Abstract
Computational methods in materials science have made huge strides in recent years and parallel computing methodologies have played a major role in enabling such a progress. The goal of this chapter is to discuss the current state of the art in computational materials science as it stands today, illustrating advances in the development of parallel algorithms and the impact such algorithms have had in the area. The paper is intended to be accessible to a diverse scientific computing audience. The focus of the paper will be the Density Functional Theory methodology and the solution of the eigenvalue problems that are encountered in solving the resulting equations.
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References
Feast. http://www.feast-solver.org.
Nessie. http://www.nessie-code.org.
Octopus. http://www.tddft.org/programs/octopus/.
Pexsi. https://math.berkeley.edu/~linlin/pexsi/download/doc_v0.6.0/.
X. ANDRADE, J. ALBERDI-RODRIGUEZ, D. A. STRUBBE, M. J. T. OLIVEIRA, F. NOGUEIRA, A. CASTRO, J. MUGUERZA, A. ARRUABARRENA, S. G. LOUIE, A. ASPURU-GUZIK, A. RUBIO, AND M. A. L. MARQUES, Time-dependent density-functional theory in massively parallel computer architectures: the octopus project, Journal of Physics: Condensed Matter, 24 (2012), p. 233202.
X. ANDRADE, D. A. STRUBBE, U. D. GIOVANNINI, A. H. LARSEN, M. J. T. OLIVEIRA, J. ALBERDI-RODRIGUEZ, A. VARAS, I. THEOPHILOU, N. HELBIG, M. VERSTRAETE, L. STELLA, F. NOGUEIRA, A. ASPURU-GUZIK, A. CASTRO, M. A. L. MARQUES, AND A. RUBIO, Real-space grids and the octopus code as tools for the development of new simulation approaches for electronic systems, Physical Chemistry Chemical Physics, 17 (2015), pp. 31371–31396.
S. BARONI AND P. GIANNOZZI, Towards very large-scale electronic-structure calculations, Europhysics Letters (EPL), 17 (1992), pp. 547–552.
T. L. BECK, Real-space mesh techniques in density functional theory, Rev. Mod. Phys., 74 (2000), pp. 1041–1080.
E. L. BRIGGS, D. J. SULLIVAN, AND J. BERNHOLC, Large-scale electronic-structure calculations with multigrid acceleration, Phys. Rev. B, 52 (1995), pp. R5471–R5474.
A. BRILEY, M. R. PEDERSON, K. A. JACKSON, D. C. PATTON, AND D. V. POREZAG, Vibrational frequencies and intensities of small molecules: All-electron, pseudopotential, and mixed-potential methodologies, Phys. Rev. B, 58 (1997), pp. 1786–1793.
K. BURKE, Perspective on density functional theory., The Journal of chemical physics, 136 15 (2012), p. 150901.
R. CAR AND M. PARRINELLO, Unified approach for molecular dynamics and density functional theory, Phys. Rev. Lett., 55 (1985), pp. 2471–2474.
J. R. CHELIKOWSKY AND M. L. COHEN, Pseudopotentials for semiconductors, in Handbook of Semiconductors, T. S. Moss and P. T. Landsberg, eds., Elsevier, Amsterdam, 2nd edition, 1992.
J. R. CHELIKOWSKY AND S. G. LOUIE, First-principles linear combination of atomic orbitals method for the cohesive and structural properties of solids: Application to diamond, Phys. Rev. B, 29 (1984), pp. 3470–3481.
J. R. CHELIKOWSKY, N. TROULLIER, X. JING, D. DEAN, N. BIGGELI, K. WU, AND Y. SAAD, Algorithms for the structural properties of clusters, Comp. Phys. Comm., 85 (1995), pp. 325–335.
J. R. CHELIKOWSKY, N. TROULLIER, AND Y. SAAD, The finite-difference-pseudopotential method: Electronic structure calculations without a basis, Phys. Rev. Lett., 72 (1994), pp. 1240–1243.
J. R. CHELIKOWSKY, N. TROULLIER, K. WU, AND Y. SAAD, Higher order finite difference pseudopotential method: An application to diatomic molecules, Phys. Rev. B, 50 (1994), pp. 11355–11364.
E. DI NAPOLI, E. POLIZZI, AND Y. SAAD, Efficient estimation of eigenvalue counts in an interval, Numerical Linear Algebra with Applications, 23 (2016), pp. 674–692.
J.-L. FATTEBERT AND J. BERNHOLC, Towards grid-based O(N) density-functional theory methods: Optimized nonorthogonal orbitals and multigrid acceleration, Phys. Rev. B, 62 (2000), pp. 1713–1722.
——, Towards grid-based O(N) density-functional theory methods: Optimized nonorthogonal orbitals and multigrid acceleration, Phys. Rev. B, 62 (2000), pp. 1713–1722.
C. Y. FONG, Topics in Computational Materials Science, World Scientific, 1998.
B. FORNBERG AND D. M. SLOAN, A review of pseudospectral methods for solving partial differential equations, Acta Numer., 94 (1994), pp. 203–268.
S. GüTTEL, E. POLIZZI, P. TANG, AND G. VIAUD, Zolotarev quadrature rules and load balancing for the feast eigensolver, SIAM Journal on Scientific Computing, 37 (2015), pp. A2100–A2122.
F. GYGI AND G. GALLI, Real-space adaptive-coordinate electronic-structure calculations, Phys. Rev. B, 52 (1995), pp. R2229–R2232.
Y. HASEGAWA, J. IWATA, M. TSUJI, D. TAKAHASHI, A. OSHIYAMA, K. MINAMI, T. BOKU, F. SHOJI, A. UNO, M. KUROKAWA, H. INOUE, I. MIYOSHI, AND M. YOKOKAWA, First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the k computer, in Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’11, New York, NY, USA, 2011, ACM, pp. 1:1–1:11.
A. HAUG, Theoretical Solid State Physics, Pergamon Press, 1972.
M. HEIKANEN, T. TORSTI, M. J. PUSKA, AND R. M. NIEMINEN, Multigrid method for electronic structure calculations, Phys. Rev. B, 63 (2001), pp. 245106–245113.
P. HOHENBERG AND W. KOHN, Inhomogeneous electron gas, Phys. Rev., 136 (1964), pp. B864–B871.
K. A. JACKSON, M. R. PEDERSON, D. V. POREZAG, Z. HAJNAL, AND T. FRAUNHEIM, Density-functional-based predictions of Raman and IR spectra for small Si clusters, Phys. Rev. B, 55 (1997), pp. 2549–2555.
R. W. JANSEN AND O. F. SANKEY, Ab initio linear combination of pseudo-atomic-orbital scheme for the electronic properties of semiconductors: Results for ten materials, Phys. Rev. B, 36 (1987), pp. 6520–6531.
J. KESTYN, V. KALANTZIS, E. POLIZZI, AND Y. SAAD, PFEAST: A high performance sparse eigenvalue solver using distributed-memory linear solvers, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’16, Piscataway, NJ, USA, 2016, IEEE Press, pp. 16:1–16:12.
Y. H. KIM, I. H. LEE, AND R. M. MARTIN, Object-oriented construction of a multigrid electronic structure code with Fortran, Comp. Phys. Comm., 131 (2000), pp. 10–25.
W. KOHN AND L. J. SHAM, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), pp. A1133–A1138.
L. KRONIK, A. MAKMAL, M. L. TIAGO, M. M. G. ALEMANY, M. JAIN, X. HUANG, Y. SAAD, AND J. R. CHELIKOWSKY, PARSEC – the pseudopotential algorithm for real-space electronic structure calculations: recent advances and novel applications to nano-structure, Phys. Stat. Sol. (B), 243 (2006), p. 1063–1079.
C. LANCZOS, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), pp. 255–282.
R. M. LARSEN, PROPACK: A software package for the symmetric eigenvalue problem and singular value problems on Lanczos and Lanczos bidiagonalization with partial reorthogonalization, SCCM, Stanford University URL: http://sun.stanford.edu/~rmunk/PROPACK/.
——, Efficient Algorithms for Helioseismic Inversion, PhD thesis, Dept. Computer Science, University of Aarhus, DK-8000 Aarhus C, Denmark, October 1998.
I. H. LEE, Y. H. KIM, AND R. M. MARTIN, One-way multigrid method in electronic-structure calculations, Phys. Rev. B, 61 (2000), p. 4397.
L. LEHTOVAARA, V. HAVU, AND M. PUSKA, All-electron density functional theory and time-dependent density functional theory with high-order finite elements, The Journal of Chemical Physics, 131 (2009), p. 054103.
A. R. LEVIN, D. ZHANG, AND E. POLIZZI, Feast fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem, Computer Physics Communications, 183 (2012), pp. 2370 – 2375.
R. LI, Y. XI, L. ERLANDSON, AND Y. SAAD, The EigenValues Slicing Library (EVSL): Algorithms, implementation, and software, Tech. Report ys-2018-02, Dept. Computer Science and Engineering, University of Minnesota, Minneapolis, MN, 2018. Submitted. ArXiv: https://arxiv.org/abs/1802.05215.
R. LI, Y. XI, E. VECHARYNSKI, C. YANG, AND Y. SAAD, A Thick-Restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems, SIAM Journal on Scientific Computing, 38 (2016), pp. A2512–A2534.
L. LIN, Y. SAAD, AND C. YANG, Approximating spectral densities of large matrices, SIAM Review, 58 (2016), pp. 34–65.
L. LIN, C. YANG, J. C. MEZA, J. LU, L. YING, AND E. WEINAN, Selinv - an algorithm for selected inversion of a sparse symmetric matrix, ACM Trans. Math. Softw., 37 (2011), pp. 40:1–40:19.
S. LUNDQVIST AND N. H. MARCH, eds., Theory of the Inhomogeneous Electron Gas, Plenum, 1983.
R. MARTIN, R. MARTIN, AND C. U. PRESS, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, 2004.
J. L. MARTINS AND M. COHEN, Diagonalization of large matrices in pseudopotential band-structure calculations: Dual-space formalism, Phys. Rev. B, 37 (1988), pp. 6134–6138.
J. L. MARTINS, N. TROULLIER, AND S.-H. WEI, Pseudopotential plane-wave calculations for ZnS, Phys. Rev. B, 43 (1991), pp. 2213–2217.
R. B. MORGAN AND D. S. SCOTT, Generalizations of Davidson’s method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Comput., 7 (1986), pp. 817–825.
T. ONO AND K. HIROSE, Timesaving double-grid method for real-space electronic structure calculations, Phys. Rev. Lett., 82 (1999), pp. 5016–5019.
——, Timesaving double-grid method for real-space electronic structure calculations, Phys. Rev. Lett., 82 (1999), pp. 5016–5019.
C. C. PAIGE, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, PhD thesis, University of London, 1971.
B. N. PARLETT, The Symmetric Eigenvalue Problem, no. 20 in Classics in Applied Mathematics, SIAM, Philadelphia, 1998.
P. PETER TANG AND E. POLIZZI, Feast as a subspace iteration eigensolver accelerated by approximate spectral projection, SIAM Journal on Matrix Analysis and Applications, 35 (2014), pp. 354–390.
E. POLIZZI, A density matrix-based algorithm for solving eigenvalue problems, phys. rev. B, 79 (2009).
E. POLIZZI AND S. DATTA, Multidimensional nanoscale device modeling: the finite element method applied to the non-equilibrium Green’s function formalism, in 2003 Third IEEE Conference on Nanotechnology, 2003. IEEE-NANO 2003., vol. 1, Aug 2003, pp. 40–43 vol.2.
H. RUTISHAUSER, Simultaneous iteration for symmetric matrices, in Handbook for automatic computations (linear algebra), J. Wilkinson and C. Reinsch, eds., New York, 1971, Springer Verlag, pp. 202–211.
Y. SAAD, Numerical Methods for Large Eigenvalue Problems, John Wiley, New York, 1992.
Y. SAAD, A. STATHOPOULOS, J. R. CHELIKOWSKY, K. WU, AND S. OGUT, Solution of large eigenvalue problems in electronic structure calculations, BIT, 36 (1996), pp. 563–578.
T. SAKURAI AND H. SUGIURA, A projection method for generalized eigenvalue problems using numerical integration, Journal of Computational and Applied Mathematics, 159 (2003), pp. 119–128. Japan-China Joint Seminar on Numerical Mathematics; In Search for the Frontier of Computational and Applied Mathematics toward the 21st Century.
T. SAKURAI AND H. TADANO, Cirr: a Rayleigh-Ritz type method with contour integral for generalized eigenvalue problems, Hokkaido Mathematical Journal, 36 (2007), p. 745–757.
L. J. SHAM, Theoretical and computational development some efforts beyond the local density approximation, International Journal of Quantum Chemistry, Vol. 56, 4 , pp 345–350, (2004).
H. D. SIMON, Analysis of the symmetric Lanczos algorithm with reorthogonalization methods, Linear Algebra Appl., 61 (1984), pp. 101–132.
——, The Lanczos algorithm with partial reorthogonalization, Math. Comp., 42 (1984), pp. 115–142.
J. M. SOLER, E. ARTACHO, J. D. GALE, A. GARCIA, J. JUNQUERA, P. ORDEJóN, AND D. SáNCHEZ-PORTAL, The SIESTA method for ab-initio order-N materials simulation, J. Phys.: Condens. Matter, 14 (2002), pp. 2745–2779.
D. C. SORENSEN, Implicit application of polynomial filters in ak-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385.
E. M. STOUDENMIRE, L. O. WAGNER, S. R. WHITE, AND K. BURKE, One-dimensional continuum electronic structure with the density matrix renormalization group and its implications for density functional theory, Phys. Rev. Lett. 109, I5, 056402, (2012).
J. TAYLOR, H. GUO, AND J. WANG, Ab initio modeling of quantum transport properties of molecular electronic devices, Phys. Rev. B, 63 (2001), p. 245407.
T. TORSTI, M. HEISKANEN, M. PUSKA, AND R. NIEMINEN, MIKA: A multigrid-based program package for electronic structure calculations, Int. J. Quantum Chem., 91 (2003), pp. 171–176.
L.-W. WANG AND A. ZUNGER, Electronic structure pseudopotential calculations of large ( 1000 atoms) SI quantum dots, J. Phys. Chem., 98 (1994), pp. 2158–2165.
S. R. WHITE, J. W. WILKINS, AND M. P. TETER, Finite-element method for electronic structure, Phys. Rev. B, 39 (1989), pp. 5819–5833.
K. WU AND H. SIMON, A parallel Lanczos method for symmetric generalized eigenvalue problems, Tech. Report 41284, Lawrence Berkeley National Laboratory, 1997. Available on line at http://www.nersc.gov/research/SIMON/planso.html.
R. ZELLER, J. DEUTZ, AND P. DEDERICHS, Application of complex energy integration to self-consistent electronic structure calculations, Solid State Communications, 44 (1982), pp. 993–997.
D. ZHANG AND E. POLIZZI, Linear scaling techniques for first-principle calculations of large nanowire devices, 2008 NSTI Nanotechnology Conference and Trade Show. Technical Proceedings, Vol. 1 pp12–15, (2008).
Y. ZHOU, Y. SAAD, M. L. TIAGO, AND J. R. CHELIKOWSKY, Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration, Phy. rev. E, 74 (2006), p. 066704.
J. M. ZIMAN, Electrons and Phonons, Oxford University Press, 1960.
G. ZUMBACH, N. A. MODINE, AND E. KAXIRAS, Adaptive coordinate, real-space electronic structure calculations on parallel computers, Solid State Commun., 99 (1996), pp. 57–61.
Acknowledgements
The work of the author “Eric Polizzi” was supported by NSF awards CCF-1510010 and SI2-SSE-1739423. The work of the author “Yousef Saad” was supported by NSF award CCF-1505970.
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Polizzi, E., Saad, Y. (2020). Computational Materials Science and Engineering. In: Grama, A., Sameh, A. (eds) Parallel Algorithms in Computational Science and Engineering. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-43736-7_5
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