publication . Preprint . Article . 2017

Solution of the k-th eigenvalue problem in large-scale electronic structure calculations

Takeo Hoshi;
Open Access English
  • Published: 14 Oct 2017
Abstract
Abstract We consider computing the k -th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n × n large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index k . We present a three-stage algorithm for computing the k -th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the k -th eigenvalue ( 1 ≪ k ≪ n ) with a non-standard application of the Lanczos method. In the second stage, spectral bi...
Subjects
free text keywords: Mathematics - Numerical Analysis, Physics and Astronomy (miscellaneous), Computer Science Applications, Applied mathematics, Lanczos resampling, Divide-and-conquer eigenvalue algorithm, Sparse matrix, Eigenvalues and eigenvectors, Electronic structure, Hermitian matrix, Linear solver, Mathematical analysis, Inverse iteration, Mathematics
Related Organizations
43 references, page 1 of 3

[1] C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear di erential and integral operators. J. Res. Natl. Bur. Stand., 45(4):255{282, 1950. [OpenAIRE]

[2] A. V. Knyazev. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput., 23(2):517{541, 2001. [OpenAIRE]

[3] T. Ericsson and A. Ruhe. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput., 35(152):1251{1268, 1980.

[4] G. L. G. Sleijpen and H. A. van der Vorst. A Jacobi{Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17(2):401{425, 1996.

[5] T. Sakurai and H. Sugiura. A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math., 159(1):119{128, 2003.

[6] E. Polizzi. Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B, 79:115112, 2009. [OpenAIRE]

[7] R. Li, Y. Xi, E. Vecharynski, C. Yang, and Y. Saad. A thick-restart Lanczos algorithm with polynomial ltering for Hermitian eigenvalue problems. SIAM J. Sci. Comput., 38(4):A2512{A2534, 2016.

[8] T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, and S.-L. Zhang. An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system. J. Phys.: Condens. Matter, 24(16):165502, 2012. [OpenAIRE]

[9] A. Marek, V. Blum, R. Johanni, V. Havu, B. Lang, T. Auckenthaler, A. Heinecke, H.-J. Bungartz, and H. Lederer. The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science. J. Phys.: Condens. Matter, 26(21):213201, 2014.

[10] L. S. Blackford, J. Choi, A. Cleary, E. D'Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley. ScaLAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, 1997.

[11] T. Imamura, S. Yamada, and M. Machida. Development of a high-performance eigensolver on a peta-scale next-generation supercomputer system. Prog. Nucl. Sci. Technol., 2:643{650, 2011.

[12] H. Imachi and T. Hoshi. Hybrid numerical solvers for massively parallel eigenvalue computations and their benchmark with electronic structure calculations. J. Inf. Process., 24(1):164{172, 2016.

[13] T. Hoshi, H. Imachi, K. Kumahata, M. Terai, K. Miyamoto, K. Minami, and F. Shoji. Extremely scalable algorithm for 108-atom quantum material simulation on the full system of the K computer. In Proceedings of the 7th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, pages 33{40, 2016. [OpenAIRE]

[14] D. R. Bowler and T. Miyazaki. O(N ) methods in electronic structure calculations. Rep. Prog. Phys., 75(3):036503, 2012.

[15] D. Lee, T. Miyata, T. Sogabe, T. Hoshi, and S.-L. Zhang. An interior eigenvalue problem from electronic structure calculations. Jpn. J. Ind. Appl. Math., 30(3):625{633, 2013.

43 references, page 1 of 3
Related research
Abstract
Abstract We consider computing the k -th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n × n large sparse matrices. In electronic structure calculations, several properties of materials, such as those of optoelectronic device materials, are governed by the eigenpair with a material-specific index k . We present a three-stage algorithm for computing the k -th eigenpair with validation of its index. In the first stage of the algorithm, we propose an efficient way of finding an interval containing the k -th eigenvalue ( 1 ≪ k ≪ n ) with a non-standard application of the Lanczos method. In the second stage, spectral bi...
Subjects
free text keywords: Mathematics - Numerical Analysis, Physics and Astronomy (miscellaneous), Computer Science Applications, Applied mathematics, Lanczos resampling, Divide-and-conquer eigenvalue algorithm, Sparse matrix, Eigenvalues and eigenvectors, Electronic structure, Hermitian matrix, Linear solver, Mathematical analysis, Inverse iteration, Mathematics
Related Organizations
43 references, page 1 of 3

[1] C. Lanczos. An iteration method for the solution of the eigenvalue problem of linear di erential and integral operators. J. Res. Natl. Bur. Stand., 45(4):255{282, 1950. [OpenAIRE]

[2] A. V. Knyazev. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput., 23(2):517{541, 2001. [OpenAIRE]

[3] T. Ericsson and A. Ruhe. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comput., 35(152):1251{1268, 1980.

[4] G. L. G. Sleijpen and H. A. van der Vorst. A Jacobi{Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl., 17(2):401{425, 1996.

[5] T. Sakurai and H. Sugiura. A projection method for generalized eigenvalue problems using numerical integration. J. Comput. Appl. Math., 159(1):119{128, 2003.

[6] E. Polizzi. Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B, 79:115112, 2009. [OpenAIRE]

[7] R. Li, Y. Xi, E. Vecharynski, C. Yang, and Y. Saad. A thick-restart Lanczos algorithm with polynomial ltering for Hermitian eigenvalue problems. SIAM J. Sci. Comput., 38(4):A2512{A2534, 2016.

[8] T. Hoshi, S. Yamamoto, T. Fujiwara, T. Sogabe, and S.-L. Zhang. An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system. J. Phys.: Condens. Matter, 24(16):165502, 2012. [OpenAIRE]

[9] A. Marek, V. Blum, R. Johanni, V. Havu, B. Lang, T. Auckenthaler, A. Heinecke, H.-J. Bungartz, and H. Lederer. The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science. J. Phys.: Condens. Matter, 26(21):213201, 2014.

[10] L. S. Blackford, J. Choi, A. Cleary, E. D'Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling, G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley. ScaLAPACK Users' Guide. Society for Industrial and Applied Mathematics, Philadelphia, 1997.

[11] T. Imamura, S. Yamada, and M. Machida. Development of a high-performance eigensolver on a peta-scale next-generation supercomputer system. Prog. Nucl. Sci. Technol., 2:643{650, 2011.

[12] H. Imachi and T. Hoshi. Hybrid numerical solvers for massively parallel eigenvalue computations and their benchmark with electronic structure calculations. J. Inf. Process., 24(1):164{172, 2016.

[13] T. Hoshi, H. Imachi, K. Kumahata, M. Terai, K. Miyamoto, K. Minami, and F. Shoji. Extremely scalable algorithm for 108-atom quantum material simulation on the full system of the K computer. In Proceedings of the 7th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, pages 33{40, 2016. [OpenAIRE]

[14] D. R. Bowler and T. Miyazaki. O(N ) methods in electronic structure calculations. Rep. Prog. Phys., 75(3):036503, 2012.

[15] D. Lee, T. Miyata, T. Sogabe, T. Hoshi, and S.-L. Zhang. An interior eigenvalue problem from electronic structure calculations. Jpn. J. Ind. Appl. Math., 30(3):625{633, 2013.

43 references, page 1 of 3
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publication . Preprint . Article . 2017

Solution of the k-th eigenvalue problem in large-scale electronic structure calculations

Takeo Hoshi;