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A block Chebyshev-Davidson method for linear response eigenvalue problems

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Abstract

We present a Chebyshev-Davidson method to compute a few smallest positive eigenvalues and corresponding eigenvectors of linear response eigenvalue problems. The method is applicable to more general linear response eigenvalue problems where some purely imaginary eigenvalues may exist. For the Chebyshev filter, a tight upper bound is obtained by a computable bound estimator that is provably correct under a reasonable condition. When the condition fails, the estimated upper bound may not be a true one. To overcome that, we develop an adaptive strategy for updating the estimated upper bound to guarantee the effectiveness of our new Chebyshev-Davidson method. We also obtain an estimate of the rate of convergence for the Ritz values by our algorithm. Finally, we present numerical results to demonstrate the performance of the proposed Chebyshev-Davidson method.

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References

  1. Anderson, C.R.: A Rayleigh-Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices. J. Comput. Phys. 229(19), 7477–7487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z., Li, R.C.: Minimization principle for linear response eigenvalue problem, I theory. SIAM J. Matrix Anal. Appl. 33(4), 1075–1100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z., Li, R.C.: Minimization principles for the linear response eigenvalue problem II Computation. SIAM J. Matrix Anal. Appl. 34(2), 392–416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Banerjee, A.S., Elliott, R.S., James, R.D.: A spectral scheme for Kohn-Sham density functional theory of clusters. J. Comput. Phys. 287, 226–253 (2015)

    Article  MathSciNet  Google Scholar 

  5. Cao, Z. -H., Xie, J. -J., Li, R. -C.: A sharp version of Kahan’s theorem on clustered eigenvalues. Linear Algebra Appl. 245, 147–155 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheney, E.W.: Introduction to approximation theory, 2nd edn. Chelsea Publishing Company, New York (1982)

  7. Davis, T., Hu, Y.: The university of florida sparse matrix collection. ACM t. Math. Softw 38(1), 1:1–1:25 (2011)

    MathSciNet  Google Scholar 

  8. Demmel, J.W.: Applied numerical linear algebra. SIAM (1997)

  9. Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G. L., Cococcioni, M., Dabo, I., et al.: QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. Condens. Matter 21(39), 395502 (2009)

    Article  Google Scholar 

  10. Knyazev, A.V.: Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem. Soviet J. Numer. Anal. Math. Modelling 2(5), 371–396 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Levitt, A., Torrent, M.: Parallel eigensolvers in plane-wave density functional theory. Comp. Phys. Comm. 187, 98–105 (2015)

    Article  MathSciNet  Google Scholar 

  12. Morgan, R.B.: GMRESwith deflated restarting. SIAMJ Sci. Comput. 24(1), 20–37 (2002)

    Article  MATH  Google Scholar 

  13. Motamarri, P., Gavini, V.: A subquadratic-scaling subspace projection method for large-scale Kohn-Sham density functional theory calculations using spectral finite-element discretization. Phys. Rev. B 90, 115127 (2014)

    Article  Google Scholar 

  14. Oliveira, S.: On the convergence rate of a preconditioned subspace eigensolver. Computing 63(3), 219–231 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ovthinnikov, E.: Convergence estimates for the generalized Davidson method for symmetric eigenvalue problems I: the preconditioning aspect. SIAM J. Numer. Anal. 41(1), 258–271 (2003)

    Article  MathSciNet  Google Scholar 

  16. Ovthinnikov, E.: Convergence estimates for the generalized Davidson method for symmetric eigenvalue problems II: the subspace acceleration. SIAM J. Numer. Anal. 41(1), 272–286 (2003)

    Article  MathSciNet  Google Scholar 

  17. Papakonstantinou, P.: Reduction of the RPA eigenvalue problem and a generalized Cholesky decomposition for real-symmetric matrices. EPL (Europhysics Letters) 78 (1), 12001 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Parlett, B.N.: The symmetric eigenvalue problem. Number 20 in classics in applied mathematics. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  19. Rocca, D.: Time-dependent density functional perturbation theory: new algorithms with applications to molecular spectra. PhD thesis, The International School for Advanced Studies, Trieste (2007)

    Google Scholar 

  20. Rocca, D., Bai, Z., Li, R. -C., Galli, G.: A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices. J. Chem. Phys. 136, 034111 (2012)

    Article  Google Scholar 

  21. Saad, Y.: Numerical methods for large eigenvalue problems. Wiley (1992)

  22. Stewart, G.W.: Matrix algorithms, volume II: eigensystems. SIAM, Philadephia (2001)

    Book  MATH  Google Scholar 

  23. Teng, Z., Li, R.-C.: Convergence analysis of Lanczos-type methods for the linear response eigenvalue problem. J. Comput. Appl. Math. 247, 17–33 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Teter, M.P., Payne, M.C., Allan, D.C.: Solution of Schrödinger’s equation for large systems. Phys. Rev. B 40(18), 12255–12263 (1989)

    Article  Google Scholar 

  25. Thouless, D.J.: Vibrational states of nuclei in the random phase approximation. Nucl. Phys. 22(1), 78–95 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  26. Thouless, D.J.: The quantum mechanics of Many-Body systems. Academic (1972)

  27. Tsiper, E.V.: A classical mechanics technique for quantum linear response. J. Phys. B Atomic Mol. Phys. 34(12), L401–L407 (2001)

    Article  Google Scholar 

  28. Yamazaki, I., Bai, Z.J., Simon, H., Wang, L.W., Wu, K.S.: Adaptive projection subspace dimension for the thick-restart Lanczos method. ACM T. Math Software 37(3), 27:1–27:18 (2010)

    Article  MathSciNet  Google Scholar 

  29. Zhang, L.-H., Lin, W.-W., Li, R.-C.: Backward perturbation analysis and residual-based error bounds for the linear response eigenvalue problem. BIT Numer. Math. 55(3), 869–896 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, Y.: A block Chebyshev-Davidson method with inner-outer restart for large eigenvalue problems. J. Comput. Phys. 229(24), 9188–9200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhou, Y., Chelikowsky, J.R., Saad, Y.: Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation. J. Comput. Phys. 274, 770–782 (2014)

    Article  Google Scholar 

  32. Zhou, Y., Li, R. -C.: Bounding the spectrum of large Hermitian matrices. Linear Algebra Appl. 435(3), 480–493 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou, Y., Saad, Y.: A Chebyshev-Davidson algorithm for large symmetric eigenproblems. SIAM J Matrix Anal. Appl. 29(3), 954–971 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou, Y., Saad, Y., Tiago, M. L., Chelikowsky, J. R.: Parallel self-consistent-field calculations using Chebyshev-filtered subspace acceleration. Phys. Rev. E. 74(6), 066704 (2006)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou, 350002, People’s Republic of China

    Zhongming Teng

  2. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Yunkai Zhou

  3. Department of Mathematics, University of Texas at Arlington, P. O. Box 19408, Arlington, TX, 76019, USA

    Ren-Cang Li

Authors
  1. Zhongming Teng
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  2. Yunkai Zhou
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  3. Ren-Cang Li
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Corresponding author

Correspondence to Yunkai Zhou.

Additional information

Communicated by: Carlos Garcia - Cervera

Zhongming Teng was supported in part by China Scholarship Council and Natural Science Foundation of Fujian province No. 2015J01580. Part of this work was done while this author was a visiting student at Department of Mathematics, University of Texas at Arlington.

Yunkai Zhou was supported in part by the National Science Foundation grants DMS-1228271 and DMS-1522587.

Ren-Cang Li was supported in part by the National Science Foundation grants DMS-1317330 and CCF-1527104, NSFC grant 11428104.

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Teng, Z., Zhou, Y. & Li, RC. A block Chebyshev-Davidson method for linear response eigenvalue problems. Adv Comput Math 42, 1103–1128 (2016). https://doi.org/10.1007/s10444-016-9455-2

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  • DOI: https://doi.org/10.1007/s10444-016-9455-2

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