Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and hamiltonian systems with hidden slow dynamics VIA flow averaging

Multiscale Modeling and Simulation

Volumn 8, Issue 4, 2010, Pages 1269-1324

  Tao, Molei ^a   Owhadi, Houman ^a   Marsden, Jerrold E ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

F convergence; Flow averaging; Hamiltonian; Langevin; Multiscale integration; Nonintrusive; Structure preservation

Indexed keywords

F-CONVERGENCE; FLOW AVERAGING; LANGEVIN; MULTISCALES; NON-INTRUSIVE; STRUCTURE PRESERVATION;

DYNAMICAL SYSTEMS; ORDINARY DIFFERENTIAL EQUATIONS; TIME MEASUREMENT;

HAMILTONIANS;

EID: 77956805470     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/090771648     Document Type: Article

References (120)

1. A. Abdulle, Fourth order Chebyshev methods with recurrence relation, SIAMJ. Sci. Comput., 23 (2002), pp. 2041-2054.
2. A. Abdulle and S. Cirilli, S-ROCK: Chebyshev methods for stiff stochastic differential equations, SIAM J. Sci. Comput., 30 (2008), pp. 997-1014.
3. G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), pp. 1482-1518.
4. H. Anderson, RATTLE: A velocity version of the SHAKE algorithm for molecular dynamics calculations, J. Comput. Phys., 52 (1983), pp. 24-34.
5. G. Ariel, B. Engquist, and Y.-H. Tsai, A multiscale method for highly oscillatory ordinary differential equations with resonance, Math. Comp., 78 (2009), pp. 929-956.
6. G. Ariel, B. Engquist, and Y.-H. Tsai, A reversible multiscale integration method, Commun. Math. Sci., 7 (2009), pp. 595-610.
7. A. Armaou and I. Kevrekidis, Equation-free optimal switching policies for bistable reacting systems, Internat.J. Robust Nonlinear Control, 15 (2005), pp. 713-726.
8. S. Artem'ev and K. Shurts, Zhestkie sistemy stokhasticheskikh differentsial'nykh uravneniǐ s malym shumom i ikh chislennoe reshenie, Preprint 1039, Ross. Akad. Nauk Sibirsk. Otdel., Vychisl. Tsentr, Novosibirsk, 1995.
9. Z. Artstein, I.G. Kevrekidis, M. Slemrod, and E.S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity, 20 (2007), pp. 2463-2481.
10. Z. Artstein, J. Linshiz, and E.S. Titi, Young measure approach to computing slowly advancing fast oscillations, Multiscale Model. Simul., 6 (2007), pp. 1085-1097.
11. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.
12. J. Bismut, Mécanique aléatoire, Springer-Verlag, Berlin, New York, 1981.
13. N. Bogolyubov, Problemy dinamičeskoǐ teorii v statističeskoǐ fizike, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, Leningrad, 1946.
14. F.A. Bornemann and C. Schütte, A Mathematical Approach to Smoothed Molecular Dynamics: Correcting Potentials for Freezing Bond Angles, Preprint SC 9-30, Zuse Institute Berlin, Berlin, 1995.
15. F.A. Bornemann and C. Schütte, Homogenization of Hamiltonian systems with a strong constraining potential, Phys. D, 102 (1997), pp. 57-77.
16. N. Bou-Rabee and J. Marsden, Hamilton-Pontryagin integrators on Lie groups.I. Introduction and structure-preserving properties, Found. Comput. Math., 9 (2009), pp. 197-219.
17. N. Bou-Rabee and H. Owhadi, Long-run accuracy of variational integrators in the stochastic context, SIAM J. Numer. Anal., 48 (2010), pp. 278-297.
18. N. Bou-Rabee and H. Owhadi, Stochastic variational integrators, IMA J. Numer. Anal., 29 (2009), pp. 421-443.
19. N. Bou-Rabee and E. Vanden-Eijnden, Pathwise accuracy and ergodicity of metropolized integrators for SDEs, Comm. Pure Appl. Math., 63 (2010), pp. 655-696.
20. C. Le Bris and F. Legoll, Integrators for highly oscillatory Hamiltonian systems: An homogenization approach, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), pp. 347-373.
21. B. Brooks, R. Bruccoleri, B. Olafson, D. States, S. Swaminathan, and M. Karplus, CHARMM: A program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4 (1983), pp. 187-217.
22. K. Burrage and T. Tian, Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations, Comput. Phys. Comm., 142 (2001), pp. 186-190. (Pubitemid 33142512)
23. K. Burrage and T. Tian, The composite Euler method for stiff stochastic differential equations, J. Comput. Appl. Math., 131 (2001), pp. 407-426. (Pubitemid 32621157)
24. M.P. Calvo and J.M. Sanz-Serna, Heterogeneous multiscale methods for mechanical systems with vibrations, SIAM J. Sci. Comput., 32 (2010), pp. 2029-2046.
25. M.P. Calvo and J.M. Sanz-Serna, Instabilities and inaccuracies in the integration of highly oscillatory problems, SIAM J. Sci. Comput., 31 (2009), pp. 1653-1677.
26. F. Castella, P. Chartier, and E. Faou, An averaging technique for highly oscillatory Hamiltonian problems, SIAM J. Numer. Anal., 47 (2009), pp. 2808-2837.
27. A. Chorin, O. Hald, and R. Kupferman, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 2968-2973.
28. A. Chorin, O. Hald, and R. Kupferman, Optimal prediction with memory, Phys. D, 166 (2002), pp. 239-257.
29. A. Chorin, A. Kast, and R. Kupferman, Optimal prediction of underresolved dynamics, Proc. Natl. Acad. Sci. USA, 95 (1998), pp. 4094-4098.
30. G. Ciccotti, T. Lelievre, and E. Vanden-Eijnden, Projections of diffusions on submanifolds: Application to mean force computation, Comm. Pure Appl. Math., 61 (2008), pp. 371-408.
31. D. Cohen, T. Jahnke, K. Lorenz, and C. Lubich, Numerical integrators for highly oscillatory Hamiltonian systems: A review, in Analysis, Modeling and Simulation of Multiscale Problems, Springer-Verlag, Berlin, 2006, pp. 553-576.
32. M. Condon, A. Deaño, and A. Iserles, On highly oscillatory problems arising in electronic engineering, M2AN Math. Model. Numer. Anal., 43 (2009), pp. 785-804.
33. W. Cornell, P. Cieplak, C. Bayly, I. Gould, K. Merz, D. Ferguson, D. Spellmeyer, T. Fox, J. Caldwell, and P. Kollman, A second generation force field for the simulation of proteins, nucleic acids, and organic molecules, J. Am. Chem. Soc., 117 (1995), pp. 5179-5197.
34. F. Cucker and A.G. Corbalan, An alternate proof of the continuity of the roots of a polynomial, Amer. Math. Monthly, 96 (1989), pp. 342-345.
35. G. Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand., 4 (1956), pp. 33-53.
36. W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci., 1 (2003), pp. 423-436.
37. W. E, B. Engquist, X. Li, W. Ren, and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), pp. 367-450.
38. W. E, D. Liu, and E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), pp. 1544-1585.
39. W. E, D. Liu, and E. Vanden-Eijnden, Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales, J. Comput. Phys., 221 (2007), pp. 158-180.
40. W. E, W. Ren, and E. Vanden-Eijnden, A general strategy for designing seamless multiscale methods, J. Comput. Phys., 228 (2009), pp. 5437-5453.
41. B. Engquist and Y.-H. Tsai, Heterogeneous multiscale methods for stiff ordinary differential equations, Math. Comp., 74 (2005), pp. 1707-1742.
42. L. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57 (2004), pp. 445-480.
43. E. Fermi, J. Pasta, and S. Ulam, Studies of Nonlinear Problems, Tech. report LA-1940, Los Alamos Scientific Laboratory, Los Alamos, NM, 1955.
44. M. Fixman, Classical statistical mechanics of constraints: A theorem and application to polymers, Proc. Nat. Acad. Sci.U.S.A., 71 (1974), pp. 3050-3053.
45. S. Flach, M.V. Ivanchenko, and O.I. Kanakov, q-breathers and the Fermi-Pasta-Ulam problem, Phys. Rev. Lett., 95 (2005), 064102.
46. J. Ford, The Fermi-Pasta-Ulam problem: Paradox turns discovery, Phys. Rep., 213 (1992), pp. 271-310.
47. B. García-Archilla, J.M. Sanz-Serna, and R.D. Skeel, Long-time-step methods for oscillatory differential equations, SIAM J. Sci. Comput., 20 (1998), pp. 930-963.
48. W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), pp. 381-397.
49. C. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice- Hall, Englewood Cliffs, NJ, 1971.
50. C. Gear and K. Gallivan, Automatic methods for highly oscillatory ordinary differential equations, in Numerical Analysis (Dundee, 1981), Lecture Notes in Math. 912, Springer- Verlag, Berlin, 1982, pp. 115-124.
51. C.W. Gear and I.G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comput., 24 (2003), pp. 1091- 1106.
52. I. Gihman, On the theory of differential equations of stochastic processes. I, II, Amer. Math. Soc. Transl. (2), 1 (1955), pp. 111-137, 139-161.
53. D. Givon, I.G. Kevrekidis, and R. Kupferman, Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Commun. Math. Sci., 4 (2006), pp. 707-729.
54. H. Grubmuller, H. Heller, A. Windemuth, and K. Schulten, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Mol. Simul., 6 (1991), pp. 121-142.
55. S. Gusev, Algoritm peremennogo shaga dlya chislennogo resheniya zhestkikh sistem stokhasticheskikh differentsial'nykh uravneniǐ, Preprint 1094, Rossiǐskaya Akademiya Nauk Sibirskoe Otdelenie, Institut Vychislitel'noǐ Matematiki i Matematicheskoǐ Geofiziki, Novosibirsk, 1997.
56. E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration illustrated by the St?ormer-Verlet method, Acta Numer., 12 (2003), pp. 399-450.
57. E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure- Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer-Verlag, Heidelberg, 2004.
58. E. Hairer, S. Nørsett, and G. Wanner, Solving Ordinary Differential Equations.I. Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math. 8, Springer-Verlag, Berlin, 1993.
59. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, 2nd ed., Springer Ser. Comput. Math. 14, Springer- Verlag, Berlin, 1996.
60. C. Hartmann, An ergodic sampling scheme for constrained Hamiltonian systems with applications to molecular dynamics, J. Stat. Phys., 130 (2008), pp. 687-711.
61. A. Iserles, On the global error of discretization methods for highly-oscillatory ordinary differential equations, BIT, 42 (2002), pp. 561-599.
62. A. Iserles, On the numerical analysis of rapid oscillation, in Group Theory and Numerical Analysis, CRM Proc. Lecture Notes 39, AMS, Providence, RI, 2005, pp. 149-163.
63. A. Iserles, Think globally, act locally: Solving highly-oscillatory ordinary differential equations, Appl. Numer. Math., 43 (2002), pp. 145-160.
64. A. Iserles, S.P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, in Numerical Mathematics and Advanced Applications, Springer-Verlag, Berlin, 2006, pp. 97-118.
65. V. Jikov, S. Kozlov, and O. Oleinǐk, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
66. P. Kapitza, Collected Papers of P.L. Kapitza, Vol.II, D. Ter Haar, ed., Pergamon Press, Oxford, UK, 1965.
67. I. Kevrekidis, C. Gear, J. Hyman, P. Kevrekidis, O. Runborg, and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1 (2003), pp. 715-762.
68. I. Kevrekidis and G. Samaey, Equation-free multiscale computation: Algorithms and applications, Annu. Rev. Phys. Chem., 60 (2009), pp. 321-344.
69. H.-O. Kreiss, Problems with different time scales, in Acta Numerica, 1992, Cambridge University Press, Cambridge, UK, 1992, pp. 101-139.
70. B. Kryloff and N. Bogoliouboff, La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math. (2), 38 (1937), pp. 65-113.
71. N. Kryloff and N. Bogoliouboff, On some problems in the ergodic theory of stochastic systems, Zap. Kafedr. Mat. Fiz. Inst. Budivel. Mat. Akad. Nauk. Ukrain. SSR, 4 (1939), pp. 243-287.
72. J.-A. Lázaro-Camí and J. Ortega, Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), pp. 65-122.
73. V. Lebedev and S. Finogenov, The use of ordered Čebyšev parameters in iteration methods, Ž. Vyčisl. Mat. i Mat. Fiz., 16 (1976), pp. 895-907, 1084.
74. B. Leimkuhler and S. Reich, A reversible averaging integrator for multiple time-scale dynamics, J. Comput. Phys., 171 (2001), pp. 95-114.
75. B. Leimkuhler and S. Reich, Simulating Hamiltonian Dynamics, Cambridge Monogr. Appl. Comput. Math. 14, Cambridge University Press, Cambridge, UK, 2004.
76. B.J. Leimkuhler, S. Reich, and R.D. Skeel, Integration methods for molecular dynamics, in Mathematical Approaches to Biomolecular Structure and Dynamics (Minneapolis, MN, 1994), IMA Vol. Math. Appl. 82, Springer-Verlag, New York, 1996, pp. 161-185.
77. A. Lew, J. Marsden, M. Ortiz, and M. West, Asynchronous variational integrators, Arch. Ration. Mech. Anal., 167 (2003), pp. 85-146.
78. A. Lew, J. Marsden, M. Ortiz, and M. West, Variational time integrators, Internat.J. Numer. Methods Engrg., 60 (2004), pp. 153-212.
79. T. Li, A. Abdulle, andW. E, Effectiveness of implicit methods for stiff stochastic differential equations, Commun. Comput. Phys., 3 (2008), pp. 295-307.
80. R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7 (1994), pp. 1623-1643.
81. S.J.A. Malham and A. Wiese, Stochastic Lie group integrators, SIAM J. Sci. Comput., 30 (2008), pp. 597-617.
82. J. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), pp. 357-514.
83. R. McLachlan and D. O'Neale, Comparison of Integrators for the Fermi-Pasta-Ulam Problem, Preprint NI07052-HOP, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, 2007; also available online from http://www.newton.ac.uk/preprints/NI07052.pdf.
84. R. McLachlan and M. Perlmutter, Conformal Hamiltonian systems, J. Geom. Phys., 39 (2001), pp. 276-300.
85. R. McLachlan, G. Reinout, and W. Quispel, Splitting methods, Acta Numer., 11 (2002), pp. 341-434.
86. G.N. Milstein, Yu.M. Repin, and M.V. Tretyakov, Numerical methods for stochastic systems preserving symplectic structure, SIAM J. Numer. Anal., 40 (2002), pp. 1583-1604.
87. G.N. Milstein, Yu.M. Repin, and M.V. Tretyakov, Symplectic integration of Hamiltonian systems with additive noise, SIAM J. Numer. Anal., 39 (2002), pp. 2066-2088.
88. G. Milstein and M. Tretyakov, Quasi-symplectic methods for Langevin-type equations, IMA J. Numer. Anal., 23 (2003), pp. 593-626.
89. G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin, 2004.
90. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), pp. 608-623.
91. G. Papanicolaou and W. Kohler, Asymptotic theory of mixing stochastic ordinary differential equations, Comm. Pure Appl. Math., 27 (1974), pp. 641-668.
92. G.A. Pavliotis and A.M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts Appl. Math. 53, Springer-Verlag, New York, 2008.
93. D. Perchak, J. Skolnick, and R. Yaris, Dynamics of rigid and flexible constraints for polymers. Effect of the Fixman potential, Macromolecules, 18 (1985), pp. 519-525.
94. L. Petzold, L. Jay, and J. Yen, Problems with different time scales, in Acta Numerica, 1997, Cambridge University Press, Cambridge, UK, 1997, pp. 437-483.
95. S. Reich, Smoothed Langevin dynamics of highly oscillatory systems, Phys. D, 138 (2000), pp. 210-224.
96. R. Rosenberg, B. Berne, and D. Chandler, Isomerization dynamics in liquids by molecular dynamics, Chem. Phys. Lett., 75 (1980), pp. 162-168.
97. J. Ryckaert and A. Bellemans, Molecular dynamics of liquid n-butane near its boiling point, Chem. Phys. Lett., 30 (1975), pp. 123-125.
98. J. Ryckaert, G. Ciccotti, and H. Berendsen, Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-Alkanes, J. Comput. Phys., 23 (1977), pp. 327-341.
99. J.A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59, Springer-Verlag, New York, 1985.
100. J.M. Sanz-Serna, Modulated Fourier expansions and heterogeneous multiscale methods, IMA J. Numer. Anal., 29 (2009), pp. 595-605.
101. J.M. Sanz-Serna, Mollified impulse methods for highly oscillatory differential equations, SIAM J. Numer. Anal., 46 (2008), pp. 1040-1059.
102. J.M. Sanz-Serna, Stabilizing with a hammer, Stoch. Dyn., 8 (2008), pp. 47-57.
103. J.M. Sanz-Serna, G. Ariel, and Y.-H. Tsai, Multiscale Methods for Stiff and Constrained Mechanical Systems, preprint, 2009.
104. R. Scheid, The accurate numerical solution of highly oscillatory ordinary differential equations, Math. Comp., 41 (1983), pp. 487-509.
105. C. Schütte and F.A. Bornemann, Homogenization approach to smoothed molecular dynamics, Nonlinear Anal., 30 (1997), pp. 1805-1814.
106. R. Sharp, Y.-H. Tsai, and B. Engquist, Multiple time scale numerical methods for the inverted pendulum problem, in Multiscale Methods in Science and Engineering, Lect. Notes Comput. Sci. Eng. 44, Springer-Verlag, Berlin, 2005, pp. 241-261.
107. R. Skeel and J.A. Izaguirre, An impulse integrator for Langevin dynamics, Mol. Phys., 100 (2002), pp. 3885-3891.
108. A. Skorokhod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Transl. Math. Monogr. 78, AMS, Providence, RI, 1989.
109. A. Stern and E. Grinspun, Implicit-explicit variational integration of highly oscillatory problems, Multiscale Model. Simul., 7 (2009), pp. 1779-1794.
110. W.B. Streett, D.J. Tildesley, and G. Saville, Multiple time-step methods in molecular dynamics, Mol. Phys., 35 (1978), pp. 639-648.
111. F. Takens, Motion under the influence of a strong constraining force, in Global Theory of Dynamical Systems, Z. Nitecki and C. Robinson, eds., Springer-Verlag, Berlin, Heidelberg, 1980, pp. 425-445.
112. T. Tian and K. Burrage, Implicit Taylor methods for stiff stochastic differential equations, Appl. Numer. Math., 38 (2001), pp. 167-185. (Pubitemid 32498432)
113. M. Tuckerman, B.J. Berne, and G.J. Martyna, Reversible multiple time scale molecular dynamics, J. Chem. Phys., 97 (1992), pp. 1990-2001.
114. E. Vanden-Eijnden, On HMM-like integrators and projective integration methods for systems with multiple time scales, Commun. Math. Sci., 5 (2007), pp. 495-505.
115. E. Vanden-Eijnden and G. Ciccotti, Second-order integrators for Langevin equations with holonomic constraints, Chem. Phys. Lett., 429 (2006), pp. 310-316.
116. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 2nd ed., Springer- Verlag, Berlin, Heidelberg, 1996.
117. L. Verlet, Computer "experiments" on classical fluids.I. Thermodynamical properties of Lennard-Jones molecules, Phys. Rev., 159 (1967), pp. 98-103.
118. D. Viswanath, Global errors of numerical ODE solvers and Lyapunov's theory of stability, IMA J. Numer. Anal., 21 (2001), pp. 387-406.
119. T. Yanao, W.S. Koon, and J.E. Marsden, Intramolecular energy transfer and the driving mechanisms for large-Amplitude collective motions of clusters, J. Chem. Phys., 130 (2009), 144111.
120. G. Zhang and T. Schlick, LIN: A new algorithm to simulate the dynamics of biomolecules by combining implicit-integration and normal mode techniques, J. Comput. Chem., 14 (1993), pp. 1212-1233.