Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation. (English) Zbl 1351.82098
Summary: First-principles density functional theory (DFT) calculations for the electronic structure problem require a solution of the Kohn-Sham equation, which requires one to solve a nonlinear eigenvalue problem. Solving the eigenvalue problem is usually the most expensive part in DFT calculations. Sparse iterative diagonalization methods that compute explicit eigenvectors can quickly become prohibitive for large scale problems. The Chebyshev-filtered subspace iteration (CheFSI) method avoids most of the explicit computation of eigenvectors and results in a significant speedup over iterative diagonalization methods for the DFT self-consistent field (SCF) calculations. However, the original formulation of the CheFSI method utilizes a sparse iterative diagonalization at the first SCF step to provide initial vectors for subspace filtering at latter SCF steps. This diagonalization is expensive for large scale problems. We develop a new initial filtering step to avoid completely this diagonalization, thus making the CheFSI method free of sparse iterative diagonalizations at all SCF steps. Our new approach saves memory usage and can be two to three times faster than the original CheFSI method.
MSC:
| 82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |
Keywords:
density functional theory; electronic structure problem; real space pseudopotentials; self-consistency; Chebyshev filters; Hamiltonian; diagonalization; nonlinear eigenvalue problem; subspace filteringReferences:
| [1] | Hohenberg, P.; Kohn, W., Inhomogeneous electron gas, Phys. Rev., 136, B864-871 (1964) |
| [2] | Kohn, W.; Sham, L. J., Self-consistent equations including exchange and correlation effects, Phys. Rev., 140, A1133-1138 (1965) |
| [3] | Phillips, J. C.; Kleinman, L., New method for calculating wave functions in crystals and molecules, Phys. Rev., 116, 287-294 (1959) · Zbl 0095.44601 |
| [4] | Chelikowsky, J. R.; Cohen, M. L., Ab initio pseudopotentials for semiconductors, (Handbook on Semiconductors, vol. 1 (1992), Elsevier: Elsevier Amsterdam), 59 |
| [5] | Martin, R. M., Electronic Structure: Basic Theory and Practical Methods (2004), Cambridge University Press · Zbl 1152.74303 |
| [6] | Payne, M. C.; Teter, M. P.; Allan, D. C.; Arias, T. A.; Joannopoulos, J. D., Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients, Rev. Mod. Phys., 64, 1045-1097 (1992) |
| [7] | Kresse, G.; Furthmüller, J., Efficient iterative schemes for ab initio total-energy calculations using a plane wave basis set, Phys. Rev. B, 54, 11169-11186 (1996) |
| [8] | Chelikowsky, J. R.; Troullier, N.; Saad, Y., Finite-difference-pseudopotential method: electronic structure calculations without a basis, Phys. Rev. Lett., 72, 1240-1243 (1994) |
| [9] | Chelikowsky, J. R.; Troullier, N.; Wu, K.; Saad, Y., Higher-order finite-difference pseudopotential method: an application to diatomic molecules, Phys. Rev. B, 50, 11355-11364 (1994) |
| [10] | Seitsonen, A. P.; Puska, M. J.; Nieminen, R. M., Real-space electronic-structure calculations: combination of the finite-difference and conjugate-gradient methods, Phys. Rev. B, 51, 14057-14061 (1995) |
| [11] | Beck, T. L., Real-space mesh techniques in density-functional theory, Rev. Mod. Phys., 72, 1041-1080 (2000) |
| [12] | Kronik, L.; Makmal, A.; Tiago, M. L.; Alemany, M. M.G.; Jain, M.; Huang, X.; Saad, Y.; Chelikowsky, J. R., PARSEC—the pseudopotential algorithm for real space electronic structure calculations: recent advances and novel applications to nano-structures, Phys. Status Solidi B, 243, 1063 (2006) |
| [13] | Saad, Y.; Stathopoulos, A.; Chelikowsky, J. R.; Wu, K.; Öğüt, S., Solution of large eigenvalue problems in electronic structure calculations, BIT, 36, 563-578 (1996) · Zbl 0862.65059 |
| [14] | Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK User’s Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (1998), SIAM · Zbl 0901.65021 |
| [15] | Wu, K.; Canning, A.; Simon, H. D.; Wang, L.-W., Thick-restart Lanczos method for electronic structure calculations, J. Comput. Phys., 154, 156-173 (1999) · Zbl 0963.82048 |
| [16] | Knyazev, A. V., Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method, SIAM J. Sci. Comput., 23, 517-541 (2001) · Zbl 0992.65028 |
| [17] | Zhou, Y.; Saad, Y., A Chebyshev-Davidson algorithm for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 29, 954-971 (2007) · Zbl 1151.65321 |
| [18] | Zhou, Y.; Saad, Y.; Tiago, M. L.; Chelikowsky, J. R., Self-consistent-field calculation using Chebyshev-filtered subspace iteration, J. Comput. Phys., 219, 172-184 (2006) · Zbl 1105.65111 |
| [19] | Zhou, Y.; Saad, Y.; Tiago, M. L.; Chelikowsky, J. R., Parallel self-consistent-field calculations using Chebyshev-filtered subspace acceleration, Phys. Rev. E, 74, 066704 (2006) |
| [20] | Tiago, M. L.; Zhou, Y.; Alemany, M.; Saad, Y.; Chelikowsky, J. R., Evolution of magnetism in iron from the atom to the bulk, Phys. Rev. Lett., 97, 147201 (2006) |
| [21] | Zhou, Y., A block Chebyshev-Davidson method with inner-outer restart for large eigenvalue problems, J. Comput. Phys., 229, 9188-9200 (2010) · Zbl 1203.65077 |
| [22] | García-Cervera, C. J.; Lu, J.; Xuan, Y.; Weinan, E., Linear-scaling subspace-iteration algorithm with optimally localized nonorthogonal wave functions for Kohn-Sham density functional theory, Phys. Rev. B, 79, 1-13 (2009) |
| [23] | Weinan, E.; Li, T.; Lu, J., Localized bases of eigensubspaces and operator compression, Proc. Natl. Acad. Sci. USA, 107, 4, 1273-1278 (2010) · Zbl 1205.15017 |
| [24] | Goedecker, S.; Teter, M. P., Tight-binding electronic-structure calculations and tight-binding molecular dynamics with localized orbitals, Phys. Rev. B, 51, 9455-9464 (1995) |
| [25] | Ozaki, T., Efficient low-order scaling method for large-scale electronic structure calculations with localized basis functions, Phys. Rev. B, 82, 075131 (2010) |
| [26] | Lin, L.; Lu, J.; Ying, L.; Weinan, E., Optimized local basis set for Kohn-Sham density functional theory, J. Comput. Phys., 231, 4515-4529 (2012) · Zbl 1250.82002 |
| [27] | Lin, L.; Lu, J.; Ying, L.; Weinan, E., Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: total energy calculation, J. Comput. Phys., 231, 2140-2154 (2012) · Zbl 1251.82008 |
| [28] | Ozaki, T., Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations, Phys. Rev. B, 75, 035123 (2007) |
| [29] | Sidje, R. B.; Saad, Y., Rational approximation to the Fermi-Dirac function with applications in density functional theory, Numer. Algorithms, 56, 455-479 (2011) · Zbl 1211.65026 |
| [30] | Lin, L.; Lu, J.; Ying, L.; Weinan, E., Pole-based approximation of the Fermi-Dirac function, Chin. Ann. Math., Ser. B, 30, 729-742 (2009) · Zbl 1188.41007 |
| [31] | Perdew, J. P.; Burke, K.; Ernzerhof, M., Generalized gradient approximation made simple, Phys. Rev. Lett., 77, 3865-3868 (1996) |
| [32] | Kümmel, S.; Perdew, J. P., Optimized effective potential made simple: orbital functionals, orbital shifts, and the exact Kohn-Sham exchange potential, Phys. Rev. B, 68, 035103 (2003) |
| [33] | Anisimov, V. I.; Aryasetiawan, F.; Lichtenstein, A. I., First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA + U method, J. Phys. Condens. Matter, 9, 767-808 (1997) |
| [34] | Cococcioni, M.; de Gironcoli, S., Linear response approach to the calculation of the effective interaction parameters in the LDA + U method, Phys. Rev. B, 71, 035105 (2005) |
| [35] | Kulik, H. J.; Cococcioni, M.; Scherlis, D. A.; Marzari, N., Density functional theory in transition-metal chemistry: a self-consistent Hubbard \(u\) approach, Phys. Rev. Lett., 97, 103001 (2006) |
| [36] | Koch, W.; Holthausen, M. C., A Chemist’s Guide to Density Functional Theory (2001), Wiley-VCH |
| [37] | Engel, B.; Dreizler, R. M., Density functional theory: an advanced course, (Theoretical and Mathematical Physics (2011), Springer) · Zbl 1216.81004 |
| [38] | Axelsson, O., Iterative Solution Methods (1994), Cambridge University Press · Zbl 0795.65014 |
| [39] | Rivlin, T. J., An Introduction to the Approximation of Functions (2003), Dover · Zbl 0189.06601 |
| [40] | Zhou, Y.; Li, R.-C., Bounding the spectrum of large Hermitian matrices, Linear Algebra Appl., 435, 480-493 (2011) · Zbl 1221.15022 |
| [41] | Saad, Y., Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comput., 42, 567-588 (1984) · Zbl 0539.65013 |
| [42] | Zhou, Y., Practical acceleration for computing the HITS ExpertRank vectors, J. Comput. Appl. Math., 236, 4398-4409 (2012) · Zbl 1287.65027 |
| [43] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001 |
| [44] | Yang, C.; Meza, J. C.; Lee, B.; Wang, L.-W., KSSOLV — a MATLAB toolbox for solving the Kohn-Sham equations, ACM Trans. Math. Softw., 36, 1-35 (2009) · Zbl 1364.65112 |
| [45] | Ceperley, D. M.; Alder, B. J., Ground state of the electron gas by a stochastic method, Phys. Rev. Lett., 45, 566-569 (1980) |
| [46] | Fang, H.-R.; Saad, Y., Two classes of multisecant methods for nonlinear acceleration, Numer. Linear Algebra Appl., 16, 3, 197-221 (2009) · Zbl 1224.65134 |
| [47] | Stixrude, L.; Karki, B. B., Structure and freezing of \(MgSiO_3\) liquid in Earth’s lower mantle, Science, 310, 297-299 (2005) |
| [48] | Umemoto, K.; Wentzcovitch, R. M.; Allen, P. B., Dissociation of \(MgSiO_3\) in the cores of gas giants and terrestrial exoplanets, Science, 311, 983-986 (2006) |
| [49] | Wu, K.; Simon, H., Thick-restart Lanczos method for large symmetric eigenvalue problems, SIAM J. Matrix Anal. Appl., 22, 602-616 (2000) · Zbl 0969.65030 |
| [50] | Sleijpen, G. L.G.; van der Vorst, H. A., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl., 17, 401-425 (1996) · Zbl 0860.65023 |
| [51] | Stathopoulos, A.; Saad, Y.; Wu, K., Dynamic thick restarting of the Davidson and the implicitly restarted Arnoldi methods, SIAM J. Sci. Comput., 19, 227-245 (1998) · Zbl 0924.65028 |
| [52] | Stathopoulos, A.; McCombs, J. R., Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: seeking many eigenvalues, SIAM J. Sci. Comput., 29, 2162-2188 (2007) · Zbl 1151.65320 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
