The moment map: Nonlinear dynamics of density evolution via a few moments

SIAM Journal on Applied Dynamical Systems

Volumn 5, Issue 3, 2006, Pages 403-434

  Barkley, D ^a   Kevrekidis, I G ^b   Stuart, A M ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b Princeton University   (United States)

Author keywords

Equation free; Metastable states; Moment map; Multiscale

Indexed keywords

BIFURCATION ANALYSES; DENSITY EVOLUTIONS; DETERMINISTIC SYSTEMS; DIMENSIONAL DYNAMICS; EQUATION-FREE; HEAT BATHS; METASTABLE STATES; MOLECULAR DYNAMICS SIMULATIONS; MOMENT MAP; MOMENT MAPS; MULTISCALE; MULTISCALE ALGORITHMS; NON LINEARITIES; NONLINEAR DYNAMICS; NUMERICAL TOOLS; SIMPLE STOCHASTIC;

CONFORMAL MAPPING; CONTROL NONLINEARITIES; CONTROL THEORY; DIFFERENTIAL EQUATIONS; DYNAMICAL SYSTEMS; MAPS; MOLECULAR DYNAMICS; OPTICAL PROJECTORS; PROBABILITY DISTRIBUTIONS; QUANTUM CHEMISTRY; SCHRODINGER EQUATION; STOCHASTIC MODELS;

DYNAMICS;

EID: 41649118783     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/050638667     Document Type: Article

References (43)

1. Y. AIT-SAHALIA, Maximum-likelihood estimation of discretely sampled diffusions: A closed-form approximation approach, Econometrica, 70 (2002), pp. 223-262.
2. A. ANGELETTI, C. CASTAGNARI, AND F. ZIRILLI, Asymptotic eigenvalue degeneracy for a class of one-dimensional Fokker-Planck operators, J. Math. Phys., 26 (1985), pp. 678-691.
3. A. CHORIN, O. H. HALD, AND R. KUPFERMAN, Optimal prediction and the Mori-Zwanzig representation of irreversible processes, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 2968-2973.
4. J. CLSTERNAS, C. W. GEAR, S. LEVIN, AND I. G. KEVREKIDIS, Equation-free modeling of evolving diseases: Coarse-grained computations with individual-based models, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), pp. 27621-2779; also available online at http://arXiv.org/abs/nlin.AO/0310011.
5. P. DEUFLHARD, W. HUISINGA, W. FISCHER, AND C. SCHÜTTE, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra Appl., 315 (2000), pp. 39-59.
6. A. DOUCET, N. DE FREITAS, AND N. J. GORDON, Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, 2001.
7. W. E, D. LIU, AND E. VANDEN-EUNDEN, Analysis of' multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), pp. 1544-1585.
8. P. EHRENFEST, Collected Scientific Papers, North-Holland, Amsterdam, 1959.
9. R. ERBAN, I. G. KEVREKIDIS, AND H. G. OTHMER, An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal, Phys. D. 215 (2006), pp. 1-24.
10. G. W. FORD AND M. KAC, On the quantum Langevin equation, J. Statist. Phys., 46 (1987), pp. 803-810.
11. C. W. GARDINER, Handbook of Stochastic Methods, 2nd ed., Springer-Verlag, New York, Berlin, 1985.
12. C. W. GEAR, I. G. KEVREKIDIS, AND K. THEODOROPOULOS, "Coarse" integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods, Comp. Chem. Engrg., 26 (2002), pp. 941-963.
13. 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.
14. C. W. GEAR AND I. G. KEVREKIDIS, Telescopic projective integrators for stiff differential equations, J. Comput. Phys., 187 (2003), pp. 95-109.
15. C. W. GEAR, Projective Integration Methods for Distributions, Technical Report NEC TR 2001130, NEC Institute, Princeton, NJ, 2001.
16. D. GIVON, R. KLTPFERMAN, AND A. M. STUART, Extracting macroscopic dynamics: Model problems and algorithms, Nonliriearity, 17 (2004), pp. R55-R127.
17. A. N. GORBAN, I. V. KARLIN, H. C. OETTINGER, AND L. L. TATARINOVA, Ehrenfest's argument extended to a formalism, of nonequilibrium thermodynamics, Phys. Rev. E (3), 63 (2001), paper 066124.
18. M. HAATAJA, D. SROLOVITZ, AND I. G. KEVREKIDIS, Apparent hysteresis in a driven system with self-organized drag, Phys. Rev. Lett., 92 (2004), paper 160603; also available online at http://arXiv.org/abs/cond-mat/0310460.
19. O. HALD, Optimal prediction of the Klein-Gordon equation, Proc. Natl. Acad. Sci. USA, 96 (1999), pp. 4774-4779.
20. C. HILLERMEIER, N. KUNSTMANN, B. RABUS, AND P. TAVAN, Topological feature maps with self-organized lateral connections: A population coded, one-layer model of associative memory, Biol. Cybernet., 72 (1994), pp. 103-117.
21. W. HUISINGA, Metastability of Markovian Systems, Ph.D. thesis, Mathematics Department, Free University, Berlin, 2003; available online at http://page.mi.fu-berlin.de/~huisinga/publications/index.html.
22. G. HUMMER AND I. G. KEVREKIDIS, Coarse molecular dynamics of a peptide fragment: Free energy, kinetics and long time dynamics computations, J. Chem. Phys., 118 (2003), pp. 10762-10773; also available online at http://arXiv.org/abs/physics/0212108.
23. C. T. KELLEY, I. G. KEVREKIDIS, AND L. QIAO, Newton-Krylov solvers for time-steppers, SIAM J. Appl. Dyn. Syst., submitted; also available online at http://arXiv.org/abs/math.DS/0404374.
24. C. T. KELLEY, Iterative Methods for Linear and Nonlinear Equations, Frontiers Appl. Math. 16, SIAM, Philadelphia, 1995.
25. I. G. KEVREKIDIS, C. W. GEAR, J. M. HYMAN, P. G. KEVREKIDIS, O. RUNBORG, AND K. THEODOR-OPOULOS, Equation-free coarse-grained, multiscale computation: Enabling microscopic simulators to perform system-level tasks, Commun. Math. Sci., 1 (2003), pp. 715-762; original version available online at http://arXiv.org/abs/physics/ 0209043.
26. I. G. KEVREKIDIS, C. W. GEAR, AND G. HUMMER, Equation-free: The computer-assisted analysis of complex, multiscale systems, AIChE J., 50 (2004), pp. 1346-1354.
27. D. I. KOPELEVICH, A. Z. PANAGIOTOPOULOS, AND I. G. KEVREKIDIS, Coarse-grained kinetic computations of rare events: Application to micelle formation, J. Chem. Phys., 122 (2005), paper 044908; also available online at http://arXiv.org/abs/cond-mat/0407220.
28. R. KUPFERMAN, A. M. STUART, J. TERRY, AND P. TUPPER, Long time behaviour of large mechanical systems with random initial data, Stoch. Dyn., 2 (2002), pp. 533-562.
29. A. MAKEEV, D. MAROUDAS, AND I. G. KEVREKIDIS, "Coarse" stability and bifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples, J. Chem. Phys., 116 (2002), pp. 10083-10091.
30. A. G. MAKEEV, D. MAROUDAS, A. Z. PANAGIOTOPOULOS, AND I. G. KEVREKIDIS, Coarse bifurcation analysis of kinetic Monte Carlo simulations: A lattice gas model with lateral interactions, J. Chem. Phys., 117 (2002), pp. 8229-8240.
31. A. G. MAKEEV AND I. G. KEVREKIDIS, Equation-free multiscale computations for a lattice-gas model: Coarse-grained bifurcation analysis of the NO+CO reaction on, Pt(100), Chem. Eng. Sci., 59 (2004), pp. 1733-1743.
32. B. J. MATKOWSKY AND Z. SCHUSS, Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusions in potential fields, SIAM J. Appl. Math., 40 (1981), pp. 242-254.
33. E. NELSON, Dynamical Theories of Brownian Motion; available online at http://www.math,princeton.edu/~nelson/books.html.
34. G. PAVLIOTIS AND A. M. STUART, White noise limits for inertial particles in a random field, Multiscale Model. Simul., 1 (2003), pp. 527-553.
35. R. RLCO-MARTINEZ, C. W. GEAR. AND I. G. KEVREKIDIS, Coarse projective kMC integration: Forward/reverse initial and boundary value problems, J. Comput. Phys., 196 (2004), pp. 474-489; also available online at http://arXiv.org/abs/nlin.CG/0307016.
36. C. SCHUETTE, J. WALTER, C. HARTMANN, AND W. HUISINGA, An averaging principle for fast degrees of freedom exhibiting long-term correlations, Multiscale Model. Simul., 2 (2004), pp. 501-526.
37. S. SETAYESHGAR, C. W. GEAR, H. G. OTHMER, AND I. G. KEVREKIDIS, Application of coarse integration to bacterial chemotaxis, Multiscale Model. Simul., 4 (2005), pp. 307-327; also available online at http://arXiv.org/abs/physics /0308040.
38. C. SIETTOS, M. D, GRAHAM, AND I. G. KEVREKIDIS, Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration and control via stochastic simulation, J. Cliem. Phys., 118 (2003), pp. 10149-10157; also available online at http://arXiv.org/abs/cond-mat/0211455.
39. S. SRIRAMAN, I. G. KEVREKIDIS, AND G. HUMMER, Coarse nonlinear dynamics of filling-emptying transitions: Water in carbon nanotubes, Phys. Rev. Lett., 95 (2005), paper 130603; also available online at http://arXiv.org/abs/cond-mat/0503491.
40. K. THEODOROPOULOS, Y.-H. QIAN, AND I. G. KEVREKIDIS, "Coarse" stability and bifurcation analysis using timesteppers: A reaction diffusion example, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 9840-9843.
41. K. THEODOROPOULOS, K. SANKARANARAYANAN, S. SUNDARESAN, AND I. G. KEVREKIDIS, Coarse bifurcation studies of bubble flow lattice Boltzmann simulations, Chem. Eng. Sci., 59 (2004), pp. 2357-2362; also available online at http://arXiv.org/abs/nlin.PS/ 0111040.
42. E. VANDEN ELJNDEN, Numerical techniques for multi-scale dynamical systems with stochastic effects, Commun. Math. Sci., 1 (2003), pp. 377-384.
43. R. ZWANZIG, Nonlinear generalized Langevin equations, J. Statist. Phys., 9 (1973), pp. 215-220.