Equation-free: The computer-aided analysis of complex multiscale systems

AIChE Journal

Volumn 50, Issue 7, 2004, Pages 1346-1355

  Kevrekidis, Ioannis G ^a   Gear, C William ^a,b   Hummer, Gerhard ^c  

^a Princeton University   (United States)

^b NEC LABORATORIES AMERICA   (United States)

^c NATIONAL INSTITUTE OF DIABETES AND DIGESTIVE AND KIDNEY DISEASES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER APPLICATION;

CHEMICAL REACTION; COMPLEX MULTISCALE SYSTEM; COMPUTER ANALYSIS; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICS; MOLECULAR DYNAMICS; REVIEW; SIMULATION; TECHNOLOGY;

EID: 3042720018     PISSN: 00011541     EISSN: None     Source Type: Journal    
DOI: 10.1002/aic.10106     Document Type: Review

References (83)

1. Armaou, A., C. I. Siettos, and I. G. Kevrekidis "Time-Steppers and Coarse Control of Microscopic Distributed Processes," Int. J. Robust & Nonlinear Control, 14, 89 (2004).
2. Armaou, A., and I. G. Kevrekidis, "Optimal Switching Policies Using Coarse Timesteppers," Proc. of the 2003 CDC Conf HI (2003). (Can be obtained as nlin.CG/0309024 at arXiv.org)
3. Bodenstein, M., "Eine Theorie der Photochemischen Reaktionsgeschwindigkeiten," Z. Phys. Chem., 85, 329 (1913).
4. Box, G. E. P., W. Hunter, and J. S. Hunter, Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building, Wiley, New York (1978)
5. Brown, R. A., L. E. Scriven, and W. J. Silliman "Computer-Aided Analysis of Nonlinear Problems in Transport Phenomena," New Approaches to Nonlinear Problems in Dynamics, P. J. Holmes, ed., SIAM Publications, Philadelphia, p. 298 (1980)
6. Car, R., and M. Parrinello, "Unified Approach for Molecular Dynamics and Density Functional Theory, Phys. Rev. Lett., 55, 2471 (1985).
7. Chen, L., P. G. Debenedetti, C. W. Gear, and I. G. Kevrekidis, "From Molecular Dynamics to Coarse Self-Similar Solutions: a Simple Example Using Equation-Free Computation," J N NFM., in press (2004). (Can be found as nlin.AO/0310011 at arXiv.org)
8. Coifman, R. R., S. Lafon, A. Lee, M. Maggioni, F. Warner, and S. Zucker, "Geometric Diffusions as a Tool for Harmonic Analysis and Structure Definition of Data," PNAS, submitted (2004).
9. Constantin, P., C. Foias, B. Nicolaenko, and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer Verlag, New York (1988).
10. Chorin, A., A. Kast, and R. Kupferman, "Optimal Prediction for Underresolved Dynamics," Proc. Natl. Acad. Sci. USA, 95, 4094 (1998).
11. Chorin, A., O. Hald, and R. Kupferman, "Optimal Prediction and the Mori-Zwanzig Representation of Irreversible Processes," Proc. Natl. Acad. Sci. USA, 97, 2968 (2000).
12. Cisternas, J., C. W. Gear, S. Levin, and I. G. Kevrekidis, "Equation-Free Modeling of Evolving Diseases: Coarse-Grained Computations with Individual-Based Models", Proc. Roy. Soc., London, U.K. in press (2004). (Can be found as nlin.AO/0310011 at arXiv.org)
13. Cybenko, G., "Just in Time Learning and Estimation," in Identification, Adaptation and Learning: the Science of Learning Models from Data, NATO ASI Ser. F153 423, Springer, Berlin (1996).
14. E. W., and B. Engquist, "The Heterogeneous Multiscale Methods," Comm. Math. Sci., 1, 87 (2003).
15. Ehrenfest, P., and T. Ehrenfest, in Enzyklopaedie der Mathematsichen Wissenschaften (1911). Reprinted in P. Ehrenfest, Collected Scientific Papers, North Holland (1959).
16. Gear, C. W., "Projective Integration Methods for Distributions," NEC Technical Report NECI TR 2001-130 (Nov. 2001).
17. Gear, C. W., and I. G. Kevrekidis, "Projective Methods for Stiff Differential Equations: Problems with Gaps in their Eigenvalue Spectrum," SIAM J. Sci. Comp., 24(4), 1091 (2003). (Original NEC Technical Report NECI-TR 2001-029, Apr., 2001)
18. Gear, C. W., I. G. Kevrekidis, and K. Theodoropoulos, "Coarse Integration/Bifurcation Analysis via Microscopic Simulators: Micro-Galerkin Methods," Comp. Chem. Eng., 26, 941 (2002). (Original NEC Technical Report NECI TR 2001-106, Oct 2001)
19. Gear, C. W., J. Li, and I. G. Kevrekidis, "The Gaptooth Method in Particle Simulations," Phys. Lett. A., 316, 190 (2003).
20. Gear, C. W., and I. G. Kevrekidis, "Computing in the Past with Forward Integration," Physics Letters A., 321, 335 (2004).
21. Givon, D., R. Kupferman, and A. Stuart, "Extracting Macroscopic Dynamics: Model Problems and Algorithms," Nonlinearity, in press (2004). Can be obtained as Warwick Preprint 11/2003, http://www.maths.warwick.ac.uk/ ~stuart/extract.pdf.
22. Guckenheimer, J., and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, (Appl. Math. Sci., 42), New York (1983).
23. Haenggi, P., P. Talkner and M. Borkovec, "Reaction-Rate Theory: 50 Years After Kramers," Rev. Mod. Phys. 62(2), 251 (1990).
24. Holmes, P., J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge (1998).
25. Huber, T., and W. F. van Gunsteren, "SWARM-MD: Searching Conformational Space by Cooperative Molecular Dynamics," J. Chem. Phys. A., 102(29), 5937 (1998).
26. Hummer, G., and I. G. Kevrekidis, "Coarse Molecular Dynamics of a Peptide Fragment: Free Energy, Kinetics and Long Time Dynamics Computations", J. Chem. Phys., 118(23), 10762 (2003).
27. Iannuzzi, M., A. Laio, and M. Parrinello, "Efficient Exploration of Reactive Potential Energy Surfaces Using Car-Parrinello Molecular Dynamics," Phys. Rev. Lett., 90(23) 238302 (2003).
28. Kalman, R. E., and R. S. Bucy, "New Results in Linear Filtering and Prediction Theory," Trans. ASME, Part D, J. of Basic Eng., 83, 95 (1961).
29. Katsoulakis, M. A., A. J. Majda, and D. G. Vlachos, "Coarse Grained Stochastic Srocesses for Microscopic Lattice Systems," Proc. Nat. Acad. Sci., USA 100(3) 782 (2003).
30. Kelley, C. T., Iterative Methods for Solving Linear and Nonlinear Equations, SIAM Publications, Philadelphia (1995).
31. Kevrekidis, I. G., "Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators," Plenary Lecture, CAST Division, AIChE Meeting, Los Angeles (2000). (Can be found at http://arnold.princeton.edu/ ~yannis)
32. Kevrekidis, I. G., C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and K. Theodoropoulos, "Equation-Free Coarse-Grained Multiscale Computation: Enabling Microscopic Simulators to Perform System-Level Tasks," Comm. Math. Sciences, 1(4) 715 (2003); (Original version can be obtained as physics/0209043 at arXiv.org).
33. Kopelevich, D., A. Z. Panagiotopoulos, and I. G. Kevrekidis, "Coarse Grained Computations for a Micellar System," J. Chem. Phys., submitted (2004a).
34. Kopelevich, D., A. Z. Panagiotopoulos, and I. G. Kevrekidis, "Coarse Kinetic Approach to Rare Events: Application to Micelle Formation," J. Chem. Phys., submitted (2004b).
35. Krener, A. J., "Nonlinear Observers in Control Systems, Robotics and Automation," Encyclopedia of Life Support Systems (EOLSS), H. Unbehauen, ed., Eolss Publishers, Oxford (2003).
36. Kupferman, R., and A. Stuart, "Fitting SDE Models to Nonlinear Kac-Zwanzig Heat Bath Models," Physica D., in press (2004).
37. Laio, A., and M. Parrinello, "Escaping Free Energy Minima," Proc. Natl. Acad. Sci. USA, 99(20), 12562 (2002).
38. Lall, S., J. E. Marsden, and S. Glavaski, "A Subspace Approach to Balanced Truncation for Model Reduction of Nonlinear Control Systems," Int. J. Robust & Nonl. Control, 12, 519 (2002).
39. Laso, M., and H.-C. Oettinger, "Calculation of Viscoelastic Flow Using Molecular Models: The CONNFFESSIT Approach," JNNFM, 47, 1 (1993).
40. Lehoucq, R. B., D. C. Sorensen and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia (1998).
41. Li, J., D. Liao, and S. Yip, "Imposing Field Boundary Conditions in MD Simulations of Fluids: Optimal Particle Controller and Buffer Zone Feedback," Mat. Res. Soc. Symp. Proc., 538, 473. (1998).
42. Li, J., P. G. Kevrekidis, C. W. Gear, and I. G. Kevrekidis, "Deciding the Nature of the Coarse Equation through Microscopic Simulation," SIAM MMS 1(3) 391 (2003).
43. Ljung, L., System Identification: Theory for the User, 2nd ed., Prentice Hall, New York (1999).
44. Lu, G. and E. Kaxiras, "An Overview of Multiscale Simulations of Materials," cond-mat/0401073 preprint at arXiv.org.
45. Luenberger, D. G., "Observing the State of a Linear System," IEEE Trans. Military Electronics, 8, 74 (1964).
46. Makeev, A., D. Maroudas and I. G. Kevrekidis, "Coarse Stability and Bifurcation Analysis Using Stochastic Simulators: Kinetic Monte Carlo Examples," J. Chem. Phys., 116, 10083 (2002a).
47. Makeev, A. G., 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(18), 8229 (2002b).
48. Makeev, A. G. 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, 1733 (2004).
49. Maroudas, D., "Multiscale Modeling of Hard Materials: Challenges and Opportunities for Chemical Engineering," AIChE J., 46, 878 (2000).
50. Melchior, M., and H. C. Oettinger, "Variance Reduced Simulations of Stochastic Differential Equations," J. Chem. Phys., 103(21), 9506 (1995).
51. Moeller, J., O. Runborg, P. G. Kevrekidis, K. Lust, and I. G. Kevrekidis, "Effective Equations for Discrete Systems: A Time Stepper Based Approach," Int. J. Bif. and Chaos, submitted (2003). (Can be found as physics/0307153 at arXiv.org)
52. Moore, B. C., "Principal Component Analysis in Linear Systems: Controllability, Observability and Model Reduction,"IEEE Trans. Automatic Control, 26(1), 17 (1981).
53. Oettinger, H. C., Stochastic Processes in Polymeric Fluids, Springer Verlag, New York (1996)
54. Ortiz, M., and R. Phillips, "Nanomechanics of Defects in Solids," Adv. Appl. Mech., 36, 1 (1999)
55. Ottino, J. M.,"Complex Systems," AIChE J., 49(2), 292 (2003).
56. Phillips, R., Crystals, Defects and Microstructures, Cambridge University Press, Cambridge (2001).
57. Pope, S., "Computationally Efficient Implementation of Combustion Chemistry Using In Situ Adaptive Tabulation," Comb. Theory and Modeling, 1, 41 (1997). Also Beam Technologies, Inc., ISAT-CK Users' Guide (Release 1.0), Beam Technologies, Inc., Ithaca, NY (1998)
58. Pope, S., "Computationally Efficient Implementation of Combustion Chemistry Using In Situ Adaptive Tabulation," Comb. Theory and Modeling, 1, 41 (1997). Also Beam Technologies, Inc., ISAT-CK Users' Guide (Release 1.0), Beam Technologies, Inc., Ithaca, NY (1998)
59. Rico-Martinez, R., C. W. Gear, and I. G. Kevrekidis, "Coarse Projective KMC Integration: Forward/Reverse Initial and Boundary Value Problems," J. Comp. Phys., 196, 474 (2004).
60. Ristenpart, W. D., I. A. Aksay, and D. A. Saville, "Electrically Guided Assembly of Planar Superlattices in Binary Colloidal Suspensions," Phys. Rev. Lett., 90, 12 (2003).
61. Runborg, O., I. G. Kevrekidis, and K. Theodoropoulos, "Effective Stability and Bifurcation Analysis: a Time Stepper Based Approach, Nonlinearity, 15, 491 (2002).
62. Ryckaert, J. P., G. Ciccotti, and H. Berendsen, "Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of N-alkanes," J. Comp. Phys., 23, 327 (1977).
63. nd ed., SIAM Publications, Philadelphia (2003).
64. Sakurai, T., E. Mihaliuk, F. Chirila, and K. Showalter, "Design and Control of Wave Propagation Patterns in Excitable Media," Science, 296, 2009 (2002).
65. Samaey, G., I. G. Kevrekidis, and D. Roose, "The Gap-Tooth Scheme for Homogenization Problems," SIAM MMS, submitted (2003). (Can be found as physics/031204 at arXiv.org)
66. Schuette, C., A. Fischer, W. Huisinga, and P. Deuflhard "A Direct Approach to Conformational Dynamics Based on Hybrid Monte-Carlo," J. Comp. Phys., 151, 146 (1999).
67. Setayeshgar, S., C. W. Gear, H. G. Othmer, and I. G. Kevrekidis, "Application of Coarse Integration to Bacterial Chemotaxis," SIAM MMS, in press (2004). (Can be found as physics/0308040 at arXiv.org)
68. Shroff, G. M., and H. B. Keller, "Stabilization of Unstable Procedures: A Recursive Projection Method," SIAM J. Numer. Anal., 30, 1099 (1993)
69. Siettos, C., M. D. Graham, and I. G. Kevrekidis, "Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation, Projective Integration and Control via Stochastic Simulation," J. Chem. Phys., 118(22), 10149 (2003a). (Can be obtained as cond-mat/0211455 at arXiv.org)
70. Siettos, C. I., A. Armaou, A. G. Makeev, and I. G. Kevrekidis, "Microscopic/Stochastic Timesteppers and Coarse Control: a Kinetic Monte Carlo Example," AIChE J., 49(7) 1922 (2003b); (nlin.CG/0207017 at arXiv.org).
71. Siettos, C. I., C. C. Pantelides and I. G. Kevrekidis, "Enabling Dynamic Process Simulators to Perform Alternative Tasks: A Time-Stepper Based Toolkit for Computer-Aided Analysis," Ind. Eng. Chem. Res., 42(26), 6795 (2003).
72. Siettos, C. I., N. Kazantzis, and I. G. Kevrekidis, "Coarse Feedback Linearization Using Timesteppers," SIAM MMS, submitted (2004a).
73. Siettos, C. I., D. Maroudas, and I. G. Kevrekidis, "Coarse Bifurcation Diagrams via Microscopic Simulators: a State-Feedback Control-Based Approach," Int. J. Bif. Chaos, 14, 207 (2004).
74. Snow, C. D., N. Nguyen, V. S. Pande and M. Gruebele, "Absolute Comparison of Simulated and Experimental Protein Folding," Nature, 420(6911), 102 (2002).
75. Theodoropoulos, K., Y.-H. Qian and I. G. Kevrekidis, "Coarse Stability and Bifurcation Analysis Using Timesteppers: a Reaction Diffusion Example," Proc. Natl. Acad. Sci., 97(18), 9840 (2000).
76. Theodoropoulos, K., Sankaranarayanan, S. Sundaresan, and I. G. Kevrekidis, "Coarse Bifurcation Studies of Bubble Flow Lattice Boltzmann Simulations," Chem. Eng. Sci., 59, 2357 (2004). (Can be obtained as nlin.PS/0111040 at arXiv.org)
77. Temam, R., Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York (1988).
78. Voter, A. F., "Parallel Replica Method for Dynamics of Infrequent Events," Phys. Rev. B., 57(22), R13985 (1998).
79. Wolff, J, A. G. Papathanasiou, I. G. Kevrekidis, H. H. Rotermund, and G. Ertl, "Spatiotemporal Addressing of Surface Activity," Science, 294, 134 (2001).
80. Xu, K. and K. Prendergast, "Numerical Navier-Stokes from Gas Kinetic Theory," J. Comp. Phys., 114, 9 (1994).
81. Xu, K, "A Gas-kinetic BGK Scheme for the Navier Stokes Equations and its Connection with Artificial Dissipation and the Godunov Method," J. Comp. Phys., 171, 289 (2001).
82. Yeh, I. C., and G. Hummer, "Peptide Loop-Closure Kinetics from Microsecond Molecular Dynamics Simulations in Explicit Solvent," JACS, 124(23), 6563 (2002).
83. Zwanzig, R., Nonequilibrium Statistical Mechanics, Oxford University Press, New York (2001).