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Scalable Eigen-Analysis Engine for Large-Scale Eigenvalue Problems

  • Tetsuya Sakurai
  • Yasunori Futamura
  • Akira Imakura
  • Toshiyuki Imamura
Chapter

Abstract

Our project aims to develop a massively parallel Eigen-Supercomputing Engine for post-petascale systems. Our Eigen-Engines are based on newly designed algorithms that are suited to the hierarchical architecture in post-petascale systems and show very good performance on petascale systems including K computer. In this paper, we introduce our Eigen-Supercomputing Engines: z-Pares and EigenExa and their performance.

References

  1. 1.
    Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 1, 52–55 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Asakura, J., Sakurai, T., Tadano, H., Ikegami, T., Kimura, K.: A numerical method for polynomial eigenvalue problems using contour integral. Jpn. J. Indust. Appl. Math. 27, 73–90 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beyn, W.-J.: An integral method for solving nonlinear eigenvalue problems. Linear Algebra Appl. 436, 3839–3863 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bischof, C., et al.: A framework for symmetric band reduction. ACM Trans. Math. Softw. (TOMS) 26, 581–601 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bischof, C., et al.: Algorithm 807: the SBR toolbox – software for successive band reduction. ACM Trans. Math. Softw. (TOMS) 26, 602–616 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blackford, L.S., et al.: ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia (1997)CrossRefGoogle Scholar
  7. 7.
    Chen, H., Imakura, A., Sakurai, T.: Improving backward stability of Sakurai-Sugiura method with balancing technique in polynomial eigenvalue problem. Appl. Math. 62, 357–375 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, H., Maeda, Y., Imakura, A., Sakurai, T., Tisseur, F.: Improving the numerical stability of the Sakurai-Sugiura method for quadratic eigenvalue problems. JSIAM Lett. 9, 17–20 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
  10. 10.
  11. 11.
    Fukaya, T., Imamura, T.: Performance evaluation of the EigenExa Eigensolver on Oakleaf-FX: Tridiagonalization Versus Pentadiagonalization. In: Proceedings of the 2015 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW) (PDSEC 2015), pp. 960–969 (2015)Google Scholar
  12. 12.
    Futamura, Y., Tadano, H., Sakurai, T.: Parallel stochastic estimation method of eigenvalue distribution. JSIAM Lett. 2, 127–130 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Futamura, Y., Sakurai, T., Furuya, S., Iwata, J.-I.: Efficient algorithm for linear systems arising in solutions of eigenproblems and its application to electronic-structure calculations. In: Proceedings of the 10th International Meeting on High-Performance Computing for Computational Science (VECPAR 2012), pp. 226–235 (2013)Google Scholar
  14. 14.
    Fukaya, T., Nakatsukasa, Y., Yanagisawa, Y., Yamamoto, Y.: CholeskyQR2: a simple and communication-avoiding algorithm for computing a tall-skinny QR factorization on a large-scale parallel system. In: Proceedings of the 5th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (ScalA14), pp. 31–38 (2014)Google Scholar
  15. 15.
    Güttel, S., Polizzi, E., Tang, T., Viaud, G.: Zolotarev quadrature rules and load balancing for the FEAST eigensolver. SIAM J. Sci. Comput. 37, A2100–A2122 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hasegawa, T., Imakura, A., Sakurai, T.: Recovering from accuracy deterioration in the contour integral-based eigensolver. JSIAM Lett. 8, 1–4 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hasegawa, Y., et al.: First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the K computer. In: Proceedings of the 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, Article No. 1 (2011)Google Scholar
  18. 18.
    Hirota, Y., Yamada, S., Imamura, T., Sasa, N., Machida, M.: Performance of quadruple precision eigenvalue solver libraries QPEigenK & QPEigenG on the K computer. In: Proceedings of the International Supercomputing Conference (ISC’16). HPC in Asia Poster Session (2016)Google Scholar
  19. 19.
    Ide, T., Toda, K., Futamura, Y., Sakurai, T.: Highly parallel computation of eigenvalue analysis in vibration for automatic transmission using Sakurai-Sugiura method and K computer. SAE Technical Paper, 2016-01-1378 (2016)Google Scholar
  20. 20.
    Ikegami, T., Sakurai, T., Nagashima, U.: A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method. J. Comput. Appl. Math. 233, 1927–1936 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ikegami, T., Sakurai, T.: Contour integral eigensolver for non-Hermitian systems: a Rayleigh-Ritz-type approach. Taiwan. J. Math. 14, 825–837 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Imakura, A., Du, L., Sakurai, T.: A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems. Appl. Math. Lett. 32, 22–27 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Imakura, A., Du, L., Sakurai, T.: Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems. Numer. Algorithms 71, 103–120 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Imakura, A., Du, L., Sakurai, T.: Relationships among contour integral-based methods for solving generalized eigenvalue problems. Jpn. J. Ind. Appl. Math. 33, 721–750 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Imakura, A., Sakurai, T.: Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems. Numer. Algorithms 75, 413–433 (2017)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Imakura, A., Sakurai, T.: Block SS–CAA: a complex moment-based parallel nonlinear eigensolver using the block communication-avoiding Arnoldi procedure. Parallel Comput. 74, 34–48 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Imakura, A., Futamura, Y., Sakurai, T.: Structure-preserving block SS–Hankel method for solving Hermitian generalized eigenvalue problems. In Proceedings of 12th International Conference on Parallel Processing and Applied Mathematics (PPAM2017) (2017, accepted)Google Scholar
  28. 28.
    Imakura, A., Futamura, Y., Sakurai, T.: Structure-preserving technique in the block SS–Hankel method for solving Hermitian generalized eigenvalue problems. In: Wyrzykowski, R., Dongarra, J., Deelman, E., Karczewski, K. (eds.) Parallel Processing and Applied Mathematics. PPAM 2017. Lecture Notes in Computer Science, vol. 10777, pp. 600–611. Springer, Cham (2017)Google Scholar
  29. 29.
    Imamura, T., et al.: Current status of EigenExa, high-performance parallel dense eigensolver. In: EPASA2018 (2018)Google Scholar
  30. 30.
    Imamura, T., et al.: Development of a high performance eigensolver on the Peta-Scale next generation supercomputer system. Prog. Nucl. Sci. Technol. 2, 643–650 (2011)CrossRefGoogle Scholar
  31. 31.
    Iwase, S., Futamura, Y., Imakura, A., Sakurai, T., Ono, T.: Efficient and Scalable Calculation of Complex Band Structure using Sakurai-Sugiura Method, In SC’17 proceeding of the International Conference for High Performance Computing, Networking, Storage and Analysis, 17, 2017 (accepted).Google Scholar
  32. 32.
    Kestyn, J., Kalantzis, V., Polizzi, E., Saad, Y.: PFEAST: a high performance sparse eigenvalue solver using distributed-memory linear solvers, In SC’16 proceeding of the International Conference for High Performance Computing, Networking, Storage and Analysis, vol. 16 (2016)Google Scholar
  33. 33.
    Kravanja, P., Sakurai, T., van Barel, M.: On locating clusters of zeros of analytic functions. BIT 39, 646–682 (1999)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Maeda, Y., Futamura, Y., Sakurai, T.: Stochastic estimation method of eigenvalue density for nonlinear eigenvalue problem on the complex plane. JSIAM Lett. 3, 61–64 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Orii, S., Imamura, T., Yamamoto, Y.: Performance Prediction of Large-Scale Parallel Computing by Regression Model with Non-Negative Model Parameters, IPSJ SIG Technical Reports (High Performance Computing), Vol. 2016-HPC-155, No. 9, pp. 1–9 (2016). (in Japanese)Google Scholar
  36. 36.
    Polizzi, E.: A density matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B 79, 115112 (2009)CrossRefGoogle Scholar
  37. 37.
    Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 119–128 (2003)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sakurai, T., Hayakawa, K., Sato, M., Takahashi, D.: A parallel method for large sparse generalized eigenvalue problems by OmniRPC in a grid environment. Lect. Notes Comput. Sci. 3732, 1151–1158 (2005)CrossRefGoogle Scholar
  39. 39.
    Sakurai, T., Tadano, H.: CIRR: a Rayleigh-Ritz type method with counter integral for generalized eigenvalue problems. Hokkaido Math. J. 36, 745–757 (2007)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Sakurai, T., Kodaki, Y., Tadano, H., Takahashi, D., Sato, M., Nagashima, U.: A parallel method for large sparse generalized eigenvalue problems using a grid RPC system. Fut. Gen. Comput. Syst. Appl. Distrib. Grid Comput. 24, 613–619 (2008)CrossRefGoogle Scholar
  41. 41.
    Sakurai, T., Asakura, J., Tadano, H., Ikegami, T.: Error analysis for a matrix pencil of Hankel matrices with perturbed complex moments. JSIAM Lett. 1, 76–79 (2009)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Sakurai, T., Futamura, Y., Tadano, H.: Efficient parameter estimation and implementation of a contour integral-based eigensolver. J. Algorithms Comput. Tech. 7, 249–269 (2013)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Shimizu, N., Utsuno, Y., Futamura, Y., Sakurai, T.: Stochastic estimation of nuclear level density in the nuclear shell model: an application to parity-dependent level density in 58 Ni. Phys. Lett. B 753, 13–17 (2016)CrossRefGoogle Scholar
  44. 44.
    Tang, P.T.P., Polizzi, E.: FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection. SIAM J. Matrix Anal. Appl. 35, 354–390 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Tisseur, F., et al.: A Parallel Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem on Distributed Memory Architectures. SIAM J. Sci. Comput. 20, 2223–2236 (1999)MathSciNetCrossRefGoogle Scholar
  46. 46.
    van Barel, M., Kravanja, P.: Nonlinear eigenvalue problems and contour integrals. J. Comput. Appl. Math. 292, 526–540 (2016)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Yamada, S., et al.: High-Performance Computing for Exact Numerical Approaches to Quantum Many-Body Problems on the Earth Simulator (SC06) (2006)Google Scholar
  48. 48.
    Yamazaki, I., Ikegami, T., Tadano, H., Sakurai, T.: Performance comparison of parallel eigensolvers based on a contour integral method and a Lanczos method. Parallel Comput. 39, 280–290 (2013)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Yin, G.: A randomized FEAST algorithm for generalized eigenvalue problems, arXiv:1612.03300 [math.NA] (2016)Google Scholar
  50. 50.
    Yin, G., Chan, R.H., Yeung, M.-C.: A FEAST algorithm for generalized non-Hermitian eigenvalue problems. Numer. Linear Algebra Appl. 24, e2092 (2017)CrossRefGoogle Scholar
  51. 51.
    Yokota, S., Sakurai, T.: A projection method for nonlinear eigenvalue problems using contour integrals. JSIAM Lett. 5, 41–44 (2013)MathSciNetCrossRefGoogle Scholar
  52. 52.
    z-Pares: Parallel Eigenvalue Solver. http://zpares.cs.tsukuba.ac.jp/

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Tetsuya Sakurai
    • 1
  • Yasunori Futamura
    • 1
  • Akira Imakura
    • 1
  • Toshiyuki Imamura
    • 2
  1. 1.University of TsukubaTsukubaJapan
  2. 2.RIKEN Center for Computational ScienceKobeJapan

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