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Uncertain, High-Dimensional Dynamical Systems

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  1. Uncertain, High-Dimensional Dynamical Systems Igor Mezić University of California, Santa Barbara IPAM, UCLA, February 2005

  2. Introduction • Measure of uncertainty? • Uncertainty and spectral theory of dynamical systems. • Model validation and data assimilation. • Decompositions.

  3. Dynamical evolution of uncertainty: an example x Output measure Input measure Tradeoff:Bifurcation vs. contracting dynamics

  4. Dynamical evolution of uncertainty: general set-up Skew-product system.

  5. Dynamical evolution of uncertainty: general set-up

  6. Dynamical evolution of uncertainty: general set-up f f T F(z) 1 0

  7. Dynamical evolution of uncertainty: simple examples Fw(z) 1 0 Expanding maps: x’=2x

  8. A measure of uncertainty of an observable 1 0

  9. Computation of uncertainty in CDF metric

  10. Maximal uncertainty for CDF metric 1 0

  11. Variance, Entropy and Uncertainty in CDF metric 0

  12. Uncertainty in CDF metric: Pitchfork bifurcation x Output measure Input measure

  13. Time-averaged uncertainty 1 0

  14. Conclusions

  15. Introduction PV=NkT • Example: thermodynamics from statistical mechanics “…Any rarified gas will behave that way, no matter how queer the dynamics of its particles…” Goodstein (1985) • Example:Galerkin truncation of evolution equations. • Information comes from a single observable time-series.

  16. Introduction When do two dynamical systems exhibit similar behavior?

  17. Introduction • Constructive proof that ergodic partitions and invariant • measures of systems can be compared using a • single observable (“Statistical Takens-Aeyels” theorem). • A formalism based on harmonic analysis that extends • the concept of comparing the invariant measure.

  18. Set-up Time averages and invariant measures:

  19. Set-up

  20. Pseudometrics for Dynamical Systems • Pseudometricthat captures statistics: where

  21. Ergodic partition

  22. Ergodic partition

  23. An example: analysis of experimental data

  24. Analysis of experimental data

  25. Analysis of experimental data

  26. Koopman operator, triple decomposition, MOD -Efficient representation of the flow field; can be done with vectors -Lagrangian analysis: FLUCTUATIONS MEAN FLOW PERIODIC APERIODIC Desirable: “Triple decomposition”:

  27. Embedding

  28. Conclusions • Constructive proof that ergodic partitions and invariant • measures of systems can be compared using a • single observable –deterministic+stochastic. • A formalism based on harmonic analysis that extends • the concept of comparing the invariant measure • Pseudometrics on spaces of dynamical systems. • Statistics – based, linear (but infinite-dimensional).

  29. Introduction Everson et al., JPO 27 (1997)

  30. Introduction • 4 modes -99% • of the variance! • -no dynamics! Everson et al., JPO 27 (1997)

  31. Introduction In this talk: -Account explicitly for dynamics to produce a decomposition. -Tool: lift to infinite-dimensional space of functions on attractor + consider properties of Koopman operator. -Allows for detailed comparison of dynamical properties of the evolution and retained modes. -Split into “deterministic” and “stochastic” parts: useful for prediction purposes.

  32. Example: Factors and harmonic analysis

  33. Factors and harmonic analysis

  34. Harmonic analysis: an example

  35. Evolution equations and Koopman operator

  36. Evolution equations and Koopman operator Similar:“Wold decomposition”

  37. Evolution equations and Koopman operator But how to get this from data?

  38. Evolution equations and Koopman operator is almost-periodic. -Remainder has continuous spectrum!

  39. Conclusions -Use properties of the Koopman operator to produce a decomposition -Tool: lift to infinite-dimensional space of functions on attractor. -Allows for detailed comparison of dynamical properties of the evolution and retained modes. -Split into “deterministic” and “stochastic” parts: useful for prediction purposes. -Useful for Lagrangian studies in fluid mechanics.

  40. -1 0 a1 a2 • But poor for dynamics: Irrationala’s produce the same statistics Invariant measures and time-averages Example:Probability histograms!

  41. Dynamical evolution of uncertainty: simple examples • Types of uncertainty: • Epistemic (reducible) • Aleatory (irreducible) • A-priori (initial conditions, • parameters, model structure) • A-posteriori (chaotic dynamics, • observation error) Expanding maps: x’=2x

  42. Uncertainty in CDF metric: Examples Uncertainty strongly dependent on distribution of initial conditions.

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