A sparse interpolation algorithm for dynamical simulations in computational chemistry

SIAM Journal on Scientific Computing

Volumn 37, Issue 5, 2015, Pages S137-S156

  Nance, J ^a   Kelley, C T ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Dynamics; Reaction path following; Smolyak's algorithm; Sparse grids; Sparse interpolation

Indexed keywords

COMPUTATIONAL CHEMISTRY; COMPUTATIONAL EFFICIENCY; GEOMETRY; INTERPOLATION; MATLAB; POTENTIAL ENERGY; POTENTIAL ENERGY SURFACES; QUANTUM CHEMISTRY;

DYNAMICAL SIMULATION; INTERPOLATION ALGORITHMS; PATH FOLLOWING; POTENTIAL-ENERGY SURFACES; REACTION PATH FOLLOWING; REACTION PATHS; SMOLYAK ALGORITHMS; SPARSE GRID; SPARSE GRID INTERPOLATIONS; SPARSE INTERPOLATION;

MOLECULES;

EID: 84945934493     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/140965284     Document Type: Conference Paper

References (45)

1. M. Barbatti, M. Ruckenbauer, F. Plasser, J. Pittner, G. Granucci, M. Persico, and H. Lischka, Newton-X: A surface-hopping program for nonadiabatic molecular dynamics, Wiley Interdiscip. Rev. Comput. Molec. Sci., 4 (2014), pp. 26-33.
2. V. Barthelmann, E. Novak, and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12 (2000), pp. 273-288.
3. T. Beringhelli, A. Gavezzotti, and M. Simonetta, Semi-empirical calculations on the thermal cis-trans isomerization of 2-butene, 2-pentene, beta-methylstyrene and stilbene, J. Mol. Struct., 12 (1972), pp. 333-342.
4. A. B. Birkholz and H. B. Schlegel, Coordinate reduction for exploring chemical reaction paths, Theor. Chem. Acc., 131 (2012), 1170.
5. S. M. Blinder, Basic concepts of self-consistent-field theory, Amer. J. Phys., 33 (1965), pp. 431-443.
6. J. M. Bofill, W. Quapp, and M. Caballero, Locating transition states on potential energy surfaces by the gentlest ascent dynamics, Chem. Phys. Lett., 583 (2013), pp. 203-208.
7. H.-J. Bungartz and M. Griebel, Sparse Grids, Acta Numer. 13, Cambridge University Press, Cambridge, UK, 2004.
8. H.-J. Bungartz, A. Heinecke, D. Pflüger, and S. Schraufstetter, Option pricing with a direct adaptive sparse grid approach, J. Comput. Appl. Math., 236 (2012), pp. 3741-3750.
9. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), pp. 1-43.
10. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J.M.Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox, Gaussian 09, Revision A.1, software, 2009, available at http://www.gaussian.com/.
11. J. Garcke and M. Griebel, Data mining with sparse grids using simplicial basis functions, in Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining -KDD '01, ACM, New York, 2001, pp. 87-96.
12. V. Gradinaru, Fourier transform on sparse grids: Code design and the time dependent Schrödinger equation, Computing, 80 (2007), pp. 1-22.
13. M. Griebel and V. Thurner, The efficient solution of fluid dynamics problems by the combination technique, Internat. J. Numer. Methods Heat Fluid Flow, 5 (1995), pp. 251-269.
14. P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), pp. 864-871.
15. H. Jónsson, G. Mills, and K. W. Jacobsen, Nudged elastic band method for finding mini-mum energy paths of transitions, in Classical and Quantum Dynamics in Condensed Phase Simulations, Proceedings of the International School of Physics, B. J. Berne, D. F. Coker, and G. Cicotti, eds., World Scientific Publishing, Singapore, 1998, pp. 385-404.
16. K. L. Judd, L. Maliar, S. Maliar, and R. Valero, Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain, J. Econom. Dynam. Control, 44 (2014), pp. 92-123.
17. A. Klimke, Efficient Construction of Hierarchical Polynomial Sparse Grid Interpolants Using the Fast Discrete Cosine Transform, technical report, Berichte aus dem Institut fur Angewandte Analysis und Numerische Simulation, Universität Stuttgart, 2006.
18. A. Klimke, Sparse Grid Interpolation Toolbox User's Guide, technical report, Berichte aus dem Institut fur Angewandte Analysis und Numerische Simulation, Universität Stuttgart, 2008.
19. A. Klimke and B. Wohlmuth, Algorithm 847: Spinterp: Piecewise multilinear hierarchical sparse grid interpolation in MATLAB, ACM Trans. Math. Softw., 31 (2005), pp. 561-579.
20. W. Koch and M. C. Holthausen, A Chemist's Guide to Density Functional Theory, 2nd ed., Wiley-VCH, Weinheim, 2001.
21. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), pp. A1133-A1138.
22. X.-L. Luo, C. T. Kelley, L.-Z. Liao, and H. W. Tam, Combining trust-region techniques and Rosenbrock methods to compute stationary points, J. Optim. Theory Appl., 140 (2009), pp. 265-286.
23. B. A. Malin, D. Krueger, and F. Kubler, Solving the multi-country real business cycle model using a Smolyak-collocation method, J. Econom. Dynam. Control, 35 (2011), pp. 229-239.
24. R. M. Minyaev, Reaction path as a gradient line on a potential energy surface, Int. J. Quantum Chem., 49 (1994), pp. 105-127.
25. D. S. Mokrauer, Interpolatory Surrogate Models for Light-Induced Transition Dynamics, Ph.D. thesis, Department of Mathematics, North Carolina State University, Raleigh, NC, 2012.
26. D. S. Mokrauer and C. T. Kelley, Sparse interpolatory reduced-order models for simulation of light-induced molecular transformations, Optim. Methods Softw., 29 (2014), pp. 264-273.
27. D. S. Mokrauer, C. T. Kelley, and A. Bykhovski, Simulations of light-induced molecular transformations in multiple dimensions with incremental sparse surrogates, J. Algorithms Comp. Tech., 6 (2012), pp. 577-592.
28. A. Murarasu, G. Buse, D. Pfluger, J.Weidendorfer, B. Arndt, A. Murarau, D. Püger, and A. Bode, Fastsg: A fast routines library for sparse grids, Proc. Comp. Sci., 9 (2012), pp. 354-363.
29. A. Murarasu, D. Pflüger, J. Weidendorfer, G. Buse, and D. Butnaru, Compact data structure and scalable algorithms for the sparse grid technique, in Proceedings of the ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (PPoPP), ACM, New York, 2011, pp. 1-10.
30. J. Nance, E. Jakubikova, and C. T. Kelley, Reaction path following with sparse interpolation, J. Chem. Theory Comput., 10 (2014), pp. 2942-2949.
31. E. Novak, High dimensional numerical problems, Nonlinear Anal., 30 (1997), pp. 1439-1446.
32. E. Novak and K. Ritter, Simple cubature formulas with high polynomial exactness, Constructive Approx., 15 (1999), pp. 499-522.
33. I. J. Palmer, I. N. Ragazos, F. Bernardi, M. Olivucci, and M. A. Robb, An MC-SCF study of the S1 and S2 photochemical reactions of benzene, J. Am. Chem. Soc., 115 (1993), pp. 673-682.
34. D. Pflüger, Spatially Adaptive Sparse Grids for High-Dimensional Problems, Ph.D. thesis, Institut für Informatik, Technische Universität München, Munich, Germany, 2010.
35. C. C. J. Roothaan, New developments in molecular orbital theory, Rev. Modern Phys., 23 (1951), pp. 69-89.
36. H. B. Schlegel, Optimization of equilibrium geometries and transition structures, J. Comput. Chem., 3 (1982), pp. 214-218.
37. Ch. Schwab and R. A. Todor, Sparse finite elements for stochastic elliptic problems-Higher order moments, Computing, 71 (2003), pp. 43-63.
38. L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), pp. 1-22.
39. D. Sheppard, R. Terrell, and G. Henkelman, Optimization methods for finding minimum energy paths, J. Chem. Phys., 128 (2008), pp. 1-10.
40. S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963), pp. 240-243.
41. J. C. Tully, Molecular dynamics with electronic transitions, J. Chem. Phys., 93 (1990), pp. 1061-1071.
42. G. W. Wasilkowski and H. Wozniakowski, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity, 11 (1995), pp. 1-56.
43. D. L. Woolard, E. R. Brown, M. Pepper, M. Kemp, and R. Brown, Terahertz frequency sensing and imaging: A time of reckoning future applications?, Proc. IEEE, 93 (2005), pp. 1722-1743.
44. D. Xiu, Efficient collocational approach for parametric uncertainty analysis, Comm. Comput. Phys., 2 (2007), pp. 293-309.
45. C. Zenger, Sparse grids, in Notes Numer. Fluid Mech. 31, Vieweg, Braunschweig, Germany, 1991, pp. 241-251.