Deciding the nature of the coarse equation through microscopic simulations: The baby-bathwater scheme

SIAM Review

Volumn 49, Issue 3, 2007, Pages 469-487

  Li, Ju ^a   Kevrekidis, Panayotis G ^b   Gear, C William ^c   Kevrekidis, Ioannis G ^c  

^a Ohio State University   (United States)

^b UNIVERSITY OF MASSACHUSETTS   (United States)

^c Princeton University   (United States)

Author keywords

Coarse graining; Equation free; Identification; Microscopic simulator; Multiscale computation

Indexed keywords

BOUNDARY CONDITIONS; COMPUTATIONAL METHODS; COMPUTER SIMULATION; DYNAMICS; HAMILTONIANS; STATISTICS;

COARSE GRAINING; MICROSCOPIC SIMULATOR; MICROSCOPICALLY EVOLVING PARTICLES; MULTISCALE COMPUTATION;

PARTICLE SIZE ANALYSIS;

EID: 34548490943     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/070692303     Document Type: Article

References (46)

1. N.G. VAN KAMPEN, Stochastic Processes in Physics and Chemistry, revised and enlarged ed., North-Holland, Amsterdam, 1992.
2. I.G. KEVREKIDIS, C.W. GEAR, J.M. HYMAN, P.G. KEVREKIDIS, O. RUNBORG, AND C. THEODOROPOULOS, Equation-free multiscale computation: Enabling microscopic simulators to perform system-level tasks, Commun. Math. Sci., 1 (2003), pp. 715-762.
3. A.G. 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.
4. 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.
5. 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.
6. G.M. SHROFF AND H.B. KELLER, Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal., 30 (1993), pp. 1099-1120.
7. C.W. GEAR, I.G. KEVREKIDIS, AND C. THEODOROPOULOS, Coarse integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods, Comput. Chem. Engrg., 26 (2002), pp. 941-963.
8. J. Li, K.J. VAN VLIET, T. ZHU, S. YIP, AND S. SURESH, Atomistic mechanisms governing elastic limit and incipient plasticity in crystals, Nature, 418 (2002), pp. 307-310.
9. K.J. VAN VLIET, J. LI, T. ZHU, S. YIP, AND S. SURESH, Quantifying the early stages of plasticity through nanoscale experiments and simulations, Phys. Rev. B, 67 (2003), article 104105.
10. I.G. KEVREKIDIS, Coarse Bifurcation Studies of Alternative Microscopic/Hybrid Simulators, Plenary Lecture, CAST Division, AIChE Annual Meeting, Los Angeles, 2000. Slides can be obtained at http://arnold.princeton.edu/~yannis/.
11. C.W. GEAR, J. LI, AND I.G. KEVREKIDIS, The gap-tooth method in particle simulations, Phys. Lett. A, 316 (2003), pp. 190-195.
12. W. E AND B. ENGQUIST, The heterogeneous multi-scale methods, Commun. Math. Sci., 1 (2003), pp. 87-132.
13. J. D. COLE, On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math., 9 (1951), pp. 225-236.
14. xx, Comm. Pure Appl. Math., 3 (1950), pp. 201-230.
15. S. ROBERTS, Convergence of a random walk method for the Burgers equation, Math. Comp., 52 (1989), pp. 647-673.
16. J. Li, D. LIAO, AND S. YIP, Nearly exact solution for coupled continuum/MD fluid simulation, J. Comput.-Aided Mater. Des., 6 (1999), pp. 95-102.
17. J. Li, D. LIAO, AND S. YIP, Coupling continuum to molecular-dynamics simulation: Reflecting particle method and the field estimator, Phys. Rev. E, 57 (1998), pp. 7259-7267.
18. J. EAPEN, J. Li, AND S. YIP, Statistical field estimators for multiscale simulations, Phys. Rev. E, 72 (2005), article 056712.
19. L.B. Lucy, A numerical approach to the testing of the fission hypothesis, Astronom. J., 82 (1977), pp. 1013-1024.
20. A. CHERTOCK AND D. LEVY, Particle methods for dispersive equations, J. Comput, Phys., 171 (2001), pp. 708-730.
21. A. CHERTOCK AND D. LEVY, A particle method for the KdV equation, J. Sci. Comput., 17 (2002), pp. 491-499.
22. V.I. ARNOLD, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989.
23. K.N. CHRISTODOULOU AND L.E. SCRIVEN, Finding leading modes of a viscous free surface flow: An asymmetric generalized eigenproblem, J. Sci. Comput., 3 (1988), pp. 355-406.
24. I. GOLDHIRSCH, S.A. ORSZAG, AND B.K. MAULIK, An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices, J. Sci. Comput., 2 (1987), pp. 33-58.
25. Z. BAI, J. DEMMEL, J. DONGARRA, A. RUHE, AND H. VAN DER VORST, EDS. , Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, Software Environ. Tools 11, SIAM, Philadelphia, 2000.
26. W.E. ARNOLDI, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17-29.
27. C. LANCZOS, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards, 45 (1950), pp. 255-282.
28. L.S. TUCKERMAN AND D. BARKLEY, Bifurcation analysis for time-steppers, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Vol. Math. Appl. 119, E. Doedel and L. S. Tuckerman, eds., Springer-Verlag, New York, 2000, pp. 453-466.
29. C.W. GEAR AND I.G. KEVREKIDIS, Computing in the past with forward integration, Phys. Lett. A, 321 (2004), pp. 335-343.
30. 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.
31. M.J. ABLOWITZ AND H. SEGUR, Solitons and the Inverse Scattering Transform, SIAM Stud. Appl. Math. 4, SIAM, Philadelphia, 1981.
32. P.G. KEVREKIDIS AND N.R. QUINTERO, Using the finite domain remnant of the continuous spectrum to examine integrability: Effect of boundary conditions, Phys. Rev. E, 68 (2003), article 036612.
33. P.G. KEVREKIDIS, Integrability revisited: A necessary condition, Phys. Lett. A, 285 (2001), pp. 383-389.
34. Y. AIT-SAHALIA, Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach, Econometrica, 70 (2002), pp. 223-262.
35. 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), article 044908.
36. 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.
37. A. SZELL, D. MERRITT, AND I.G. KEVREKIDIS, Core collapse via coarse dynamic renormalization, Phys. Rev. Lett., 95 (2005), article 081102.
38. S.-J. MOON, R. GHANEM, AND I.G. KEVREKIDIS, An equation-free approach to coupled oscillator dynamics, Phys. Rev. Lett., 96 (2006), article 144101.
39. G. SAMAEY, I.G. KEVREKIDIS, AND D. ROOSE, Patch dynamics with buffers for homogenization problems, J. Comput. Phys., 213 (2006), pp. 264-287.
40. D. XIU, I.G. KEVREKIDIS, AND R. GHANEM, An equation-free, multiscale approach to uncertainty quantification, IEEE Comput. Sci. Engrg., 7 (2005), pp. 16-23.
41. W. E, B. ENGQUIST, X. LI, E. REN, AND E. VANDEN-EIJNDEN, Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2 (2007), pp. 367-450.
42. 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.
43. S.L. LEE AND C.W. GEAR, Second-order accurate projective integrators for multiscale problems, J. Comput. Appl. Math., 201 (2007), pp. 258-274.
44. C. VANDEKERCKHOVE, D. ROOSE, AND K. LUST, Numerical stability analysis of an acceleration scheme for step size constrained time integrators, J. Comput. Appl. Math., 200 (2007), pp. 761-777.
45. C. VANDEKERCKHOVE AND D. ROOSE, Accuracy analysis of acceleration schemes for stiff multiscale problems, J. Comput. Appl. Math., to appear.
46. Y. Zou, I.G. KEVREKIDIS, AND R. GHANEM, Equation-free dynamic renormalization: Self-similarity in multidimensional particle system dynamics, Phys. Rev. E., 72 (2005), article 046702.