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Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems
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 |
References:
[1] |
E. H. Abed, A simple proof of stability on the center manifold for Hopf bifurcation,, SIAM Review, 30 (1988), 487.
doi: 10.1137/1030096. |
[2] |
H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics,, Multiscale Model. Simul., 3 (2005), 838.
doi: 10.1137/040603139. |
[3] |
J. Carr, Applications of Center Manifold Theory,, Springer-Verlag, (1981). Google Scholar |
[4] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Equat., 31 (1979), 53.
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.
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.
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.
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.
doi: 10.1021/j100540a008. |
[9] |
G. Guckenheimer and M. Myers, Computing Hopf bifurcations,, SIAM J. Sci. Comput., 17 (1996), 1275.
doi: 10.1137/S1064827593253495. |
[10] |
P. Holmes, Center manifolds, normal forms and bifurcations of vector fields,, Physica 2D, 2 (1981), 449.
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.
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, (1999).
doi: 10.1137/1.9781611970944. |
[13] |
C. T. Kelley, Iterative Methods for Optimization,, SIAM series on Frontiers in Applied Mathematics, (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.
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. 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.
doi: 10.1073/pnas.0608270104. |
[17] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $2^{nd}$ edition, (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.
doi: 10.1063/1.1476929. |
[19] |
A. H. Nayef, Applied Nonlinear Dynamics,, Wiley-VCH, (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. Google Scholar |
[21] |
Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992). Google Scholar |
[22] |
R. Seydel, Practical Bifurcation and Stability Analysis,, Springer-Verlag, (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. 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). 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.
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.
doi: 10.1137/1030096. |
[2] |
H. Boumediene, K. Wei and A. J. Krener, The controlled center dynamics,, Multiscale Model. Simul., 3 (2005), 838.
doi: 10.1137/040603139. |
[3] |
J. Carr, Applications of Center Manifold Theory,, Springer-Verlag, (1981). Google Scholar |
[4] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Diff. Equat., 31 (1979), 53.
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.
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.
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.
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.
doi: 10.1021/j100540a008. |
[9] |
G. Guckenheimer and M. Myers, Computing Hopf bifurcations,, SIAM J. Sci. Comput., 17 (1996), 1275.
doi: 10.1137/S1064827593253495. |
[10] |
P. Holmes, Center manifolds, normal forms and bifurcations of vector fields,, Physica 2D, 2 (1981), 449.
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.
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, (1999).
doi: 10.1137/1.9781611970944. |
[13] |
C. T. Kelley, Iterative Methods for Optimization,, SIAM series on Frontiers in Applied Mathematics, (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.
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. 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.
doi: 10.1073/pnas.0608270104. |
[17] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $2^{nd}$ edition, (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.
doi: 10.1063/1.1476929. |
[19] |
A. H. Nayef, Applied Nonlinear Dynamics,, Wiley-VCH, (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. Google Scholar |
[21] |
Y. Saad, Numerical Methods for Large Eigenvalue Problems,, Manchester University Press, (1992). Google Scholar |
[22] |
R. Seydel, Practical Bifurcation and Stability Analysis,, Springer-Verlag, (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. 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). 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.
doi: 10.1051/m2an/2009026. |
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