Free Access
Math. Model. Nat. Phenom.
Volume 10, Number 3, 2015
Model Reduction
Page(s) 149 - 167
Published online 22 June 2015
  1. M. Aoki. Some approximation methods for estimation and control of large scale systems. IEEE Trans. Automat. Control, AC-23 (1978), 173–182. [Google Scholar]
  2. P. Auger, R. Bravo de la Parra. Methods of aggregation of variables in population dynamics. C. R. Acad. Sci., Ser. III, 323 (2000), 665–674. [CrossRef] [Google Scholar]
  3. F. Büchel, N. Rodriguez, N. Swainston, C. Wrzodek, T. Czauderna, R. Keller, F. Mittag, M. Schubert, M. Glont, M. Golebiewski, M. van Iersel, S. Keating, M. Rall, M. Wybrow, H. Hermjakob, M. Hucka, D. B. Kell, W. Müller, P. Mendes, A. Zell, C. Chaouiya, J. Saez-Rodriguez, F. Schreiber, C. Laibe, A. Dräger, N. Le Novère. Path2Models: Large-scale generation of computational models from biochemical pathway maps. BMC Syst. Biol., 7 (2013), 116. [CrossRef] [PubMed] [Google Scholar]
  4. F. W. Chang, F. A. Howes. Nonlinear Singular Perturbation Phenomena: Theory and Application. Applied Mathematical Sciences. vol. 56. Springer, New York, 1984. [Google Scholar]
  5. G.-M. Côme. Radical reaction mechanisms. Mathematical theory. J. Phys. Chem., 81 (1977), 2560–2563. [CrossRef] [Google Scholar]
  6. S. M. Cox, A. J. Roberts. Initial conditions for models of dynamical systems. Physica D, 85 (1995), 126–141. [CrossRef] [Google Scholar]
  7. P. G. Coxson, K. B. Bischoff. Lumping strategy. 1. Introductory techniques and applications of cluster analysis. Ind. Eng. Chem. Res., 26 (1987), 1239–1248. [CrossRef] [Google Scholar]
  8. P. G. Coxson, K. B. Bischoff. Lumping strategy. 2. A system theoretic approach. Ind. Eng. Chem. Res., 26 (1987), 2151–2157. [CrossRef] [Google Scholar]
  9. C. F. Curtiss, J. O. Hirschfelder. Integration of stiff equations. Proc. Natl. Acad. Sci. U.S.A., 38 (1952), 235–243. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  10. M. J. Davis, R. T. Skodje. Geometric investigation of low-dimensional manifolds in systems approaching equilibrium. J. Chem. Phys., 111 (1999), 859–874. [CrossRef] [Google Scholar]
  11. G. Farkas. Kinetic lumping schemes. Chem. Eng. Sci., 54 (1999), 3909–3915. [Google Scholar]
  12. N. Fenichel. Geometric singular perturbation theory for ordinary differential equations. J. Differential Equations, 31 (1979), 53–98. [Google Scholar]
  13. S. J. Fraser. The steady state and equilibrium approximations: A geometrical picture. J. Chem. Phys., 88 (1988), 4732–4738. [CrossRef] [Google Scholar]
  14. S. J. Fraser. Symbolic methods for invariant manifolds in chemical kinetics. Int. J. Quantum Chem., 106 (2006), 228–243. [CrossRef] [Google Scholar]
  15. J.-M. Ginoux, B. Rossetto, L. O. Chua. Slow invariant manifolds as curvature of the flow of dynamical systems. Int. J. Bifurc. Chaos, 18 (2008), 3409–3430. [CrossRef] [Google Scholar]
  16. A. N. Gorban, I. V. Karlin, V. B. Zmievskii, S. V. Dymova. Reduced description in the reaction kinetics. Physica A, 275 (2000), 361–379. [CrossRef] [Google Scholar]
  17. A. N. Gorban, I. V. Karlin. Method of invariant manifolds and regularization of acoustic spectra. Transport Theory Stat. Phys., 23 (1994), 559–632. [CrossRef] [Google Scholar]
  18. A. N. Gorban, I. V. Karlin. Method of invariant manifold for chemical kinetics. Chem. Eng. Sci., 58 (2003), 4751–4768. [CrossRef] [Google Scholar]
  19. A. N. Gorban, I. V. Karlin, A. Yu. Zinovyev. Constructive methods of invariant manifolds for kinetic problems. Phys. Rep., 396 (2004), 197–403. [CrossRef] [Google Scholar]
  20. A. N. Gorban, I. V. Karlin, A. Yu. Zinovyev. Invariant grids for reaction kinetics. Physica A, 333 (2004), 106–154. [CrossRef] [Google Scholar]
  21. D. A. Goussis, M. Valorani. An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems. J. Comput. Phys., 214 (2006), 316–346. [CrossRef] [Google Scholar]
  22. H. Huang, M. Fairweather, J. F. Griffiths, A. S. Tomlin, R. B. Brad. A systematic lumping approach for the reduction of comprehensive kinetic models. Proc. Combust. Inst., 30 (2005), 1309–1316. [CrossRef] [Google Scholar]
  23. F. G. Heineken, H. M. Tsuchiya, R. Aris. On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math. Biosci., 1 (1967), 95–113. [CrossRef] [Google Scholar]
  24. Y. Iwasa, V. Andreasen, S. Levin. Aggregation in model ecosystems. I. Perfect aggregation. Ecol. Modelling, 37 (1987), 287–302. [Google Scholar]
  25. J. G. Kemeny, J. L. Snell. Finite Markov Chains. Springer, New York, 1976, pp. 123–140. [Google Scholar]
  26. J. C. W. Kuo, J. Wei. A lumping analysis in monomolecular reaction systems. Analysis of approximately lumpable systems. Ind. Eng. Chem. Fundam., 8 (1969), 124–133. [Google Scholar]
  27. S. H. Lam. Using CSP to understand complex chemical kinetics. Combust. Sci. Technol., 89 (1993), 375–404. [CrossRef] [Google Scholar]
  28. S. H. Lam, D. A. Goussis. The CSP method for simplifying kinetics. Int. J. Chem. Kinet., 26 (1994), 461–486. [CrossRef] [Google Scholar]
  29. G. Li, H. Rabitz. Combined symbolic and numerical approach to constrained nonlinear lumping—with application to an H2/O2 oxidation model. Chem. Eng. Sci., 51 (1996), 4801–4816. [CrossRef] [Google Scholar]
  30. G. Li, H. Rabitz, J. Tóth. A general analysis of exact nonlinear lumping in chemical kinetics. Chem. Eng. Sci., 49 (1994), 343–361. [CrossRef] [Google Scholar]
  31. G. Li, A. S. Tomlin, H. Rabitz, J. Tóth. Determination of approximate lumping schemes by a singular perturbation method. J. Chem. Phys., 99 (1993), 3562–3574. [CrossRef] [Google Scholar]
  32. C. C. Lin, L. A. Segel. Mathematics Applied to Deterministic Problems in the Natural Sciences. Classics in Applied Mathematics, vol. 1 SIAM, Philadelphia, 1988, ch. 9-10. [Google Scholar]
  33. Ch. Lubich, K. Nipp, D. Stoffer. Runge-Kutta solutions of stiff differential equations near stationary points. SIAM J. Numer. Anal., 32 (1995), 1296–1307. [CrossRef] [Google Scholar]
  34. U. Maas, A. S. Tomlin. Time-scale splitting-based mechanism reduction. Cleaner Combustion (F. Battin-Leclerc et al., ed.), Springer, London, 2013, pp. 467–484. [Google Scholar]
  35. A. H. Nguyen, S. J. Fraser. Geometrical picture of reaction in enzyme kinetics. J. Chem. Phys., 91 (1989), 186–193. [CrossRef] [Google Scholar]
  36. B. R. Noack, K. Afanasiev, M. Morzynski, G. Tadmor, F. Thiele. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497 (2003), 335–363. [CrossRef] [Google Scholar]
  37. B. E. Okeke. Lumping Methods for Model Reduction. Master’s thesis, University of Lethbridge, 2013. URL: [Google Scholar]
  38. M. S. Okino, M. L. Mavrovouniotis. Simplification of mathematical models of chemical reaction systems. Chem. Rev., 98 (1998), 391–408. [CrossRef] [PubMed] [Google Scholar]
  39. D. J. M. Park. The hierarchical structure of metabolic networks and the construction of efficient metabolic simulators. J. Theor. Biol., 46 (1974), 31–74. [CrossRef] [PubMed] [Google Scholar]
  40. N. Peters, B. Rogg. Reduced Kinetic Mechanism for Applications in Combustion Systems. 2nd ed., Springer, Berlin, 1993. [Google Scholar]
  41. D. Rempfer. On low-dimensional Galerkin models for fluid flow. Theor. Comput. Fluid Dyn., 14 (2000), 75–88. [CrossRef] [Google Scholar]
  42. A. J. Roberts. Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems. J. Austral. Math. Soc. B, 31 (1989), 48–75. [CrossRef] [Google Scholar]
  43. B. Rossetto, T. Lenzini, S. Ramdani, G. Suchey. Slow-fast autonomous dynamical systems. Int. J. Bifurc. Chaos, 8 (1998), 2135–2145. [CrossRef] [Google Scholar]
  44. M. R. Roussel. A Rigorous Approach to Steady-State Kinetics Applied to Simple Enzyme Mechanisms. Ph.D. thesis, University of Toronto, 1994. [Google Scholar]
  45. M. R. Roussel. Forced-convergence iterative schemes for the approximation of invariant manifolds. J. Math. Chem., 21 (1997), 385–393. [CrossRef] [Google Scholar]
  46. M. R. Roussel. Approximating state-space manifolds which attract solutions of systems of delay-differential equations. J. Chem. Phys., 109 (1998), 8154–8160. [CrossRef] [Google Scholar]
  47. M. R. Roussel, S. J. Fraser. Geometry of the steady-state approximation: Perturbation and accelerated convergence methods. J. Chem. Phys., 93 (1990), 1072–1081. [CrossRef] [Google Scholar]
  48. M. R. Roussel, S. J. Fraser. On the geometry of transient relaxation. J. Chem. Phys., 94 (1991), 7106–7113. [CrossRef] [Google Scholar]
  49. M. R. Roussel, S. J. Fraser. Global analysis of enzyme inhibition kinetics. J. Phys. Chem., 97 (1993), 8316–8327. Errata, ibid. 98 (1994), 5174. [CrossRef] [Google Scholar]
  50. M. R. Roussel, S. J. Fraser. Invariant manifold methods for metabolic model reduction. Chaos, 11 (2001), 196–206. [CrossRef] [PubMed] [Google Scholar]
  51. D. Shear. An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions. J. Theor. Biol., 16 (1967), 212–228. [CrossRef] [PubMed] [Google Scholar]
  52. A. Stagni, A. Cuoci, A. Frassoldati, T. Faravelli, E. Ranzi. Lumping and reduction of detailed kinetic schemes: an effective coupling. Ind. Eng. Chem. Res., 53 (2014), 9004–9016. [CrossRef] [Google Scholar]
  53. A. N. Tikhonov, A. B. Vasil’eva, A. G. Sveshnikov. Differential Equations. Springer, Berlin, 1985, pp. 186–213. [Google Scholar]
  54. A. S. Tomlin, T. Turányi, M. J. Pilling. Mathematical tools for the construction, investigation and reduction of combustion mechanisms. Compr. Chem. Kinet., 35 (1997), 293–437. [CrossRef] [Google Scholar]
  55. K. Uldall Kristiansen, M. Brøns, J. Starke. An iterative method for the approximation of fibers in slow-fast systems. SIAM J. Appl. Dyn. Syst., 13 (2014), 861–900. [CrossRef] [Google Scholar]
  56. J. Warnatz, U. Maas, R.W. Dibble. Combustion: Physical and Chemical Fundamentals, Modeling and Simulation, Experiments, Pollutant Formation, 2nd ed., Springer, Berlin, 1999. [Google Scholar]
  57. J. Wei, J. C. W. Kuo. A lumping analysis in monomolecular reaction systems. Analysis of the exactly lumpable system. Ind. Eng. Chem. Fundam., 8 (1969), 114–123. [CrossRef] [Google Scholar]
  58. L. E. Whitehouse, A. S. Tomlin, M. J. Pilling. Systematic reduction of complex tropospheric chemical mechanisms, Part I: Sensitivity and time-scale analyses. Atmos. Chem. Phys., 4 (2004), 2025–2056. [CrossRef] [Google Scholar]
  59. A. Zagaris, H. G. Kaper, T. J. Kaper. Analysis of the computational singular perturbation reduction method for chemical kinetics. J. Nonlinear Sci., 14 (2004), 59–91. [CrossRef] [MathSciNet] [Google Scholar]
  60. A. Zagaris, H. G. Kaper, T. J. Kaper, Fast and slow dynamics for the computational singular perturbation method. Multiscale Model. Simul., 2 (2004), 613–638. [CrossRef] [Google Scholar]

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