Efficient Contour Integral-based Eigenvalue Computation Using an Iterative Linear Solver with Shift-Invert Preconditioning

ABSTRACT

Contour integral-based (CI) eigenvalue solvers are one of the efficient and robust approaches for sparse eigenvalue problems. They have attracted attention owing to their inherent parallelism. For implementing a CI eigensolver, the inner linear systems arising in the algorithm need to be solved using an efficient method. One widely-used method is to use a sparse direct linear solver provided by a well-established numerical library; it is numerically robust and presents good load balancing of parallel execution of the CI eigensolver. However, owing to high total computational and memory cost, the performance of the direct solver approach is suboptimal. In this study, we propose an alternative method that utilizes a block Krylov iterative linear solver and shift-invert preconditioning that can take advantage of the shift-invariance of the block Krylov subspace. Our approach adaptively sets a preconditioning parameter according to the number of parallel processes to reduce the iteration counts. Several numerical examples confirm that our method outperforms the direct solver approach.

References

1. [n.d.]. CONQUEST: Linear Scaling DFT. http://www.order-n.org/.
2. [n.d.]. ELSES matrix library. http://www.elses.jp/matrix/.
3. [n.d.]. FEAST Eigenvalue Solver. http://www.ecs.umass.edu/~polizzi/feast/.
4. [n.d.]. MUMPS: MUltifrontal Massively Parallel sparse direct Solver. http://mumps.enseeiht.fr/.
5. [n.d.]. z-Pares: Parallel Eigenvalue Solver. https://zpares.cs.tsukubaa.ac.jp.
6. Timothy A. Davis and Yifan Hu. 2011. The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1, Article 1 (Dec. 2011), 25 pages. https://doi.org/10.1145/2049662.2049663
7. Yasunori Futamura, Tetsuya Sakurai, Shinnosuke Furuya, and Jun-Ichi Iwata. 2013. Efficient Algorithm for Linear Systems Arising in Solutions of Eigenproblems and Its Application to Electronic-Structure Calculations. In High Performance Computing for Computational Science - VECPAR 2012, Michel Daydé, Osni Marques, and Kengo Nakajima (Eds.). Lecture Notes in Computer Science, Vol. 7851. Springer Berlin Heidelberg, 226–235. https://doi.org/10.1007/978-3-642-38718-0_23
8. M.J. Gillan, D.R. Bowler, A.S. Torralba, and T. Miyazaki. 2007. Order-N first-principles calculations with the conquest code. Computer Physics Communications 177, 1 (2007), 14–18. https://doi.org/10.1016/j.cpc.2007.02.075 Proceedings of the Conference on Computational Physics 2006.
9. Gene H. Golub and Charles F. Van Loan. 2013. Matrix Computations(4th ed.). Johns Hopkins University Press, Baltimore, MD.
10. Roger G. Grimes, John G. Lewis, and Horst D. Simon. 1994. A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems. SIAM J. Matrix Anal. Appl. 15, 1 (1994), 228–272. https://doi.org/10.1137/S0895479888151111
11. Martin H. Gutknecht and Thomas Schmelzer. 2009. The block grade of a block Krylov space. Linear Algebra Appl. 430, 1 (2009), 174–185. https://doi.org/10.1016/j.laa.2008.07.008
12. Takeo Hoshi, Hiroto Imachi, Akiyoshi Kuwata, Kohsuke Kakuda, Takatoshi Fujita, and Hiroyuki Matsui. 2019. Numerical aspect of large-scale electronic state calculation for flexible device material. Japan Journal of Industrial and Applied Mathematics 36 (2019), 685–698. https://doi.org/10.1007/s13160-019-00358-2
13. Takanori Ide, Yuto Inoue, Yasunori Futamura, and Tetsuya Sakurai. 2017. Highly Parallel Computation of Generalized Eigenvalue Problem in Vibration for Automatic Transmission of Vehicles Using the Sakurai–Sugiura Method and Supercomputers. In Mathematical Analysis of Continuum Mechanics and Industrial Applications, Hiromichi Itou, Masato Kimura, Vladimír Chalupecký, Kohji Ohtsuka, Daisuke Tagami, and Akira Takada (Eds.). Springer Singapore, Singapore, 207–218.
14. Tsutomu Ikegami and Tetsuya Sakurai. 2010. Contour integral eigensolver for non-Hermitian systems: a Rayleigh-Ritz-type approach. Taiwanese J. Math. 14(2010), 825–837.
15. Tsutomu Ikegami, Tetsuya Sakurai, and Umpei Nagashima. 2010. A Filter Diagonalization for Generalized Eigenvalue Problems Based on the Sakurai-Sugiura Projection Method. J. Comput. Appl. Math. 233, 8 (Feb. 2010), 1927–1936. https://doi.org/10.1016/j.cam.2009.09.029
16. Akira Imakura, Lei Du, and Tetsuya Sakurai. 2014. A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems. Applied Mathematics Letters 32 (2014), 22–27. https://doi.org/10.1016/j.aml.2014.02.007
17. Akira Imakura, Lei Du, and Hiroto Tadano. 2013. A Weighted Block GMRES method for solving linear systems with multiple right-hand sides. JSIAM Letters 5(2013), 65–68. https://doi.org/10.14495/jsiaml.5.65
18. Akira Imakura and Tetsuya Sakurai. 2018. Block SS–CAA: A complex moment-based parallel nonlinear eigensolver using the block communication-avoiding Arnoldi procedure. Parallel Comput. 74(2018), 34–48. https://doi.org/10.1016/j.parco.2017.11.007 Parallel Matrix Algorithms and Applications (PMAA’16).
19. J. Kestyn, V. Kalantzis, E. Polizzi, and Y. Saad. 2016. PFEAST: A High Performance Sparse Eigenvalue Solver Using Distributed-Memory Linear Solvers. In SC ’16: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis. 178–189.
20. Karl Meerbergen. 2003. The Solution of Parametrized Symmetric Linear Systems. SIAM J. Matrix Anal. Appl. 24, 4 (2003), 1038–1059. https://doi.org/10.1137/S0895479800380386
21. Karl Meerbergen and Zhaojun Bai. 2010. The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides. SIAM J. Matrix Anal. Appl. 31, 4 (2010), 1642–1662. https://doi.org/10.1137/08073144X
22. Ayako Nakata, Jack S. Baker, Shereif Y. Mujahed, Jack T. L. Poulton, Sergiu Arapan, Jianbo Lin, Zamaan Raza, Sushma Yadav, Lionel Truflandier, Tsuyoshi Miyazaki, and David R. Bowler. 2020. Large scale and linear scaling DFT with the CONQUEST code. The Journal of Chemical Physics 152, 16 (2020), 164112. https://doi.org/10.1063/5.0005074
23. Ayako Nakata, Yasunori Futamura, Tetsuya Sakurai, David R Bowler, and Tsuyoshi Miyazaki. 2017. Efficient Calculation of Electronic Structure Using O(N) Density Functional Theory. Journal of Chemical Theory and Computation 13, 9 (2017), 4146–4153. https://doi.org/10.1021/acs.jctc.7b00385
24. Dianne P. O’Leary. 1980. The block conjugate gradient algorithm and related methods. Linear Algebra Appl. 29(1980), 293–322. https://doi.org/10.1016/0024-3795(80)90247-5 Special Volume Dedicated to Alson S. Householder.
25. C. C. Paige and M. A. Saunders. 1975. Solution of Sparse Indefinite Systems of Linear Equations. SIAM J. Numer. Anal. 12, 4 (1975), 617–629. https://doi.org/10.1137/0712047
26. Eric Polizzi. 2009. Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79(2009), 115112. Issue 11. https://doi.org/10.1103/PhysRevB.79.115112
27. Yousef Saad. 2011. Numerical Methods for Large Eigenvalue Problems (revised ed.). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611970739
28. Youcef Saad and Martin H. Schultz. 1986. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. Statist. Comput. 7, 3 (1986), 856–869.
29. Tetsuya Sakurai and Hiroshi Sugiura. 2003. A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math. 159, 1 (2003), 119–128.
30. Tetsuya Sakurai and Hiroto Tadano. 2007. CIRR: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems. Hokkaido Math. J. 36(2007), 745–757.
31. Tetsuya Sakurai, Hiroto Tadano, and Yasunori Futamura. 2013. Efficient parameter estimation and implementation of a contour integral-based eigensolver. J. Algo. Comput. Tech. 7(2013), 249–269.
32. Olaf Schenk and Klaus Gärtner. 2011. PARDISO. In Encyclopedia of Parallel Computing, David Padua(Ed.). Springer US, Boston, MA, 1458–1464. https://doi.org/10.1007/978-0-387-09766-4_90
33. Kirk M. Soodhalter. 2015. A block MINRES algorithm based on the band Lanczos method. Numerical Algorithms 69, 3 (2015), 473–494. https://doi.org/10.1007/s11075-014-9907-z
34. Kirk M. Soodhalter. 2016. Block Krylov Subspace Recycling for Shifted Systems with Unrelated Right-Hand Sides. SIAM Journal on Scientific Computing 38, 1 (2016), A302–A324. https://doi.org/10.1137/140998214
35. Brigitte Vital. 1990. Étude de quelques méthodes de résolution de problèmes linéaires de grande taille sur multiprocesseur. Ph. D. thesis, Université de Rennes I(1990).
36. Yusaku Yamamoto, Yuji Nakatsukasa, Yuka Yanagisawa, and Takeshi Fukaya. 2016. Roundoff error analysis of the CholeskyQR2 algorithm in an oblique inner product. JSIAM Letters 8(2016), 5–8. https://doi.org/10.14495/jsiaml.8.5

Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation

    • Symbolic and algebraic algorithms

      • Linear algebra algorithms

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Computations on matrices