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June  2014, 1(2): 377-389. doi: 10.3934/jcd.2014.1.377

Equation-free computation of coarse-grained center manifolds of microscopic simulators

Constantinos Siettos 1,

1. 

School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Athens, GR-157 80, Greece

Received  April 2012 Revised  May 2012 Published  December 2014

An algorithm, based on the Equation-free concept, for the approximation of coarse-grained center manifolds of microscopic simulators is addressed. It is assumed that the macroscopic equations describing the emergent dynamics are not available in a closed form. Appropriately initialized short runs of the microscopic simulators, which are treated as black box input-output maps provide a polynomial estimate of a local coarse-grained center manifold; the coefficients of the polynomial are obtained by wrapping around the microscopic simulator an optimization algorithm. The proposed method is demonstrated through kinetic Monte Carlo simulations, of simple reactions taking place on catalytic surfaces, exhibiting coarse-grained turning points and Andronov-Hopf bifurcations.
Keywords: multiscale dynamics., nonlinear dynamics, Equation-free, microscopic simulators, center manifolds.
Mathematics Subject Classification: Primary: 37M20, 65P30; Secondary: 82-0.
Citation: Constantinos Siettos. Equation-free computation of coarse-grained center manifolds of microscopic simulators. Journal of Computational Dynamics, 2014, 1 (2) : 377-389. doi: 10.3934/jcd.2014.1.377
References:
[1]

E. H. Abed, A simple proof of stability on the center manifold for Hopf bifurcation, SIAM Review, 30 (1988), 487-491. doi: 10.1137/1030096.

[2]

H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics, Multiscale Model. Simul., 3 (2005), 838-852. doi: 10.1137/040603139.

[3]

J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.

[4]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[5]

C. W. Gear and I. G. Kevrekidis, Constraint-defined manifolds: A legacy code approach to low-dimensional computation, J. Scientific Comput., 25 (2005), 17-28. doi: 10.1007/s10915-004-4630-x.

[6]

C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst., 4 (2005), 711-732. doi: 10.1137/040608295.

[7]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.

[8]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[9]

G. Guckenheimer and M. Myers, Computing Hopf bifurcations, SIAM J. Sci. Comput., 17 (1996), 1275-1301. doi: 10.1137/S1064827593253495.

[10]

P. Holmes, Center manifolds, normal forms and bifurcations of vector fields, Physica 2D, 2 (1981), 449-481. doi: 10.1016/0167-2789(81)90022-1.

[11]

N. Kazantzis and T. Good, Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs, Comp. Chem. Eng., 26 (2002), 999-1012. doi: 10.1016/S0098-1354(02)00022-4.

[12]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970944.

[13]

C. T. Kelley, Iterative Methods for Optimization, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970920.

[14]

I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level tasks, Comm. Math. Sciences, 1 (2003), 715-762. doi: 10.4310/CMS.2003.v1.n4.a5.

[15]

I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: the computer-assisted analysis of complex, multiscale systems, A.I.Ch.E.J., 50 (2004), 1346-1354.

[16]

A. Kolpas, J. Moehlis and I. G. Kevrekidis, Coarse-grained analysis of stochasticity-induced switching between collective motion states, Proc. Nat. Acad. Sci. USA, 104 (2007), 5931-5935. doi: 10.1073/pnas.0608270104.

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998.

[18]

A. Makeev, D. Maroudas and I. G. Kevrekidis, Coarse stability and biifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples, J. Chem. Phys., 116 (2002), 10083-10091. doi: 10.1063/1.1476929.

[19]

A. H. Nayef, Applied Nonlinear Dynamics, Wiley-VCH, New York, 2007. doi: 10.1002/9783527617548.

[20]

C. I. Siettos, M. Graham and I. G. Kevrekidis, Coarse brownian dynamics for nematic liquid crystals: Bifurcation diagrams via stochastic simulation, J. Chem. Phys., 118 (2003), 10149-10156.

[21]

Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Oxford-Manchester, 1992.

[22]

R. Seydel, Practical Bifurcation and Stability Analysis, Springer-Verlag, New York, 1994.

[23]

C. I. Siettos, R. Rico-Martinez and I. G. Kevrekidis, A systems-based approach to multiscale computation: EquationfFree detection of coarse-grained bifurcations, Comp. Chem. Eng., 30 (2006), 1632-1642.

[24]

I. Yammaguchi, Y. Ogawa, Y. Jimbo, H. Nakao and K. Kotani, Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-indeced oscillations, PLoS ONE, 6 (2011), e26497.

[25]

A. Zagaris, C. W. Gear, T. J. Kapper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 757-784. doi: 10.1051/m2an/2009026.

show all references

References:
[1]

E. H. Abed, A simple proof of stability on the center manifold for Hopf bifurcation, SIAM Review, 30 (1988), 487-491. doi: 10.1137/1030096.

[2]

H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics, Multiscale Model. Simul., 3 (2005), 838-852. doi: 10.1137/040603139.

[3]

J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981.

[4]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[5]

C. W. Gear and I. G. Kevrekidis, Constraint-defined manifolds: A legacy code approach to low-dimensional computation, J. Scientific Comput., 25 (2005), 17-28. doi: 10.1007/s10915-004-4630-x.

[6]

C. W. Gear, T. J. Kaper, I. G. Kevrekidis and A. Zagaris, Projecting to a slow manifold: Singularly perturbed systems and legacy codes, SIAM J. Appl. Dyn. Syst., 4 (2005), 711-732. doi: 10.1137/040608295.

[7]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comput. Phys., 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.

[8]

D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[9]

G. Guckenheimer and M. Myers, Computing Hopf bifurcations, SIAM J. Sci. Comput., 17 (1996), 1275-1301. doi: 10.1137/S1064827593253495.

[10]

P. Holmes, Center manifolds, normal forms and bifurcations of vector fields, Physica 2D, 2 (1981), 449-481. doi: 10.1016/0167-2789(81)90022-1.

[11]

N. Kazantzis and T. Good, Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs, Comp. Chem. Eng., 26 (2002), 999-1012. doi: 10.1016/S0098-1354(02)00022-4.

[12]

C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970944.

[13]

C. T. Kelley, Iterative Methods for Optimization, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970920.

[14]

I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level tasks, Comm. Math. Sciences, 1 (2003), 715-762. doi: 10.4310/CMS.2003.v1.n4.a5.

[15]

I. G. Kevrekidis, C. W. Gear and G. Hummer, Equation-free: the computer-assisted analysis of complex, multiscale systems, A.I.Ch.E.J., 50 (2004), 1346-1354.

[16]

A. Kolpas, J. Moehlis and I. G. Kevrekidis, Coarse-grained analysis of stochasticity-induced switching between collective motion states, Proc. Nat. Acad. Sci. USA, 104 (2007), 5931-5935. doi: 10.1073/pnas.0608270104.

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998.

[18]

A. Makeev, D. Maroudas and I. G. Kevrekidis, Coarse stability and biifurcation analysis using stochastic simulators: Kinetic Monte Carlo examples, J. Chem. Phys., 116 (2002), 10083-10091. doi: 10.1063/1.1476929.

[19]

A. H. Nayef, Applied Nonlinear Dynamics, Wiley-VCH, New York, 2007. doi: 10.1002/9783527617548.

[20]

C. I. Siettos, M. Graham and I. G. Kevrekidis, Coarse brownian dynamics for nematic liquid crystals: Bifurcation diagrams via stochastic simulation, J. Chem. Phys., 118 (2003), 10149-10156.

[21]

Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Oxford-Manchester, 1992.

[22]

R. Seydel, Practical Bifurcation and Stability Analysis, Springer-Verlag, New York, 1994.

[23]

C. I. Siettos, R. Rico-Martinez and I. G. Kevrekidis, A systems-based approach to multiscale computation: EquationfFree detection of coarse-grained bifurcations, Comp. Chem. Eng., 30 (2006), 1632-1642.

[24]

I. Yammaguchi, Y. Ogawa, Y. Jimbo, H. Nakao and K. Kotani, Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-indeced oscillations, PLoS ONE, 6 (2011), e26497.

[25]

A. Zagaris, C. W. Gear, T. J. Kapper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 757-784. doi: 10.1051/m2an/2009026.

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