# American Institute of Mathematical Sciences

August  2012, 32(8): 2759-2803. doi: 10.3934/dcds.2012.32.2759

## Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold

 1 Department of Applied Mathematics, University of Twente, Enschede, 7500 AE, Netherlands 2 Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, B-3001, Belgium 3 Department of Chemical and Biological Egineering, Princeton University, Princeton, NJ 08544, United States 4 Department of Mathematics and Statistics and Center for Biodynamics, Boston University, Boston, MA 02215 5 Department of Chemical and Biological Engineering, and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, United States

Received  August 2011 Revised  November 2011 Published  March 2012

In [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], we developed the family of constrained runs algorithms to find points on low-dimensional, attracting, slow manifolds in systems of nonlinear differential equations with multiple time scales. For user-specified values of a subset of the system variables parametrizing the slow manifold (which we term observables and denote collectively by $u$), these iterative algorithms return values of the remaining system variables $v$ so that the point $(u,v)$ approximates a point on a slow manifold. In particular, the $m-$th constrained runs algorithm ($m = 0, 1, \ldots$) approximates a point $(u,v_m)$ that is the appropriate zero of the $(m+1)-$st time derivative of $v$. % The accuracy with which $(u,v_m)$ approximates the corresponding point on the slow manifold with the same value of the observables has been established in [A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, ESAIM: M2AN 43(4) (2009) 757--784] for systems for which the observables $u$ evolve exclusively on the slow time scale. There, we also determined explicit conditions under which the $m-$th constrained runs scheme converges to the fixed point $(u,v_m)$ and identified conditions under which it fails to converge. Here, we consider the questions of stability and stabilization of these iterative algorithms for the case in which the observables $u$ are also allowed to evolve on a fast time scale. The stability question in this case is more complicated, since it involves a generalized eigenvalue problem for a pair of matrices encoding geometric and dynamical characteristics of the system of differential equations. We determine the conditions under which these schemes converge or diverge in a series of cases in which this problem is explicitly solvable. We illustrate our main stability and stabilization results for the constrained runs schemes on certain planar systems with multiple time scales, and also on a more-realistic sixth order system with multiple time scales that models a network of coupled enzymatic reactions. Finally, we consider the issue of stabilization of the $m-$th constrained runs algorithm when the functional iteration scheme is divergent or converges slowly. In that case, we demonstrate on concrete examples how Newton's method and Broyden's method may replace functional iteration to yield stable iterative schemes.
Citation: 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 and Continuous Dynamical Systems, 2012, 32 (8) : 2759-2803. doi: 10.3934/dcds.2012.32.2759
##### References:
 [1] G. Browning and H.-O. Kreiss, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math., 42 (1982), 704-718. doi: 10.1137/0142049. [2] M. S. Calder and D. Siegel, Properties of the Michaelis-Menten mechanism in phase space, J. Math. Anal. Appl., 339 (2008), 1044-1064. doi: 10.1016/j.jmaa.2007.06.078. [3] A. Ciliberto, F. Capuani and J. J. Tyson, Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation, PLoS Comp. Biol., 3 (2007), 463-472. [4] J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26. doi: 10.1016/0771-050X(80)90013-3. [5] C. W. Gear, I. G. Kevrekidis and C. Theodoropoulos, "Coarse" integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods, Comp. Chem. Eng., 26 (2002), 941-963. doi: 10.1016/S0098-1354(02)00020-0. [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] C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comp., 24 (2003), 1091-1106. doi: 10.1137/S1064827501388157. [8] A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, PNAS U.S.A., 78 (1981), 6840-6844. [9] G. H. Golub and C. F. van Loan, "Matrix Computations,'' 3rd edition, Johns Hopkins Studies in the Mathematical Sciences, John Hopkins University Press, Baltimore, MD, 1996. [10] H. M. Härdin, A. Zagaris, K. Kraab and H. V. Westerhoff, Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations, FEBS J., 276 (2009), 5491-5506. doi: 10.1111/j.1742-4658.2009.07233.x. [11] H. M. Härdin, A. Zagaris and H. V. Westerhoff, Relaxation behavior in reactive protein networks as instantiated in the phosphotransferase system, submitted, 2011. [12] J. M. Hyman, Patch dynamics for multiscale problems, Comp. Sci. Eng., 7 (2005), 47-53. doi: 10.1109/MCSE.2005.57. [13] C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (Montecatini Terme, 1994) (ed. L. Arnold), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44-118. [14] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,'' With separately available software, Frontiers In Applied Mathematics, 16, SIAM, Philadelphia, PA, 1995. [15] 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 analysis, Comm. Math. Sci., 1 (2003), 715-762. [16] H.-O. Kreiss, Problems with different time scales for ordinary differential equations, SIAM J. Numer. Anal., 16 (1979), 980-998. doi: 10.1137/0716072. [17] H.-O. Kreiss, Problems with different time scales, in "Multiple Time Scales'' (eds. J. H. Brackbill and B. I. Cohen), Comput. Tech., 3, Academic Press, Orlando, FL, (1985), 29-57. [18] A. Kumar and K. Josić, Reduced models of networks of coupled enzymatic reactions, J. Theor. Biol., 278 (2011), 87-106. doi: 10.1016/j.jtbi.2011.02.025. [19] E. N. Lorenz, Attractor sets and quasigeostrophic equilibrium, J. Atmos. Sci., 37 (1980), 1685-1699. doi: 10.1175/1520-0469(1980)037<1685:ASAQGE>2.0.CO;2. [20] R. E. O'Malley, Jr., "Singular Perturbation Methods for Ordinary Differential Equations,'' Applied Mathematical Sciences, 89, Springer-Verlag, New York, 1991. [21] J. Murdock, "Normal Forms and Unfoldings for Local Dynamical Systems,'' Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. [22] Z. Ren, S. B. Pope, A. Vladimirsky and J. M. Guckenheimer, The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics, J. Chem. Phys., 124 (2006) 114111. doi: 10.1063/1.2177243. [23] G. Samaey, I. G. Kevrekidis and D. Roose, Patch dynamics with buffers for homogenization problems, J. Comp. Phys., 213 (2006), 264-287. doi: 10.1016/j.jcp.2005.08.010. [24] G. M. Shroff and H. B. Keller, Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal., 30 (1993), 1099-1120. doi: 10.1137/0730057. [25] L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comp., 18 (1997), 1-22. doi: 10.1137/S1064827594276424. [26] C. Vandekerckhove, I. G. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for Initializing on a slow manifold, J. Sci. Comput., 39 (2009), 167-188. doi: 10.1007/s10915-008-9256-y. [27] A. Zagaris, H. G. Kaper and T. J. Kaper, Two perspectives on reduction of ordinary differential equations, Mathematische Nachrichten, 278 (2005), 1629-1642. doi: 10.1002/mana.200410328. [28] A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, M2AN Math. Model. Num. Anal., 43 (2009), 757-784. doi: 10.1051/m2an/2009026. [29] A. Zagaris, "Analysis of Reduction Methods for Multiscale Problems,'' Ph.D thesis, Boston University, 2005.

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##### References:
 [1] G. Browning and H.-O. Kreiss, Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math., 42 (1982), 704-718. doi: 10.1137/0142049. [2] M. S. Calder and D. Siegel, Properties of the Michaelis-Menten mechanism in phase space, J. Math. Anal. Appl., 339 (2008), 1044-1064. doi: 10.1016/j.jmaa.2007.06.078. [3] A. Ciliberto, F. Capuani and J. J. Tyson, Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation, PLoS Comp. Biol., 3 (2007), 463-472. [4] J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26. doi: 10.1016/0771-050X(80)90013-3. [5] C. W. Gear, I. G. Kevrekidis and C. Theodoropoulos, "Coarse" integration/bifurcation analysis via microscopic simulators: Micro-Galerkin methods, Comp. Chem. Eng., 26 (2002), 941-963. doi: 10.1016/S0098-1354(02)00020-0. [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] C. W. Gear and I. G. Kevrekidis, Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM J. Sci. Comp., 24 (2003), 1091-1106. doi: 10.1137/S1064827501388157. [8] A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, PNAS U.S.A., 78 (1981), 6840-6844. [9] G. H. Golub and C. F. van Loan, "Matrix Computations,'' 3rd edition, Johns Hopkins Studies in the Mathematical Sciences, John Hopkins University Press, Baltimore, MD, 1996. [10] H. M. Härdin, A. Zagaris, K. Kraab and H. V. Westerhoff, Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations, FEBS J., 276 (2009), 5491-5506. doi: 10.1111/j.1742-4658.2009.07233.x. [11] H. M. Härdin, A. Zagaris and H. V. Westerhoff, Relaxation behavior in reactive protein networks as instantiated in the phosphotransferase system, submitted, 2011. [12] J. M. Hyman, Patch dynamics for multiscale problems, Comp. Sci. Eng., 7 (2005), 47-53. doi: 10.1109/MCSE.2005.57. [13] C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (Montecatini Terme, 1994) (ed. L. Arnold), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44-118. [14] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,'' With separately available software, Frontiers In Applied Mathematics, 16, SIAM, Philadelphia, PA, 1995. [15] 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 analysis, Comm. Math. Sci., 1 (2003), 715-762. [16] H.-O. Kreiss, Problems with different time scales for ordinary differential equations, SIAM J. Numer. Anal., 16 (1979), 980-998. doi: 10.1137/0716072. [17] H.-O. Kreiss, Problems with different time scales, in "Multiple Time Scales'' (eds. J. H. Brackbill and B. I. Cohen), Comput. Tech., 3, Academic Press, Orlando, FL, (1985), 29-57. [18] A. Kumar and K. Josić, Reduced models of networks of coupled enzymatic reactions, J. Theor. Biol., 278 (2011), 87-106. doi: 10.1016/j.jtbi.2011.02.025. [19] E. N. Lorenz, Attractor sets and quasigeostrophic equilibrium, J. Atmos. Sci., 37 (1980), 1685-1699. doi: 10.1175/1520-0469(1980)037<1685:ASAQGE>2.0.CO;2. [20] R. E. O'Malley, Jr., "Singular Perturbation Methods for Ordinary Differential Equations,'' Applied Mathematical Sciences, 89, Springer-Verlag, New York, 1991. [21] J. Murdock, "Normal Forms and Unfoldings for Local Dynamical Systems,'' Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. [22] Z. Ren, S. B. Pope, A. Vladimirsky and J. M. Guckenheimer, The invariant constrained equilibrium edge preimage curve method for the dimension reduction of chemical kinetics, J. Chem. Phys., 124 (2006) 114111. doi: 10.1063/1.2177243. [23] G. Samaey, I. G. Kevrekidis and D. Roose, Patch dynamics with buffers for homogenization problems, J. Comp. Phys., 213 (2006), 264-287. doi: 10.1016/j.jcp.2005.08.010. [24] G. M. Shroff and H. B. Keller, Stabilization of unstable procedures: The recursive projection method, SIAM J. Numer. Anal., 30 (1993), 1099-1120. doi: 10.1137/0730057. [25] L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comp., 18 (1997), 1-22. doi: 10.1137/S1064827594276424. [26] C. Vandekerckhove, I. G. Kevrekidis and D. Roose, An efficient Newton-Krylov implementation of the constrained runs scheme for Initializing on a slow manifold, J. Sci. Comput., 39 (2009), 167-188. doi: 10.1007/s10915-008-9256-y. [27] A. Zagaris, H. G. Kaper and T. J. Kaper, Two perspectives on reduction of ordinary differential equations, Mathematische Nachrichten, 278 (2005), 1629-1642. doi: 10.1002/mana.200410328. [28] A. Zagaris, C. W. Gear, T. J. Kaper and I. G. Kevrekidis, Analysis of the accuracy and convergence of equation-free projection to a slow manifold, M2AN Math. Model. Num. Anal., 43 (2009), 757-784. doi: 10.1051/m2an/2009026. [29] A. Zagaris, "Analysis of Reduction Methods for Multiscale Problems,'' Ph.D thesis, Boston University, 2005.
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