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GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method

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Abstract

In this paper, we introduce some strategies to improve the efficiency and scalability of the generalized conjugate gradient algorithm and build a package GCGE for solving large scale eigenvalue problems. This method is the combination of damping idea, subspace projection method and inverse power algorithm with dynamic shifts. To reduce the dimensions of projection subspaces, a moving mechanism is developed when the number of desired eigenpairs is large. The numerical methods, implementing techniques and the structure of the package are presented. Plenty of numerical results are provided to demonstrate the efficiency, stability and scalability of the concerned eigensolver and the package GCGE for computing many eigenpairs of large symmetric matrices arising from applications.

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  1. https://sparse.tamu.edu.

References

  • Amestoy, P.. R., Buttari, A., L’Excellent, J.. Y., Mary, T.: Performance and scalability of the block low-rank multifrontal factorization on multicore architectures. ACM Trans. Math. Software 45(1), 21–226 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  • Amestoy, Patrick. R., Duff, Iain S., Koster, Jacko, L’Excellent, Jean Yves: fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal on Matrix Analysis & Applications 23(1), 15–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., GreenBaum, A., Hammarling, S., Mckenney, A., Sorensen, D.: LAPACK Users’ Guide. SIAM (1999)

  • Bai, Zhaojun, Demmel, James, Dongarra, Jack, Ruhe, Axel, van der Vorst, Henk: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Vol. 11. SIAM (2000)

  • Balay, Satish, Gropp, William D., McInnes, Lois Curfman, Smith, Barry F.: Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing. . Birkhäuser Press (1997)

    MATH  Google Scholar 

  • Duersch, Meiyue, amd Shao, Jed A., Yang, Chao: Robust and Efficient Implementation of LOBPCG. SIAM Journal on Scientific Computing 40(5), 655–676 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Falgout, Robert D , Jones, Jim E, Yang, Ulrike Meier: The design and implementation of hypre, a library of parallel high performance preconditioners. In Numerical Solution of Partial Differential Equations on Parallel Computers. Springer, 267–294 (2006)

  • Hernandez, Vicente, Roman, Jose E., Vidal, Vicente: SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software 31(3), 351–362 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Hetmaniuk, U., Lehoucq, R.: Basis selection in LOBPCG. J. Comput. Phys. 218(1), 324–332 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Knyazev, Andrew V.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM Journal on Scientific Computing 23(2), 517–541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Knyazev, Andrew V., Argentati, Merico E., Lashuk, Ilya, Ovtchinnikov, Evgueni E.: Block locally optimal preconditioned eigenvalue xolvers (BLOPEX) in HYPRE and PETSc. SIAM Journal on Scientific Computing 29(5), 2224–2239 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Knyazev, Andrew V., Neymeyr, Klaus: Efficient solution of symmetric eigenvalue problems using multigrid preconditioners in the locally optimal block conjugate gradient method. Electronic Transactions on Numerical Analysis 15(2003), 38–55 (2003)

    MathSciNet  MATH  Google Scholar 

  • Kronik, Leeor, Makmal, Adi, Tiago, Murilo L., Alemany, M..M..G., Jain, Manish, Huang, Xiangyang, Saad, Yousef, Chelikowsky, James R.: PARSEC – the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nano-structures. Physica Status Solidi 243(5), 1063–1079 (2006)

    Article  Google Scholar 

  • Li, Yu., Xie, Hehu, Ran, Xu., You, Chun’Guang., Zhang, Ning: A parallel generalized conjugate gradient method for large scale eigenvalue problems. CCF Transactions on High Performance Computing 2(2020), 111–122 (2020)

    Article  Google Scholar 

  • Natan, Amir, Benjamini, Ayelet, Naveh, Doron, Kronik, Leeor, Tiago, Murilo L., Beckman, Scott P., Chelikowsky, James R.: Real-space pseudopotential method for first principles calculations of general periodic and partially periodic systems. Physical Review B 78(7), 75–109 (2008)

    Article  Google Scholar 

  • Saad, Youcef : Numerical Methods for Large Eigenvalue Problems. Vol. 158. SIAM (1992)

  • Saad, Yousef, Chelikowsky, James R., Shontz, Suzanne M.: Numerical methods for electronic structure calculations of materials. SIAM Rev. 52(1), 3–54 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Stewart, G.W.: Block Gram-Schmidt Orthogonalization. SIAM Journal on Entific Computing 31(1), 761–775 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Yokozawa, Takuya, Takahashi, Daisuke, Boku, Taisuke, Sato, Mitsuhisa: Efficient parallel implementation of classical Gram-Schmidt orthogonalization using matrix multiplication. In Proceedings of Fourth International Workshop on Parallel matrix Algorithms and Applications (PMAA’06). 37–38 (2006)

  • Zhang, Linbo: A parallel algorithm for adaptive local refinement of tetrahedral meshes using bisection. Numerical Mathematics: Theory, Methods and Applications 2(2009), 65–89 (2009)

    MathSciNet  MATH  Google Scholar 

  • Zhang, N.Y., Li, H.X., Ran, X., You, C.: A generalized conjugate gradient method for eigenvalue problems. Sci. Sin. Math. 50(12), 1–24 (2020)

    Google Scholar 

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Authors and Affiliations

  1. Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, 300222, China

    Yu Li

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zijing Wang & Hehu Xie

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Zijing Wang & Hehu Xie

Authors
  1. Yu Li
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  2. Zijing Wang
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  3. Hehu Xie
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Correspondence to Hehu Xie.

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All authors declare that they have no conflict of interest.

Additional information

This research is supported partly by National Key R &D Program of China 2019YFA0709600, 2019YFA0709601, Science Challenge Project (No. TZ2016002), the National Center for Mathematics and Interdisciplinary Science, CAS, and Tianjin Education Commission Scientific Research Plan (2017KJ236).

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Li, Y., Wang, Z. & Xie, H. GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method. CCF Trans. HPC 5, 171–190 (2023). https://doi.org/10.1007/s42514-023-00135-1

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