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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem
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by Zhen-Chen Guo, Eric King-Wah Chu and Wen-Wei Lin HTML | PDF
Math. Comp. 88 (2019), 2325-2350 Request permission

Abstract:

The discretized Bethe-Salpeter eigenvalue problem arises in the Green’s function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb {C}^{2n \times 2n}$ with a Hamiltonian-like structure. After an appropriate transformation of $H$ to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and the structure-preserving nature of the algorithm.
References
  • Zhaojun Bai and Ren-Cang Li, Minimization principles for the linear response eigenvalue problem I: Theory, SIAM J. Matrix Anal. Appl. 33 (2012), no. 4, 1075–1100. MR 3023465, DOI 10.1137/110838960
  • Zhaojun Bai and Ren-Cang Li, Minimization principles for the linear response eigenvalue problem II: Computation, SIAM J. Matrix Anal. Appl. 34 (2013), no. 2, 392–416. MR 3046810, DOI 10.1137/110838972
  • Peter Benner, Sergey Dolgov, Venera Khoromskaia, and Boris N. Khoromskij, Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation, J. Comput. Phys. 334 (2017), 221–239. MR 3606226, DOI 10.1016/j.jcp.2016.12.047
  • P. Benner, H. Fassbender, and C. Yang, Some remarks on the complex $J$-symmetric eigenvalue problem, Preprint, MPIMD/15-12, Max Planck Institute Magdeburg, 2015 (available at www.mpi-magdeburg.mpg.de/preprints).
  • P. Benner, V. Khoromskaia, and B. N. Khoromskij, A reduced basis approach for calculation of the Bethe-Salpeter excitation energies using low-rank tensor factorizations, Molecular Phys., 114 (2016) 1148–1161.
  • M. E. Casida, Time-dependent density-functional response theory for molecules, Recent Advances in Density Functional Methods, Part I, D.P. Chong (Ed.), World Scientific, Singapore, 155 (1995) 1207–1216.
  • E. K.-W. Chu, H.-Y. Fan, and W.-W. Lin, A structure-preserving doubling algorithm for continuous-time algebraic Riccati equations, Linear Algebra Appl. 396 (2005), 55–80. MR 2112199, DOI 10.1016/j.laa.2004.10.010
  • E. K.-W. Chu, H.-Y. Fan, W.-W. Lin, and C.-S. Wang, Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations, Internat. J. Control 77 (2004), no. 8, 767–788. MR 2072208, DOI 10.1080/00207170410001714988
  • Eric King-Wah Chu, Tsung-Min Hwang, Wen-Wei Lin, and Chin-Tien Wu, Vibration of fast trains, palindromic eigenvalue problems and structure-preserving doubling algorithms, J. Comput. Appl. Math. 219 (2008), no. 1, 237–252. MR 2437709, DOI 10.1016/j.cam.2007.07.016
  • Gene H. Golub and Charles F. Van Loan, Matrix computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR 1417720
  • T.-M. Huang, R.-C. Li, W.-W. Lin, and L. Lu., Optimal parameters for doubling algorithms, Technical Report 2017-03, Department of Mathematics, University of Texas at Arlington, May 2017. Available at http://www.uta.edu/math/preprint/.
  • Tsung-Ming Huang and Wen-Wei Lin, Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations, Linear Algebra Appl. 430 (2009), no. 5-6, 1452–1478. MR 2490689, DOI 10.1016/j.laa.2007.08.043
  • V. Khoromskaia, B. N. Khoromskij, and R. Schneider, Tensor-structured factorized calculation of two-electron integrals in a general basis, SIAM J. Sci. Comput. 35 (2013), no. 2, A987–A1010. MR 3040965, DOI 10.1137/120884067
  • S. Körbel, P. Boulanger, I. Duchemin, X. Blase, M. AL Marques, and S. Botti, Benchmark many-body GW and Bethe-Salpeter calculations for small transition metal molecules, J. Chemical Theory Comp., 10 (2014) 3934–3943.
  • X. Leng, F. Jin, M. Wei, and Y. Ma, GW method and Bethe-Salpeter equation for calculating electronic excitations, Wiley Interdisciplinary Reviews: Computation Molecular Science, Wiley Online Library, 2016.
  • Tiexiang Li, Chun-Yueh Chiang, Eric King-wah Chu, and Wen-Wei Lin, The palindromic generalized eigenvalue problem $A^*x=\lambda Ax$: numerical solution and applications, Linear Algebra Appl. 434 (2011), no. 11, 2269–2284. MR 2776795, DOI 10.1016/j.laa.2009.12.020
  • Tiexiang Li, Eric King-wah Chu, Jong Juang, and Wen-Wei Lin, Solution of a nonsymmetric algebraic Riccati equation from a one-dimensional multistate transport model, IMA J. Numer. Anal. 31 (2011), no. 4, 1453–1467. MR 2846762, DOI 10.1093/imanum/drq034
  • Tiexiang Li, Eric King-wah Chu, Jong Juang, and Wen-Wei Lin, Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model, Linear Algebra Appl. 434 (2011), no. 1, 201–214. MR 2737242, DOI 10.1016/j.laa.2010.09.006
  • Wen-Wei Lin and Shu-Fang Xu, Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations, SIAM J. Matrix Anal. Appl. 28 (2006), no. 1, 26–39. MR 2218940, DOI 10.1137/040617650
  • G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body Green’s-function approaches, Rev. Mod. Phys., 74 (2002) 601–659.
  • R.M. Parrish, E.G. Hohenstein, N. Schunck, C. Sherrill, and T. J. Martinez, Exact tensor hypercontraction: A universal technique for the resolution of matrix elements of local finite-range N-body potentials in many-body quantum problems, Phys. Rev. Lett., 111 (2013) 132505.
  • Y. Ping, D. Rocca, and G. Galli, Electronic excitations in light absorbers for photo-electrochemical energy conversion: First principles calculations based on many body perturbation theory, Chem. Soc. Rev., 42 (2013) 2437–2469.
  • S. Reine, T. Helgaker, and R. Lindh, Multi-electron integrals, WIREs Comput. Mol. Sci., 2 (2012) 290–303.
  • L. Reining, V. Olevano, A. Rubio, and G. Onida, Excitonic effects in solids described by time-dependent density functional theory, Phys. Rev. Lett., 88 (2002) 066404.
  • E. Ribolini, J. Toulouse, and A. Savin, Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of H2 molecule, Concepts and Methods in Modern Theoretical Chemistry, S. Ghosh and P. Chattaraj (eds), Vol. 1: Electronic Structure and Reactivity, 367 (2013) 367–390.
  • E. Ribolini, J. Toulouse, and A. Savin, Electronic excitations from a linear-response range-separated hybrid scheme, Molecular Phys., 111 (2013) 1219–1234.
  • M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles. Phys. Rev. B, 62 (2000) 4927–4944.
  • E. Runge and E. Gross, Density-function theory for time-dependent systems, Phys. Rev. Lett., 52 (1984) 997–1000.
  • E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Phys. Rev. (2) 84 (1951), 1232–1242. MR 52996
  • Meiyue Shao, Felipe H. da Jornada, Chao Yang, Jack Deslippe, and Steven G. Louie, Structure preserving parallel algorithms for solving the Bethe-Salpeter eigenvalue problem, Linear Algebra Appl. 488 (2016), 148–167. MR 3419779, DOI 10.1016/j.laa.2015.09.036
  • G. W. Stewart and Ji Guang Sun, Matrix perturbation theory, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1990. MR 1061154
  • R. E. Stratmann, G. E. Scuseria, and M. J. Frisch, An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules, J. Chem. Phys., 109 (1998) 8218–8224.
  • S. Wilson, Universal basis sets and Cholesky decomposition of the two-electron integral matrix, Comput. Phys. Commun., 58 (1990) 71–81.
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Additional Information
  • Zhen-Chen Guo
  • Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
  • MR Author ID: 1095300
  • Email: guozhch06@gmail.com
  • Eric King-Wah Chu
  • Affiliation: School of Mathematics, Monash University, 9 Rainforest Walk, Victoria 3800, Australia
  • MR Author ID: 49125
  • Email: eric.chu@monash.edu
  • Wen-Wei Lin
  • Affiliation: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan
  • MR Author ID: 232126
  • Email: wwlin@math.nctu.edu.tw
  • Received by editor(s): November 20, 2017
  • Received by editor(s) in revised form: May 6, 2018, July 25, 2018, and August 21, 2018
  • Published electronically: January 9, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: Math. Comp. 88 (2019), 2325-2350
  • MSC (2010): Primary 15A18, 65F15
  • DOI: https://doi.org/10.1090/mcom/3398
  • MathSciNet review: 3957895