24. George A., Liu D. Numerical solution of large sparse systems of equations [Russian translation]. Moscow: Mir, 1984. 334 p.

  25. Parlet B.N. The symmetric eigenvalue problem. SIAM, 1998. 391 p. https://doi.org/10.1137/1.9781611971163 .

  26. Popov O.V., Rudych O.V. Towards solving systems of linear equations on hybrid architecture computers. Mathematical and computer modeling. Ser. Physical and mathematical sciences. 2017. Iss. 15. P. 158–164.

  27. SKIT supercomputer complex. URL: http://icybcluster.org.ua .

  28. The SuiteSparse Matrix Collection. URL: cise.ufl.edu/research/sparse/matrices .




Editorial Board Announcements Abstracts Authors Archive
about scope office editorial board instructions for authors subscriptions contacts
© 2023 Kibernetika.org. All rights reserved.