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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:371

Article

Lee, Dongjin¹  Hoshi, Takeo²  Sogabe, Tomohiro¹  Miyatake, Yuto¹  Zhang, Shao-Liang¹ 

[1] Nagoya Univ, Grad Sch Engn, Chikusa Ku, Furo Cho, Nagoya, Aichi 4648603, Japan

[2] Tottori Univ, Dept Appl Math & Phys, 4-101 Koyama Minami, Tottori 6808550, Japan

关键词: Generalized eigenvalue problem;    Electronic structure calculations;    Spectral bisection;    Lanczos method;    Sparse direct linear solver;   

DOI  :  10.1016/j.jcp.2018.06.002

来源: Elsevier

【 摘 要 】

We consider computing the k-th eigenvalue and its corresponding eigenvector of a generalized Hermitian eigenvalue problem of n x 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 bisection for large-scale problems is realized using a sparse direct linear solver to narrow down the interval of the k-th eigenvalue. In the third stage, we switch to a modified shift-and-invert Lanczos method to reduce bisection iterations and compute the k-th eigenpair with validation. Numerical results with problem sizes up to 1.5 million are reported, and the results demonstrate the accuracy and efficiency of the three-stage algorithm. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free