Advertisement

Computing Manifolds

  • Christian Kuehn
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 191)

Abstract

We have extensively discussed the properties of invariant manifolds and their relevance for fast–slow systems in previous chapters. However, we usually used explicit algebraic expressions or asymptotic expansions to deal with critical and slow manifolds. For a general multiple time scale system, there are several complications. They may not be in standard form, and even if they are, then calculating a slow manifold analytically may be intractable. This chapter deals with algorithms to find and compute invariant manifolds for fast–slow systems numerically.

Bibliography

  1. [Abr13b]
    R.V. Abramov. A simple stochastic parameterization for reduced models of multiscale dynamics. arXiv:1302.4132v1, pages 1–23, 2013.Google Scholar
  2. [ACC+07]
    A. Adrover, F. Creta, S. Cerbelli, M. Valorani, and M. Giona. The structure of slow invariant manifolds and their bifurcational routes in chemical kinetic models. Comput. Chem. Eng., 31(11): 1456–1474, 2007.CrossRefGoogle Scholar
  3. [ACGV07]
    A. Adrover, F. Creta, M. Giona, and M. Valorani. Stretching-based diagnostics and reduction of chemical kinetic models with diffusion. J. Comp. Phys., 225(2):1442–1471, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [ASST12]
    G. Ariel, J.M. Sanz-Serna, and R. Tsai. A multiscale technique for finding slow manifolds of stiff mechanical systems. Multiscale Model. Simul., 10(4):1180–1203, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [BBS96]
    J.A. Borghans, R.J. De Boer, and L.A. Segel. Extending the quasi-steady state approximation by changing variables. Bull. Math. Biol., 58(1):43–63, 1996.CrossRefzbMATHGoogle Scholar
  6. [BCGFS13]
    T. Berry, J.R. Cressman, Z. Greguric-Ferencek, and T. Sauer. Time-scale separation from diffusion-mapped delay coordinates. SIAM J. Appl. Dyn. Syst., 12(2):618–649, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [BG13c]
    V. Bykov and V. Gol’dshtein. Fast and slow invariant manifolds in chemical kinetics. Comput. Math. Appl., 65(10):1502–1515, 2013.Google Scholar
  8. [BGG08]
    S. Borok, I. Goldfarb, and V. Gol’dshtein. About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM). Comm. Nonl. Sci. Numer. Simul., 13(6):1029–1038, 2008.Google Scholar
  9. [BGGM06]
    V. Bykov, I. Goldfarb, V. Gol’dshtein, and U. Maas. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J. Appl. Math., 71(3):359–382, 2006.Google Scholar
  10. [BGM08]
    V. Bykov, V. Gol’dshtein, and U. Maas. Simple global reduction technique based on decomposition approach. Combust. Theor. Model., 12(2):389–405, 2008.Google Scholar
  11. [BH25]
    G.E. Briggs and J.B.S. Haldane. A note on the kinetics of enzyme action. Biochem. J., 19(2):338–339, 1925.Google Scholar
  12. [BHV07]
    H.W. Broer, A. Hagen, and G. Vegter. Numerical continuation of normally hyperbolic invariant manifolds. Nonlinearity, 20(6):1499–1534, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [BM07a]
    V. Bykov and U. Maas. The extension of the ILDM concept to reaction–diffusion manifolds. Comust. Theor. Model., 11(6):839–862, 2007.CrossRefzbMATHGoogle Scholar
  14. [BM07b]
    V. Bykov and U. Maas. Extension of the ILDM method to the domain of slow chemistry. Proceed. Comust. Inst., 31(1):465–472, 2007.CrossRefGoogle Scholar
  15. [BYS10]
    E.M. Bollt, C. Yao, and I.B. Schwartz. Dimensional implications of dynamical data on manifolds to empirical KL analysis. Physica D, 239(23):2039–2049, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [Chi12]
    E. Chiavazzo. Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized relaxation redistribution method. J. Comp. Phys., 231(4):1751–1765, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [CRK05]
    R. Clewley, H.G. Rotstein, and N. Kopell. A computational tool for the reduction of nonlinear ODE systems possessing mutltiple scales. Multiscale Model. Simul., 4(3):732–759, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [CS11]
    M.S. Calder and D. Siegel. Properties of the Lindemann mechanism in phase space. Electron. J. Differential Equat., 2011(8):1–31, 2011.Google Scholar
  19. [CS12]
    M.J. Capinski and C. Simo. Computer assisted proof for normally hyperbolic invariant manifolds. Nonlinearity, 25:1997–2026, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [CTS12]
    R. Chachra, M.K. Transtrum, and J.P. Sethna. Structural susceptibility and separation of time scales in the van der Pol oscillator. Phys. Rev. E, 86:026712, 2012.CrossRefGoogle Scholar
  21. [DCD+07]
    E.J. Doedel, A. Champneys, F. Dercole, T. Fairgrieve, Y. Kuznetsov, B. Oldeman, R. Paffenroth, B. Sandstede, X. Wang, and C. Zhang. Auto 2007p: Continuation and bifurcation software for ordinary differential equations (with homcont). http://cmvl.cs.concordia.ca/auto, 2007.
  22. [DH96]
    M. Dellnitz and A. Hohmann. The computation of unstable manifolds using subdivision and continuation. In H.W. Broer, S.A. Van Gils, I. Hoveijn, and F. Takens, editors, Nonlinear Dynamical Systems and Chaos PNLDE 19, pages 449–459. Birkhäuser, 1996.Google Scholar
  23. [DH97]
    M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numer. Math., 75:293–317, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [DGK+12]
    M. Desroches, J. Guckenheimer, C. Kuehn, B. Krauskopf, H. Osinga, and M. Wechselberger. Mixed-mode oscillations with multiple time scales. SIAM Rev., 54(2):211–288, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [DKO08a]
    M. Desroches, B. Krauskopf, and H.M. Osinga. The geometry of slow manifolds near a folded node. SIAM J. Appl. Dyn. Syst., 7(4):1131–1162, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [DKO10]
    M. Desroches, B. Krauskopf, and H.M. Osinga. Numerical continuation of canard orbits in slow–fast dynamical systems. Nonlinearity, 23(3):739–765, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [DR96a]
    P. Duchene and P. Rouchon. Kinetic scheme reduction via geometric singular perturbation techniques. Chem. Engineer. Sci., 51(20):4661–4672, 1996.CrossRefGoogle Scholar
  28. [DS99b]
    M.J. Davis and R.T. Skodje. Geometric investigation of low-dimensional manifolds in systems approaching equilibrium. J. Chem. Phys., 111:859–874, 1999.CrossRefGoogle Scholar
  29. [EKO07]
    J.P. England, B. Krauskopf, and H.M. Osinga. Computing two-dimensional global invariant manifolds in slow–fast systems. Int. J. Bif. Chaos, 17(3):805–822, 2007.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [Fen79]
    N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equat., 31:53–98, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [FH06]
    D. Flockerzi and W. Heineken. Comment on: Chaos 9, 108–123 (1999). Identification of low order manifolds: validating the algorithm of Maas and Pope. Chaos, 16(4):048101, 2006.Google Scholar
  32. [Fra88b]
    S.J. Fraser. The steady state and equilibrium approximations: a geometrical picture. J. Chem. Phys., 88:4732–4738, 1988.CrossRefGoogle Scholar
  33. [GDH04]
    Z.P. Gerdtzen, P. Daoutidis, and W.S. Hu. Non-linear reduction for kinetic models of metabolic reaction networks. Metabolic Engineering, 6(2):140–154, 2004.CrossRefGoogle Scholar
  34. [GGM04]
    I. Goldfarb, V. Gol’dshtein, and U. Maas. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J. Appl. Math., 69(4):353–374, 2004.Google Scholar
  35. [GH83]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY, 1983.CrossRefzbMATHGoogle Scholar
  36. [GJM12]
    J. Guckenheimer, T. Johnson, and P. Meerkamp. Rigorous enclosures of a slow manifold. SIAM J. Appl. Dyn. Syst., 11(3):831–863, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  37. [GK09a]
    J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [GK10b]
    J. Guckenheimer and C. Kuehn. Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system. SIAM J. Appl. Dyn. Syst., 9:138–153, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [GKKZ05]
    C.W. Gear, T.J. Kaper, I.G. Kevrikidis, and A. Zagaris. Projecting to a slow manifold: singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst., 4(3):711–732, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [GKS04]
    D. Givon, R. Kupferman, and A.M. Stuart. Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity, 17:55–127, 2004.CrossRefMathSciNetGoogle Scholar
  41. [GKZ04]
    A.N. Gorban, I.V. Karlin, and A.Yu. Zinovyev. Constructive methods of invariant manifolds for kinetic problems. Physics Reports, 396:197–403, 2004.CrossRefMathSciNetGoogle Scholar
  42. [GV04]
    J. Guckenheimer and A. Vladimirsky. A fast method for approximating invariant manifolds. SIAM J. Appl. Dyn. Syst., 3(3):232–260, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [GV06]
    D.A. Goussis and M. Valorani. An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems. J. Comp. Phys., 214:316–346, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [GvL96]
    G.H. Golub and C. van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, MD, 1996.zbMATHGoogle Scholar
  45. [GW93]
    J. Guckenheimer and P. Worfolk. Dynamical systems: some computational problems. In D. Schlomiuk, editor, Bifurcations and Periodic Orbits of Vector Fields, pages 241–277. Kluwer, 1993.Google Scholar
  46. [Hal03]
    B.C. Hall. Lie Groups, Lie Algebras, and Representations. Springer, 2003.Google Scholar
  47. [Hen03]
    M.E. Henderson. Computing invariant manifolds by integrating fat trajectories. Technical Report RC22944, IBM Research, 2003.Google Scholar
  48. [HMKS01]
    S. Handrock-Meyer, L.V. Kalachev, and K.R. Schneider. A method to determine the dimension of long-time dynamics in multi-scale systems. J. Math. Chem., 30(2):133–160, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [HR02]
    E.L. Haseltine and Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117(15):6959–6969, 2002.Google Scholar
  50. [HZKW09]
    H.M. Hädin, A. Zagaris, K. Krab, and H.V. Westerhoff. Simplified yet highly accurate enzyme kinetics for cases of low substrate concentrations. FEBS J., 276(19):5491–5506, 2009.CrossRefGoogle Scholar
  51. [Jan89]
    J.A.M. Janssen. The elimination of fast variables in complex chemical reactions. I. Macroscopic level. J. Stat. Phys., 57(1):157–169, 1989.Google Scholar
  52. [JJK97]
    M.E. Johnson, M.S. Jolly, and I.G. Kevrekidis. Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies. Num. Alg., 14(1):125–140, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  53. [KAWC80]
    P.V. Kokotovic, J.J. Allemong, J.R. Winkleman, and J.H. Chow. Singular perturbation and iterative separation of time scales. Automatica, 16:23–33, 1980.CrossRefzbMATHGoogle Scholar
  54. [Kaz00a]
    N. Kazantzis. Singular PDEs and the problem of finding invariant manifolds for nonlinear dynamical systems. Phys. Lett. A, 272(4):257–263, 2000.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [KBS12]
    K.U. Kristiansen, M. Brøns, and J. Starke. An iterative method for the approximation of fibers in slow–fast systems. arXiv:1208.6420, pages 1–28, 2012.Google Scholar
  56. [KG02]
    N. Kazantzis and T. Good. Invariant manifolds and the calculation of the long-term asymptotic response of nonlinear processes using singular PDEs. Comput. Chem. Engineer., 26(7):999–1012, 2002.CrossRefGoogle Scholar
  57. [KJ11]
    A. Kumar and K. Josić. Reduced models of networks of coupled enzymatic reactions. J. Theor. Biol., pages 87–106, 2011.Google Scholar
  58. [KK02]
    H.G. Kaper and T.J. Kaper. Asymptotic analysis of two reduction methods for systems of chemical reactions. Physica D, 165:66–93, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  59. [KK13]
    H.-W. Kang and T.G. Kurtz. Separation of time scales and model reduction for stochastic reaction networks. Ann. Appl. Prob., 23(2):529–583, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  60. [KKK+07]
    L.V. Kalachev, H.G. Kaper, T.J. Kaper, N. Popovic, and A. Zagaris. Reduction for Michaelis–Menten–Henri kinetics in the presence of diffusion. Electronic J. Diff. Eq., 16:155–184, 2007.MathSciNetGoogle Scholar
  61. [KKS10]
    N. Kazantzis, C. Kravaris, and L. Syrou. A new model reduction method for nonlinear dynamical systems. Nonlinear Dyn., 59(1):183–194, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  62. [Kna04]
    A.W. Knapp. Lie Groups Beyond an Introduction. Birkhäuser, 2004.Google Scholar
  63. [KO99]
    B. Krauskopf and H.M. Osinga. Two-dimensional global manifolds of vector fields. Chaos, 9(3): 768–774, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [KO03]
    B. Krauskopf and H.M. Osinga. Computing geodesic level sets on global (un)stable manifolds of vector fields. SIAM J. Appl. Dyn. Syst., 4(2):546–569, 2003.CrossRefMathSciNetGoogle Scholar
  65. [KOD+05]
    B. Krauskopf, H.M. Osinga, E.J. Doedel, M.E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge. A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bif. Chaos, 15(3):763–791, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  66. [KSG+98]
    B.N. Kholdenko, S. Schuster, J. Garcia, H.V. Westerhoff, and M. Cascante. Control analysis of metabolic systems involving quasi-equilibrium reactions. Biochimica et Biophysica Acta, 1379(3): 337–352, 1998.CrossRefGoogle Scholar
  67. [KSG10]
    P.D. Kourdis, R. Steuer, and D.A. Goussis. Physical understanding of complex multiscale biochemical models via algorithmic simplification: glycolysis in Saccharomyces cerevisiae. Physica D, 239(18):1798–1817, 2010.Google Scholar
  68. [Leb04]
    D. Lebiedz. Computing minimal entropy production trajectories: an approach to model reduction in chemical kinetics. J. Chem. Phys., 120:6890–6897, 2004.CrossRefGoogle Scholar
  69. [Lee06]
    J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006.Google Scholar
  70. [LL09b]
    C.H. Lee and R. Lui. A reduction method for multiple time scale stochastic reaction networks. J. Math. Chem., 46(4):1292–1321, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  71. [LL10]
    C.H. Lee and R. Lui. A reduction method for multiple time scale stochastic reaction networks with non-unique equilibrium probability. J. Math. Chem., 47(2):750–770, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  72. [LO10]
    C.H. Lee and H.G. Othmer. A multi-time-scale analysis of chemical reaction networks: I. Deterministic systems. J. Math. Biol., 60:387–450, 2010.Google Scholar
  73. [LS13]
    D. Lebiedz and J. Siehr. A continuation method for the efficient solution of parametric optimization problems in kinetic model reduction. arXiv:1301.5815, pages 1–19, 2013.Google Scholar
  74. [LSU11]
    D. Lebiedz, J. Siehr, and J. Unger. A variational principle for computing slow invariant manifolds in dissipative dynamical systems. SIAM J. Sci. Comput., 33(2):703–720, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  75. [Maa98]
    U. Maas. Efficient calculation of intrinsic low-dimensional manifolds for simplification of chemical kinetics. Comp. Vis. Sci., 1:69–82, 1998.CrossRefzbMATHGoogle Scholar
  76. [Mas02]
    M. Massot. Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Comptes Rendus Math., 335:93–98, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  77. [Mea95]
    K.D. Mease. Geometry of computational singular perturbations. Nonlinear Contr. Syst. Design, 2: 855–861, 1995.Google Scholar
  78. [Mei78]
    W. Meiske. An approximate solution of the Michaelis–Menten mechanism for quasi-steady and state quasi-equilibrium. Math. Biosci., 42:63–71, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  79. [MP92a]
    U. Maas and S.B. Pope. Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds. In Proceedings of the 24th International Symposium on Combustion, pages 103–112. The Combustion Institute, 1992.Google Scholar
  80. [MP92b]
    U. Maas and S.B. Pope. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space. Combust. Flame, 88:239–264, 1992.CrossRefGoogle Scholar
  81. [MP13]
    J.D. Mengers and J.M. Powers. One-dimensional slow invariant manifolds for fully coupled reaction and micro-scale diffusion. SIAM J. Appl. Dyn. Syst., 12(2):560–595, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  82. [NF89]
    A.H. Nguyen and S.J. Fraser. Geometrical picture of reaction in enzyme kinetics. J. Chem. Phys., 91:186, 1989.CrossRefGoogle Scholar
  83. [NF13]
    P. Nicolini and D. Frezzato. Features in chemical kinetics. II. A self-emerging definition of slow manifolds. J. Chem. Phys., 138:234102, 2013.Google Scholar
  84. [NLCK06]
    B. Nadler, S. Lafon, R.R. Coifman, and I.G. Kevrekidis. Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmonic Anal., 21(1):113–127, 2006.CrossRefzbMATHMathSciNetGoogle Scholar
  85. [RA03]
    C.V. Rao and A.P. Arkin. Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J. Chem. Phys., 118(11):4999–5010, 2003.CrossRefGoogle Scholar
  86. [RF91a]
    M.R. Roussel and S.J. Fraser. Accurate steady-state approximations: implications for kinetics experiments and mechanism. J. Phys. Chem., 95(22):8762–8770, 1991.CrossRefGoogle Scholar
  87. [RF91b]
    M.R. Roussel and S.J. Fraser. Geometry of steady-state approximation: perturbation and accelerated convergence methods. J. Chem. Phys., 95:8762–8770, 1991.CrossRefGoogle Scholar
  88. [RF91c]
    M.R. Roussel and S.J. Fraser. On the geometry of transient relaxation. J. Chem. Phys., 94:7106, 1991.CrossRefGoogle Scholar
  89. [RF01]
    M.R. Roussel and S.J. Fraser. Invariant manifold methods for metabolic model reduction. Chaos, 11(1):196–206, 2001.CrossRefzbMATHGoogle Scholar
  90. [RGZL08]
    O. Radulescu, A.N. Gorban, A. Zinovyev, and A. Lilienbaum. Robust simplifications of multiscale biochemical networks. BMC Syst. Biol., 2(1):86, 2008.Google Scholar
  91. [RMW99]
    C. Rhodes, M. Morari, and S. Wiggins. Identification of low order manifolds: validating the algorithm of Maas and Pope. Chaos, 9(1):108–123, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  92. [RPVG07]
    Z. Ren, S.B. Pope, A. Vladimirsky, and J.M. Guckenheimer. Application of the ICE-PIC method for the dimension reduction of chemical kinetics coupled with transport. Proceed. Combust. Inst., 31: 473–481, 2007.CrossRefGoogle Scholar
  93. [RSS11]
    E. Reznik, D. Segré, and W.E. Sherwood. The quasi-steady state assumption in an enzymatically open system. arXiv:1103.1200v1, pages 1–28, 2011.Google Scholar
  94. [SFMH05]
    R. Straube, D. Flockerzi, S.C. Müller, and M.J. Hauser. Reduction of chemical reaction networks using quasi-integrals. J. Phys. Chem. A, 109(3):441–450, 2005.CrossRefGoogle Scholar
  95. [SK93a]
    G.M. Shroff and H.B. Keller. Stabilization of unstable procedures: a recursive projection method. SIAM J. Numer. Anal., 30:1099–1120, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  96. [SS89]
    L.A. Segel and M. Slemrod. The quasi-steady-state assumption: a case study in perturbation. SIAM Rev., 31(3):446–477, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  97. [Ste73]
    G.W. Stewart. Introduction to Matrix Computations. Academic Press, 1973.Google Scholar
  98. [Sti98a]
    M. Stiefenhofer. Quasi-steady-state approximation for chemical reaction networks. J. Math. Biol., 36(6):593–609, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  99. [VCG+06]
    M. Valorani, F. Creta, D.A. Goussis, J.C. Lee, and H.N. Najm. An automatic procedure for the simplification of chemical kinetic mechanisms based on CSP. Combustion and Flame, 146(1):29–51, 2006.CrossRefGoogle Scholar
  100. [VGCN05]
    M. Valorani, D.A. Goussis, F. Creta, and H.N. Najm. Higher order corrections in the approximation of low-dimensional manifolds and the construction of simplified problems with the CSP method. J. Comp. Phys., 209(2):754–786, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  101. [VP09]
    M. Valorani and S. Paolucci. The G-scheme: a framework for multi-scale adaptive model reduction. J. Comp. Phys., 228(13):4665–4701, 2009.CrossRefzbMATHMathSciNetGoogle Scholar
  102. [ZGKK09]
    A. Zagaris, C.W. Gear, T.J. Kaper, and I.G. Kevrikidis. Analysis of the accuracy and convergence of equation-free projection to a slow manifold. ESAIM: Math. Model. Numer. Anal., 43(4):754–784, 2009.CrossRefGoogle Scholar
  103. [ZKK04a]
    A. Zagaris, H.G. Kaper, and T.J. Kaper. Analysis of the computational singular perturbation method for chemical kinetics. J. Nonlinear Sci., 14:59–91, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  104. [ZKK04b]
    A. Zagaris, H.G. Kaper, and T.J. Kaper. Fast and slow dynamics for the computational singular perturbation method. Multiscale Model. Simul., 2(4):613–638, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  105. [ZKK05]
    A. Zagaris, H.G. Kaper, and T.J. Kaper. Two perspectives on reduction of ordinary differential equations. Math. Nachr., 278(12):1629–1642, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  106. [ZNK13]
    A. Zakharova, Z. Nikoloski, and A. Koseka. Dimensionality reduction of bistable biological systems. Bull. Math. Biol., 75:373–392, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  107. [ZVG+12]
    A. Zagaris, C. Vanderkerckhove, 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. Discr. Cont. Dyn. Syst. A, 32(8):2759–2803, 2012.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

Personalised recommendations