A computational method to extract macroscopic variables and their dynamics in multiscale systems

SIAM Journal on Applied Dynamical Systems

Volumn 13, Issue 4, 2014, Pages 1816-1846

  Froyland, Gary ^a   Gottwald, Georg A ^b   Hammerlindl, Andy ^a,b  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Koopman operator; Multiscale systems; Slow fast systems; Transfer operator

Indexed keywords

DYNAMICAL SYSTEMS; DYNAMICS;

KOOPMAN OPERATOR; MACROSCOPIC VARIABLES; MULTI-SCALE SYSTEM; NUMERICAL TECHNIQUES; SLOW-FAST SYSTEMS; STATISTICAL FEATURES; TIME SCALE SEPARATION; TRANSFER OPERATOR;

NUMERICAL METHODS;

EID: 84920871817     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/130943637     Document Type: Article

References (72)

1. V. ARNOLD, V. V. KOZLOV, AND A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, New York, 1993.
2. I. BABUŠKA AND J. E. OSBORN, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Math. Comp., 52 (1989), pp. 275-297.
3. V. BALADI AND L.-S. YOUNG, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), pp. 355-385.
4. N. BERGLUND AND B. GENTZ, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer, New York, 2005.
5. M. BLANK, G. KELLER, AND C. LIVERANI, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), pp. 1905-1973.
6. C. BOSE AND G. FROYLAND, Strange eigenmodes for invertible measure-preserving systems, in preparation.
7. P. BOXLER, A stochastic version of the centre manifold theorem, Probab. Theory Related Fields, 83 (1989), pp. 509-545.
8. J. P. BOYD, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover, Mineola, NY, 2001.
9. J. CARR, Applications of Centre Manifold Theory, Appl. Math. Sci. 35, Springer, New York, 1981.
10. A. J. CHORIN AND O. H. HALD, Stochastic Tools in Mathematics and Science, Surv. Tutor. Appl. Math. Sci. 1, Springer-Verlag, New York, 2006.
11. A. J. 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.
12. R. COIFMAN, I. KEVREKIDIS, S. LAFON, M. MAGGIONI, AND B. NADLER, Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems, Multiscale Model. Simul., 7 (2008), pp. 842-864.
13. R. COIFMAN AND S. LAFON, Diffusion maps, Appl. Comput. Harmon. Anal., 21 (2006), pp. 5-30.
14. D. CROMMELIN AND E. VANDEN-EIJNDEN, Diffusion estimation from multiscale data by operator eigenpairs, Multiscale Model. Simul., 9 (2011), pp. 1588-1623.
15. D. T. CROMMELIN AND E. VANDEN-EIJNDEN, Reconstruction of diffusions using spectral data from time-series, Commun. Math. Sci., 4 (2006), pp. 651-668.
16. M. DELLNITZ, G. FROYLAND, AND S. SERTL, A conjecture on the existence of isolated eigenvalues of the Perron-Frobenius operator, in International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), World Scientific, River Edge, NJ, 2000, pp. 1030-1032.
17. M. DELLNITZ AND O. JUNGE, On the approximation of complicated dynamical behavior, SIAMJ. Numer. Anal., 36 (1999), pp. 491-515.
18. P. DEUFLHARD, W. HUISINGA, A. FISCHER, AND C. SCHÜTTE, Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains, Linear Algebra Appl., 315 (2000), pp. 39-59.
19. P. DEUFLHARD AND C. SCHÜTTE, Molecular conformation dynamics and computational drug design, in Applied Mathematics Entering the 21th Century, ICIAM 2003, Sydney, J. M. Hill and R. Moore, eds., 2004, pp. 91-119.
20. W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci., 1 (2003), pp. 423-436.
21. W. E, Principles of Multiscale Modelling, Cambridge University Press, Cambridge, UK, 2011.
22. 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.
23. W. E and E. VANDEN-EIJNDEN, Metastability, conformation dynamics, and transition pathways in complex systems, in Multiscale Modelling and Simulation, Lect. Notes Comput. Sci. Eng. 39, S. Attinger and P. Koumoutsakos, eds., Springer, Berlin, Heidelberg, 2004, pp. 35-68.
24. G. FROYLAND, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), pp. 237-257.
25. G. FROYLAND, Statistically optimal almost-invariant sets, Phys. D, 200 (2005), pp. 205-219.
26. G. FROYLAND, On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps, Discrete Contin. Dyn. Syst., 17 (2007), pp. 671-689.
27. G. FROYLAND, O. JUNGE, AND P. KOLTAI, Estimating long-term behavior of flows without trajectory integration: The infinitesimal generator approach, SIAM J. Numer. Anal., 51 (2013), pp. 223-247.
28. G. FROYLAND AND K. PADBERG, Almost-invariant sets and invariant manifolds-connecting probabilistic and geometric descriptions of coherent structures in flows, Phys. D, 238 (2009), pp. 1507-1523.
29. G. FROYLAND, K. PADBERG, M. ENGLAND, AND A. M. TREGUIER, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007), 224503.
30. G. FROYLAND, N. SANTITISSADEEKORN, AND A. MONAHAN, Transport in time-dependent dynamical systems: Finite-time coherent sets, Chaos, 20 (2010), 043116.
31. C. W. GARDINER, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd ed., Springer, New York, 2003.
32. B. GAVEAU AND L. S. SCHULMAN, Multiple phases in stochastic dynamics: Geometry and probabilities, Phys. Rev. E, 73 (2006), 036124.
33. 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.
34. D. GIVON, R. KUPFERMAN, AND A. STUART, Extracting macroscopic dynamics: Model problems and algorithms, Nonlinearity, 17 (2004), pp. R55-R127.
35. G. A. GOTTWALD AND J. HARLIM, The role of additive and multiplicative noise in filtering complex dynamical systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130096.
36. G. A. GOTTWALD AND I. MELBOURNE, Homogenization for deterministic maps and multiplicative noise, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20130201.
37. C. HARTMANN AND C. SCHÜTTE, A geometric approach to constrained molecular dynamics and free energy, Commun. Math. Sci., 3 (2005), pp. 1-20.
38. C. HARTMANN AND C. SCHÜTTE, Comment on two distinct notions of free energy, Phys. D, 228 (2007), pp. 59-63.
39. W. JUST, H. KANTZ, C. RÖDENBECK, AND M. HELM, Stochastic modelling: Replacing fast degrees of freedom by noise, J. Phys. A, 34 (2001), pp. 3199-3213.
40. I. KEVREKIDIS AND G. SAMAEY, Equation-free multiscale computation: Algorithms and applications, Ann. Rev. Phys. Chem., 60 (2009), pp. 321-344.
41. I. G. KEVREKIDIS, C. W. GEAR, J. M. HYMAN, G. K. PANAGIOTIS, 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.
42. R. Z. KHASMINSKY, On stochastic processes defined by differential equations with a small parameter, Theory Probab. Appl., 11 (1966), pp. 211-228.
43. P. E. KLOEDEN AND E. PLATEN, Numerical methods for stochastic differential equations, in Nonlinear Dynamics and Stochastic Mechanics, CRC Math. Model. Ser., CRC, Boca Raton, FL, 1995, pp. 437-461.
44. P. KOLTAI, Efficient Approximation Methods for the Global Long-Term Behavior of Dynamical Systems-Theory, Algorithms and Examples, Ph.D. thesis, Technische Universität München, Munich, Germany, 2010.
45. T. G. KURTZ, A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12 (1973), pp. 55-67.
46. T. LELIÈVRE, Free Energy Computations: A Mathematical Perspective, Imperial College Press, London, 2010.
47. T. LELIÈVRE, M. ROUSSET, AND G. STOLTZ, Langevin dynamics with constraints and computation of free energy differences, Math. Comp., 81 (2012), pp. 2071-2125.
48. T. Y. LI, Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture, J. Approx. Theory, 17 (1976), pp. 177-186.
49. H. W. LILLIEFORS AND C. C. MCBRIDE, Kolmogorov-Smirnov and Anderson-Darling statistics with parameters estimated by linear combinations of order statistics and some robustness properties, in Goodness-of-Fit (Debrecen, 1984), Colloq. Math. Soc. János Bolyai 45, North-Holland, Amsterdam, 1987, pp. 369-376.
50. W. LIU AND G. HALLER, Strange eigenmodes and decay of variance in the mixing of diffusive tracers, Phys. D, 188 (2004), pp. 1-39.
51. J. MACLEAN AND G. A. GOTTWALD, On convergence of the projective integration method for stiff ordinary differential equations, Commun. Math. Sci., 12 (2014), pp. 235-255.
52. A. J. MAJDA, I. TIMOFEYEV, AND E. VANDEN EIJNDEN, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), pp. 891-974.
53. MATLAB, version 7.10.0 (R2010a), The MathWorks, Inc., Natick, MA, 2010.
54. I. MELBOURNE AND A. STUART, A note on diffusion limits of chaotic skew-product flows, Nonlinearity, 24 (2011), pp. 1361-1367.
55. I. MEZIĆ, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynam., 41 (2005), pp. 309-325.
56. I. MEZIĆ, Analysis of fluid flows via spectral properties of the Koopman operator, in Annual Review of Fluid Mechanics, Volume 45, Annu. Rev. Fluid Mech. 45, Annual Reviews, Palo Alto, CA, 2013, pp. 357-378.
57. I. MEZIĆ AND A. BANASZUK, Comparison of systems with complex behavior, Phys. D, 197 (2004), pp. 101-133.
58. L. MITCHELL AND G. A. GOTTWALD, Data assimilation in slow-fast systems using homogenized climate models, J. Atmospher. Sci., 69 (2012), pp. 1359-1377.
59. H. MORI, H. FUJISAKA, AND H. SHIGEMATSU, A new expansion of the master equation, Progr. Theoret. Phys., 51 (1974), pp. 109-122.
60. N. NAMACHCHIVAYA AND G. LENG, Equivalence of stochastic averaging and stochastic normal forms, J. Appl. Mech., 57 (1990), pp. 1011-1017.
61. N. NAMACHCHIVAYA AND Y. K. LIN, Method of stochastic normal forms, Internat. J. Non-Linear Mech., 26 (1991), pp. 1011-1017.
62. G. C. PAPANICOLAOU, Some probabilistic problems and methods in singular perturbations, Rocky Mountain J. Math., 6 (1976), pp. 653-674.
63. G. A. PAVLIOTIS AND A. M. STUART, Multiscale Methods: Averaging and Homogenization, Springer, New York, 2008.
64. A. J. ROBERTS, Normal form transforms separate slow and fast modes in stochastic dynamical systems, Phys. A, 387 (2008), pp. 12-38.
65. J. A. SANDERS AND F. VERHULST, Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci. 59, Springer-Verlag, New York, 1985.
66. C. SCHÜTTE, W. HUISINGA, AND P. DEUFLHARD, Transfer operator approach to conformational dynamics in biomolecular systems, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler, ed., Springer, Berlin, 2001, pp. 191-223.
67. P. SJÖGREN, Operators associated with the Hermite semigroup-a survey, in Proceedings of the Conference Dedicated to Professor Miguel de Guzmán (El Escorial, 1996), Vol. 3, CRC Press, Boca Raton, FL, 1997, pp. 813-823.
68. P. STINIS, Higher order Mori-Zwanzig models for the Euler equations, Multiscale Model. Simul., 6 (2007), pp. 741-760.
69. S. M. ULAM, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics 8, Interscience, New York, London, 1960.
70. J. WALTER AND C. SCHÜTTE, Conditional averaging for diffusive fast-slow systems: A sketch for derivation, in Analysis, Modeling and Simulation of Multiscale Problems, A. Mielke, ed., Springer, Berlin, 2006, pp. 647-682.
71. R. ZWANZIG, Ensemble method in the theory of irreversibility, J. Chem. Phys., 33 (1960), pp. 1338-1341.
72. R. ZWANZIG, Nonequilibrium Statistical Mechanics, Oxford University Press, Oxford, UK, 2001.