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Invariant grids: method of complexity reduction in reaction networks

Invariant grids: method of complexity reduction in reaction networks. Andrei Zinovyev Institut Curie , Paris Institut des Hautes Études Scientifiques. Stoichiometric equations. a s1 A 1 + … + a sn A n  b s1 A 1 + … + b sn A n. n – number of species, s – number of reactions. c n. c1.

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Invariant grids: method of complexity reduction in reaction networks

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  1. Invariant grids:method of complexity reduction in reaction networks Andrei Zinovyev Institut Curie, Paris Institut des Hautes Études Scientifiques

  2. Stoichiometric equations as1A1 + … + asnAn bs1A1 + … + bsnAn n – number of species, s – number of reactions cn c1 c2 2 - European conference on complex systems, Paris, November 2005 -

  3. What is Model Reduction? • 1 Shorten list of species • eliminate some • create “integrated” components • 2 Shorten list of reactions • eliminate some • “freeze” fast reactions • 3 Decompose motion into fast and slow 3 - European conference on complex systems, Paris, November 2005 -

  4. Approaching steady state 4 - European conference on complex systems, Paris, November 2005 -

  5. Positively Invariant Manifold fast motion slow motion Steady state W Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium 5 - European conference on complex systems, Paris, November 2005 -

  6. Why for do we need invariant manifold? Model reduction: “Macroscopic” system description xRN – detailed description yRm – “macroscopic” description. m<<N 6 - European conference on complex systems, Paris, November 2005 -

  7. Why for do we need invariant manifold? Dynamics visualization 7 - European conference on complex systems, Paris, November 2005 -

  8. Other useful non-invariant manifolds • Quasy steady-state • Fast variables are steady • Quasi-equilibrium • Manifolds maximizing entropy • Intrinsic low-dimensional manifold • Decomposition of Jacobian fields 8 - European conference on complex systems, Paris, November 2005 -

  9. Projector Pc on (some) manifoldinduces new (reduced) dynamics induced dynamics  tangent space J Pc J W TxW  = (1-Pc)J - invariance defect 9 - European conference on complex systems, Paris, November 2005 -

  10. Class of dissipative systems Lyapunov function G c ceq 11 - European conference on complex systems, Paris, November 2005 -

  11. Thermodynamic projector J Pc J The induced dynamics is dissipative only if 12 - European conference on complex systems, Paris, November 2005 -

  12. Correction of invariance defect invariant manifold corrections C1 invariance equation 1.0 (1-Pc)J = 0 0.8 0.6 Newton iterations equilibrium 0.4 0.2 0.20 0.10 0.15 0.05 C3 initial approximation 13 - European conference on complex systems, Paris, November 2005 -

  13. Invariant grid tangent space EQUILIBRIUM J tangent space invariance defect is corrected for every node independently 14 - European conference on complex systems, Paris, November 2005 -

  14. x0+ker PC J Tx Invariant grid 15 - European conference on complex systems, Paris, November 2005 -

  15. Growing Invariant Flag Phase space 16 - European conference on complex systems, Paris, November 2005 -

  16. Entropic scalar product 2 1 0 equilibrium -1 natural parameter entropy -2 17 - European conference on complex systems, Paris, November 2005 -

  17. H2 H O2 O H2O OH Hydrogen burning model reaction 1: H2 2H 2: O2 2O 3: H2O  H + OH 4: H2 + O  H + OH 5: O2 + H  O + OH 6: H2 + O  H2O Conservation laws: 2cH2 +2cH2O+cH+cOH = bH 2cO2+cH2O+cO+cOH = bO 18 - European conference on complex systems, Paris, November 2005 -

  18. One-dimensional dynamics equilibrium 19 - European conference on complex systems, Paris, November 2005 -

  19. Separation of times l is the eigen value of symmetrised matrix 20 - European conference on complex systems, Paris, November 2005 -

  20. Two-dimensional dynamics 21 - European conference on complex systems, Paris, November 2005 -

  21. Visualizing functions:concentration of H “Fast” coordinate 22 - European conference on complex systems, Paris, November 2005 -

  22. Visualizing functions:concentration of H2 “Slow” coordinate 23 - European conference on complex systems, Paris, November 2005 -

  23. Visualizing functions:concentration of OH 24 - European conference on complex systems, Paris, November 2005 -

  24. Visualizing functions:Entropy and entropy production Entropy production Entropy 25 - European conference on complex systems, Paris, November 2005 -

  25. Visualizing functions:Separation of relaxation times 3/2 2/1 26 - European conference on complex systems, Paris, November 2005 -

  26. closed system Open system as “closed system in a flow” flow • Entropy does not increase everywhere • Non-uniqueness of stationary states, auto-oscillations, etc. • inertial manifold often exists 27 - European conference on complex systems, Paris, November 2005 -

  27. Zero-order approximation Construct the invariant manifold W for W 28 - European conference on complex systems, Paris, November 2005 -

  28. W’ First-order approximation “Fast” and “slow” flow: New invariance equation: W 29 - European conference on complex systems, Paris, November 2005 -

  29. Conclusions Invariant grids: constructive method for chemical kinetics class of dissipative systems extension to open systems Use of thermodynamics: metrics in the phase space unique thermodynamic projector Possibility to visualize and explore system dynamics globally 30 - European conference on complex systems, Paris, November 2005 -

  30. Papers Gorban A, Karlin I, Zinovyev A. Constructive Methods of Invariant Manifolds for Kinetic Problems 2004. Physics Reports 396, pp.197-403. Gorban A, Karlin I, Zinovyev A. Invariant Grids for Reaction Kinetics 2004. Physica A, V.333, pp.106-154 31 - European conference on complex systems, Paris, November 2005 -

  31. People Professor Alexander Gorban University of Leicester, UK Doctor Iliya KarlinETH, Zurich 32 - European conference on complex systems, Paris, November 2005 -

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