# American Institute of Mathematical Sciences

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

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

 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.
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.  Google Scholar [2] H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics, Multiscale Model. Simul., 3 (2005), 838-852. doi: 10.1137/040603139.  Google Scholar [3] J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.  Google Scholar [9] G. Guckenheimer and M. Myers, Computing Hopf bifurcations, SIAM J. Sci. Comput., 17 (1996), 1275-1301. doi: 10.1137/S1064827593253495.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. T. Kelley, Iterative Methods for Optimization, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970920.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [17] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998. Google Scholar [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.  Google Scholar [19] A. H. Nayef, Applied Nonlinear Dynamics, Wiley-VCH, New York, 2007. doi: 10.1002/9783527617548.  Google Scholar [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. Google Scholar [21] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Oxford-Manchester, 1992. Google Scholar [22] R. Seydel, Practical Bifurcation and Stability Analysis, Springer-Verlag, New York, 1994.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar

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.  Google Scholar [2] H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics, Multiscale Model. Simul., 3 (2005), 838-852. doi: 10.1137/040603139.  Google Scholar [3] J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York, 1981. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.  Google Scholar [9] G. Guckenheimer and M. Myers, Computing Hopf bifurcations, SIAM J. Sci. Comput., 17 (1996), 1275-1301. doi: 10.1137/S1064827593253495.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. T. Kelley, Iterative Methods for Optimization, SIAM series on Frontiers in Applied Mathematics, PA, 1999. doi: 10.1137/1.9781611970920.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [17] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, $2^{nd}$ edition, Springer-Verlag, New York, 1998. Google Scholar [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.  Google Scholar [19] A. H. Nayef, Applied Nonlinear Dynamics, Wiley-VCH, New York, 2007. doi: 10.1002/9783527617548.  Google Scholar [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. Google Scholar [21] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Oxford-Manchester, 1992. Google Scholar [22] R. Seydel, Practical Bifurcation and Stability Analysis, Springer-Verlag, New York, 1994.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar
 [1] Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies, Dirk Roose. Lifting in equation-free methods for molecular dynamics simulations of dense fluids. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 855-874. doi: 10.3934/dcdsb.2009.11.855 [2] Katherine A. Bold, Karthikeyan Rajendran, Balázs Ráth, Ioannis G. Kevrekidis. An equation-free approach to coarse-graining the dynamics of networks. Journal of Computational Dynamics, 2014, 1 (1) : 111-134. doi: 10.3934/jcd.2014.1.111 [3] Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677 [4] Yuri Latushkin, Jan Prüss, Ronald Schnaubelt. Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 595-633. doi: 10.3934/dcdsb.2008.9.595 [5] MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 [6] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [7] Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025 [8] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 [9] Antonios Zagaris, Christophe Vandekerckhove, C. William Gear, Tasso J. Kaper, Ioannis G. Kevrekidis. Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2759-2803. doi: 10.3934/dcds.2012.32.2759 [10] Samantha Erwin, Stanca M. Ciupe. Germinal center dynamics during acute and chronic infection. Mathematical Biosciences & Engineering, 2017, 14 (3) : 655-671. doi: 10.3934/mbe.2017037 [11] Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549 [12] Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923 [13] D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419 [14] Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141 [15] Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028 [16] Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Tori with hyperbolic dynamics in 3-manifolds. Journal of Modern Dynamics, 2011, 5 (1) : 185-202. doi: 10.3934/jmd.2011.5.185 [17] Andrey V. Kremnev, Alexander S. Kuleshov. Nonlinear dynamics and stability of the skateboard. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 85-103. doi: 10.3934/dcdss.2010.3.85 [18] Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013 [19] Lei Li, Jianping Wang, Mingxin Wang. The dynamics of nonlocal diffusion systems with different free boundaries. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3651-3672. doi: 10.3934/cpaa.2020161 [20] Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193

Impact Factor: