Analysis of the accuracy and convergence of equation-free projection to a slow manifold
ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 757-784.

In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis and A. Zagaris, SIAM J. Appl. Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms within the context of equation-free methods to approximate low-dimensional, attracting, slow manifolds in systems of differential equations with multiple time scales. For user-specified values of a finite number of the observables, the mth member of the class of algorithms (m=0,1,...) finds iteratively an approximation of the appropriate zero of the (m+1)st time derivative of the remaining variables and uses this root to approximate the location of the point on the slow manifold corresponding to these values of the observables. This article is the first of two articles in which the accuracy and convergence of the iterative algorithms are analyzed. Here, we work directly with fast-slow systems, in which there is an explicit small parameter, ε, measuring the separation of time scales. We show that, for each m=0,1,..., the fixed point of the iterative algorithm approximates the slow manifold up to and including terms of 𝒪(εm). Moreover, for each m, we identify explicitly the conditions under which the mth iterative algorithm converges to this fixed point. Finally, we show that when the iteration is unstable (or converges slowly) it may be stabilized (or its convergence may be accelerated) by application of the Recursive Projection Method. Alternatively, the Newton-Krylov Generalized Minimal Residual Method may be used. In the subsequent article, we will consider the accuracy and convergence of the iterative algorithms for a broader class of systems - in which there need not be an explicit small parameter - to which the algorithms also apply.

DOI : 10.1051/m2an/2009026
Classification : 35B25, 35B42, 37M99, 65L20, 65P99
Mots-clés : iterative initialization, DAEs, singular perturbations, legacy codes, inertial manifolds
@article{M2AN_2009__43_4_757_0,
     author = {Zagaris, Antonios and Gear, C. William and Kaper, Tasso J. and Kevrekidis, Yannis G.},
     title = {Analysis of the accuracy and convergence of equation-free projection to a slow manifold},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {757--784},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {4},
     year = {2009},
     doi = {10.1051/m2an/2009026},
     mrnumber = {2542876},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2009026/}
}
TY  - JOUR
AU  - Zagaris, Antonios
AU  - Gear, C. William
AU  - Kaper, Tasso J.
AU  - Kevrekidis, Yannis G.
TI  - Analysis of the accuracy and convergence of equation-free projection to a slow manifold
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 757
EP  - 784
VL  - 43
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2009026/
DO  - 10.1051/m2an/2009026
LA  - en
ID  - M2AN_2009__43_4_757_0
ER  - 
%0 Journal Article
%A Zagaris, Antonios
%A Gear, C. William
%A Kaper, Tasso J.
%A Kevrekidis, Yannis G.
%T Analysis of the accuracy and convergence of equation-free projection to a slow manifold
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 757-784
%V 43
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2009026/
%R 10.1051/m2an/2009026
%G en
%F M2AN_2009__43_4_757_0
Zagaris, Antonios; Gear, C. William; Kaper, Tasso J.; Kevrekidis, Yannis G. Analysis of the accuracy and convergence of equation-free projection to a slow manifold. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 757-784. doi : 10.1051/m2an/2009026. http://www.numdam.org/articles/10.1051/m2an/2009026/

[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. | MR | Zbl

[2] J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences 35. Springer-Verlag, New York (1981). | MR | Zbl

[3] J. Curry, S.E. Haupt and M.E. Limber, Low-order modeling, initializations, and the slow manifold. Tellus 47A (1995) 145-161.

[4] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq. 31 (1979) 53-98. | MR | Zbl

[5] C.W. Gear and I.G. Kevrekidis, Constraint-defined manifolds: a legacy-code approach to low-dimensional computation. J. Sci. Comp. 25 (2005) 17-28. | MR

[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. | MR | Zbl

[7] S.S. Girimaji, Reduction of large dynamical systems by minimization of evolution rate. Phys. Rev. Lett. 82 (1999) 2282-2285.

[8] C.K.R.T. Jones, Geometric singular perturbation theory, in Dynamical Systems, Montecatini Terme, L. Arnold Ed., Lecture Notes Math. 1609, Springer-Verlag, Berlin (1994) 44-118. | MR | Zbl

[9] H.G. Kaper and T.J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions. Physica D 165 (2002) 66-93. | MR | Zbl

[10] C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, Frontiers In Applied Mathematics 16. SIAM Publications, Philadelphia (1995). | MR | Zbl

[11] 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. Commun. Math. Sci. 1 (2003) 715-762. | MR | Zbl

[12] H.-O. Kreiss, Problems with different time scales for ordinary differential equations. SIAM J. Numer. Anal. 16 (1979) 980-998. | MR | Zbl

[13] H.-O. Kreiss, Problems with Different Time Scales, in Multiple Time Scales, J.H. Brackbill and B.I. Cohen Eds., Academic Press (1985) 29-57. | MR | Zbl

[14] E.N. Lorenz, Attractor sets and quasi-geostrophic equilibrium. J. Atmos. Sci. 37 (1980) 1685-1699. | MR

[15] U. Maas and S.B. Pope, Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame 88 (1992) 239-264.

[16] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107. Springer-Verlag, New York (1986). | MR | Zbl

[17] G.M. Shroff and H.B. Keller, Stabilization of unstable procedures: A recursive projection method. SIAM J. Numer. Anal. 30 (1993) 1099-1120. | MR | Zbl

[18] P. Van Leemput, W. Vanroose and D. Roose, Initialization of a Lattice Boltzmann Model with Constrained Runs. Report TW444, Catholic University of Leuven, Belgium (2005).

[19] P. Van Leemput, C. Vandekerckhove, W. Vanroose and D. Roose, Accuracy of hybrid Lattice Boltzmann/Finite Difference schemes for reaction-diffusion systems. Multiscale Model. Sim. 6 (2007) 838-857. | MR | Zbl

[20] A. Zagaris, H.G. Kaper and T.J. Kaper, Analysis of the Computational Singular Perturbation reduction method for chemical kinetics. J. Nonlin. Sci. 14 (2004) 59-91. | MR | Zbl

[21] A. Zagaris, H.G. Kaper and T.J. Kaper, Fast and slow dynamics for the Computational Singular Perturbation method. Multiscale Model. Sim. 2 (2004) 613-638. | MR | Zbl

[22] A. Zagaris, C. Vandekerckhove, C.W. Gear, T.J. Kaper and I.G. Kevrekidis, Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Numer. Math. (submitted).

  • Divahar, J.; Roberts, A. J.; Mattner, Trent W.; Bunder, J. E.; Kevrekidis, Ioannis G. Staggered grids for multidimensional multiscale modelling, Computers and Fluids, Volume 271 (2024), p. 18 (Id/No 106167) | DOI:10.1016/j.compfluid.2023.106167 | Zbl:7833625
  • Patsatzis, Dimitrios G.; Russo, Lucia; Siettos, Constantinos Slow invariant manifolds of fast-slow systems of ODEs with physics-informed neural networks, SIAM Journal on Applied Dynamical Systems, Volume 23 (2024) no. 4, pp. 3077-3122 | DOI:10.1137/24m1656402 | Zbl:7965867
  • Patsatzis, Dimitrios; Fabiani, Gianluca; Russo, Lucia; Siettos, Constantinos Slow invariant manifolds of singularly perturbed systems via physics-informed machine learning, SIAM Journal on Scientific Computing, Volume 46 (2024) no. 4, p. c297-c322 | DOI:10.1137/23m1602991 | Zbl:1543.65204
  • Divahar, J.; Roberts, A. J.; Mattner, Trent W.; Bunder, J. E.; Kevrekidis, Ioannis G. Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves, Computer Methods in Applied Mechanics and Engineering, Volume 413 (2023), p. 21 (Id/No 116133) | DOI:10.1016/j.cma.2023.116133 | Zbl:1539.76141
  • Malani, Saurabh; Bertalan, Tom S.; Cui, Tianqi; Avalos, José L.; Betenbaugh, Michael; Kevrekidis, Ioannis G. Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information, Computers Chemical Engineering, Volume 178 (2023), p. 108343 | DOI:10.1016/j.compchemeng.2023.108343
  • Patsatzis, Dimitrios G.; Russo, Lucia; Kevrekidis, Ioannis G.; Siettos, Constantinos Data-driven control of agent-based models: an equation/variable-free machine learning approach, Journal of Computational Physics, Volume 478 (2023), p. 25 (Id/No 111953) | DOI:10.1016/j.jcp.2023.111953 | Zbl:7660326
  • Siettos, Constantinos; Russo, Lucia A numerical method for the approximation of stable and unstable manifolds of microscopic simulators, Numerical Algorithms, Volume 89 (2022) no. 3, pp. 1335-1368 | DOI:10.1007/s11075-021-01155-0 | Zbl:1491.65162
  • Spiliotis, Konstantinos; Koutsoumaris, Constantinos Chr.; Reppas, Andreas I.; Papaxenopoulou, Lito A.; Starke, Jens; Hatzikirou, Haralampos Optimal vaccine roll-out strategies including social distancing for pandemics, iScience, Volume 25 (2022) no. 7, p. 104575 | DOI:10.1016/j.isci.2022.104575
  • Ginoux, Jean-Marc Slow invariant manifolds of slow-fast dynamical systems, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, Volume 31 (2021) no. 7, p. 17 (Id/No 2150112) | DOI:10.1142/s0218127421501121 | Zbl:1471.34114
  • Ginoux, Jean-Marc; Meucci, Riccardo Slow Invariant Manifold of Laser with Feedback, Symmetry, Volume 13 (2021) no. 10, p. 1898 | DOI:10.3390/sym13101898
  • Burby, J. W.; Klotz, T. J. INVITED: Slow manifold reduction for plasma science, Communications in Nonlinear Science and Numerical Simulation, Volume 89 (2020), p. 61 (Id/No 105289) | DOI:10.1016/j.cnsns.2020.105289 | Zbl:1451.82052
  • Burby, J. W. Guiding center dynamics as motion on a formal slow manifold in loop space, Journal of Mathematical Physics, Volume 61 (2020) no. 1, p. 012703 | DOI:10.1063/1.5119801 | Zbl:1435.78005
  • Heiter, Pascal; Lebiedz, Dirk Towards differential geometric characterization of slow invariant manifolds in extended phase space: sectional curvature and flow invariance, SIAM Journal on Applied Dynamical Systems, Volume 17 (2018) no. 1, pp. 732-753 | DOI:10.1137/16m1106353 | Zbl:1396.34041
  • Sieber, Jan; Marschler, Christian; Starke, Jens Convergence of equation-free methods in the case of finite time scale separation with application to deterministic and stochastic systems, SIAM Journal on Applied Dynamical Systems, Volume 17 (2018) no. 4, pp. 2574-2614 | DOI:10.1137/17m1126084 | Zbl:1406.65134
  • Ceccato, Alessandro; Nicolini, Paolo; Frezzato, Diego A Low-Computational-Cost Strategy to Localize Points in the Slow Manifold Proximity for Isothermal Chemical Kinetics, International Journal of Chemical Kinetics, Volume 49 (2017) no. 7, p. 477 | DOI:10.1002/kin.21091
  • Kristiansen, K. U.; Wulff, Claudia Exponential estimates of symplectic slow manifolds, Journal of Differential Equations, Volume 261 (2016) no. 1, pp. 56-101 | DOI:10.1016/j.jde.2016.03.003 | Zbl:1382.37055
  • Lebiedz, D.; Unger, J. On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models, Mathematical and Computer Modelling of Dynamical Systems, Volume 22 (2016) no. 2, pp. 87-112 | DOI:10.1080/13873954.2016.1141219 | Zbl:1342.34076
  • Scheibe, Timothy D.; Murphy, Ellyn M.; Chen, Xingyuan; Rice, Amy K.; Carroll, Kenneth C.; Palmer, Bruce J.; Tartakovsky, Alexandre M.; Battiato, Ilenia; Wood, Brian D. An Analysis Platform for Multiscale Hydrogeologic Modeling with Emphasis on Hybrid Multiscale Methods, Groundwater, Volume 53 (2015) no. 1, p. 38 | DOI:10.1111/gwat.12179
  • Maclean, John; Gottwald, Georg A. On convergence of higher order schemes for the projective integration method for stiff ordinary differential equations, Journal of Computational and Applied Mathematics, Volume 288 (2015), pp. 44-69 | DOI:10.1016/j.cam.2015.04.004 | Zbl:1320.65107
  • Kuehn, Christian Computing Manifolds, Multiple Time Scale Dynamics, Volume 191 (2015), p. 327 | DOI:10.1007/978-3-319-12316-5_11
  • Kristiansen, K. Uldall Computation of saddle-type slow manifolds using iterative methods, SIAM Journal on Applied Dynamical Systems, Volume 14 (2015) no. 2, pp. 1189-1227 | DOI:10.1137/140961948 | Zbl:1323.34025
  • Marschler, Christian; Sieber, Jan; Hjorth, Poul G.; Starke, Jens Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow, Traffic and Granular Flow '13 (2015), p. 423 | DOI:10.1007/978-3-319-10629-8_48
  • Benoît, Eric; Brøns, Morten; Desroches, Mathieu; Krupa, Martin Extending the zero-derivative principle for slow-fast dynamical systems, ZAMP. Zeitschrift für angewandte Mathematik und Physik, Volume 66 (2015) no. 5, pp. 2255-2270 | DOI:10.1007/s00033-015-0552-8 | Zbl:1330.34093
  • Siettos, Constantinos Equation-free computation of coarse-grained center manifolds of microscopic simulators, Journal of Computational Dynamics, Volume 1 (2014) no. 2, pp. 377-389 | DOI:10.3934/jcd.2014.1.377 | Zbl:1346.37065
  • Kristiansen, K. U.; Brøns, M.; Starke, J. An Iterative Method for the Approximation of Fibers in Slow-Fast Systems, SIAM Journal on Applied Dynamical Systems, Volume 13 (2014) no. 2, p. 861 | DOI:10.1137/120889666
  • Marschler, Christian; Sieber, Jan; Berkemer, Rainer; Kawamoto, Atsushi; Starke, Jens Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models, SIAM Journal on Applied Dynamical Systems, Volume 13 (2014) no. 3, p. 1202 | DOI:10.1137/130913961
  • Härdin, Hanna M.; Zagaris, Antonios; Willms, Allan R.; Westerhoff, Hans V. Clusters of reaction rates and concentrations in protein networks such as the phosphotransferase system, The FEBS Journal, Volume 281 (2014) no. 2, p. 531 | DOI:10.1111/febs.12664
  • Kan, Xingye; Duan, Jinqiao; Kevrekidis, Ioannis G.; Roberts, Anthony J. Simulating Stochastic Inertial Manifolds by a Backward-Forward Approach, SIAM Journal on Applied Dynamical Systems, Volume 12 (2013) no. 1, p. 487 | DOI:10.1137/120881968
  • Zagaris, Antonios; Vandekerckhove, Christophe; Gear, C. William; Kaper, Tasso J.; Kevrekidis, Ioannis G. Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold, Discrete and Continuous Dynamical Systems, Volume 32 (2012) no. 8, pp. 2759-2803 | DOI:10.3934/dcds.2012.32.2759 | Zbl:1245.65102
  • Ariel, G.; Sanz-Serna, J. M.; Tsai, R. A Multiscale Technique for Finding Slow Manifolds of Stiff Mechanical Systems, Multiscale Modeling Simulation, Volume 10 (2012) no. 4, p. 1180 | DOI:10.1137/120861461
  • Mazzi, Giacomo; De Decker, Yannick; Samaey, Giovanni Towards an efficient multiscale modeling of low-dimensional reactive systems: Study of numerical closure procedures, The Journal of Chemical Physics, Volume 137 (2012) no. 20 | DOI:10.1063/1.4764109
  • Vanderhoydonc, Y.; Vanroose, W. Lifting in hybrid lattice Boltzmann and PDE models, Computing and Visualization in Science, Volume 14 (2011) no. 2, pp. 67-78 | DOI:10.1007/s00791-011-0164-6 | Zbl:1310.82035
  • Lebiedz, Dirk; Siehr, Jochen; Unger, Jonas A Variational Principle for Computing Slow Invariant Manifolds in Dissipative Dynamical Systems, SIAM Journal on Scientific Computing, Volume 33 (2011) no. 2, p. 703 | DOI:10.1137/100790318
  • Lebiedz, Dirk Entropy-related extremum principles for model reduction of dissipative dynamical systems, Entropy, Volume 12 (2010) no. 4, pp. 706-719 | DOI:10.3390/e12040706 | Zbl:1229.37019
  • Lebiedz, Dirk; Reinhardt, Volkmar; Siehr, Jochen Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics, Journal of Computational Physics, Volume 229 (2010) no. 18, pp. 6512-6533 | DOI:10.1016/j.jcp.2010.05.008 | Zbl:1197.65070
  • Kevrekidis, Ioannis G.; Samaey, Giovanni Equation-Free Multiscale Computation: Algorithms and Applications, Annual Review of Physical Chemistry, Volume 60 (2009) no. 1, p. 321 | DOI:10.1146/annurev.physchem.59.032607.093610
  • Van Leemput, Pieter; Rheinländer, Martin; Junk, Michael Smooth initialization of lattice Boltzmann schemes, Computers Mathematics with Applications, Volume 58 (2009) no. 5, pp. 867-882 | DOI:10.1016/j.camwa.2009.02.022 | Zbl:1189.76417
  • Vandekerckhove, Christophe; Kevrekidis, Ioannis; Roose, Dirk An efficient Newton-Krylov implementation of the constrained runs scheme for initializing on a slow manifold, Journal of Scientific Computing, Volume 39 (2009) no. 2, pp. 167-188 | DOI:10.1007/s10915-008-9256-y | Zbl:1203.65154
  • Härdin, Hanna M.; Zagaris, Antonios; Krab, Klaas; Westerhoff, Hans V. Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations, The FEBS Journal, Volume 276 (2009) no. 19, p. 5491 | DOI:10.1111/j.1742-4658.2009.07233.x

Cité par 39 documents. Sources : Crossref, zbMATH