Uncertain, High-Dimensional Uncertain, High-Dimensional Dynamical SystemsDynamical Systems

Igor MezićIgor Mezić

University of California, Santa BarbaraUniversity of California, Santa Barbara

IPAM, UCLA, February 2005IPAM, UCLA, February 2005

IntroductionIntroduction

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

x

Dynamical evolution of uncertainty: an exampleDynamical evolution of uncertainty: an example

Input measureInput measure

Output measureOutput measure

Tradeoff:Tradeoff: Bifurcation Bifurcation vs.vs. contracting dynamicscontracting dynamics

Dynamical evolution of uncertainty: general set-upDynamical evolution of uncertainty: general set-up

Skew-product system.Skew-product system.

Dynamical evolution of uncertainty: general set-upDynamical evolution of uncertainty: general set-up

Dynamical evolution of uncertainty: general set-upDynamical evolution of uncertainty: general set-up

ff ff

TT

F(z)F(z)11

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Dynamical evolution of uncertainty: simple examplesDynamical evolution of uncertainty: simple examples

Expanding maps: Expanding maps: x’=2xx’=2x

FF(z)(z)11

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A measure of uncertainty of an observableA measure of uncertainty of an observable

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Computation of uncertainty in CDF metricComputation of uncertainty in CDF metric

Maximal uncertainty for CDF metricMaximal uncertainty for CDF metric

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Variance, Entropy and Uncertainty in CDF metricVariance, Entropy and Uncertainty in CDF metric

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Uncertainty in CDF metric: Pitchfork bifurcationUncertainty in CDF metric: Pitchfork bifurcation

x

Input measureInput measure

Output measureOutput measure

Time-averaged uncertaintyTime-averaged uncertainty

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ConclusionsConclusions

IntroductionIntroduction

• Example: thermodynamics from statistical mechanics

PV=NkTPV=NkT

“…“…Any rarified gas will behave that way, Any rarified gas will behave that way, no matter how queer the dynamics of its particles…”no matter how queer the dynamics of its particles…”

Goodstein (1985)Goodstein (1985)

•Example: Galerkin truncation of evolution equations.

•Information comes from a single observable time-series.

IntroductionIntroduction

When do two dynamical systemsWhen do two dynamical systems exhibit exhibit similar behaviorsimilar behavior??

IntroductionIntroduction

• Constructive proof that Constructive proof that ergodic partitionsergodic partitions and and invariant invariant measuresmeasures of systems can be compared using a of systems can be compared using a single observablesingle observable (“ (“Statistical Takens-AeyelsStatistical Takens-Aeyels” theorem).” theorem).• A A formalismformalism based on based on harmonic analysisharmonic analysis that extends that extends the concept of comparing the invariant measure.the concept of comparing the invariant measure.

Set-upSet-up

Time averages and invariant measures:

Set-upSet-up

Pseudometrics for Dynamical SystemsPseudometrics for Dynamical Systems

wherewhere

• PseudometricPseudometric that captures statisticsthat captures statistics::

Ergodic partitionErgodic partition

Ergodic partitionErgodic partition

An example: analysis of experimental dataAn example: analysis of experimental data

Go( j )

Gc( j)

+

-P ressureD isturbance

V alvecommand

P lant: 2 n d orderw it h delay

P hase-shi ftingcon trol ler

V alv e saturation

Analysis of experimental dataAnalysis of experimental data

Analysis of experimental dataAnalysis of experimental data

Koopman operator, triple decomposition, Koopman operator, triple decomposition, MODMOD

-Efficient representation of the flow field; can be done with vectors -Lagrangian analysis:

MEANMEAN FLOW FLOW FLUCTUATIONSFLUCTUATIONS

Desirable: “Triple decomposition”: PERIODICPERIODIC APERIODICAPERIODIC

EmbeddingEmbedding

Conclusions

• Constructive proof that Constructive proof that ergodic partitionsergodic partitions and and invariant invariant measuresmeasures of systems can be compared using a of systems can be compared using a single observable –deterministic+stochastic.single observable –deterministic+stochastic.• A A formalismformalism based on based on harmonic analysisharmonic analysis that extends that extends the concept of comparing the invariant measurethe concept of comparing the invariant measure• PseudometricsPseudometrics on spaces of dynamical systems. on spaces of dynamical systems.• Statistics – basedStatistics – based, , linearlinear (but (but infinite-dimensionalinfinite-dimensional).).

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

IntroductionIntroduction

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

IntroductionIntroduction

-4 modes -99% 4 modes -99% of the of the variance!variance!-no -no dynamicsdynamics!!

IntroductionIntroduction

In this talk: In this talk:

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

Factors and harmonic analysisFactors and harmonic analysis

Example:

Factors and harmonic analysisFactors and harmonic analysis

Harmonic analysis: an exampleHarmonic analysis: an example

x x a x

y y x a x

'

'

sin( ),

sin( ).

= + +

= + +

π

π

2

2

Evolution equations and Koopman Evolution equations and Koopman operatoroperator

Evolution equations and Koopman Evolution equations and Koopman operatoroperator

Similar:“Wold decomposition”Similar:“Wold decomposition”

Evolution equations and Koopman Evolution equations and Koopman operatoroperator

But how to get this from data?But how to get this from data?

Evolution equations and Koopman Evolution equations and Koopman operatoroperator

-Remainder has continuous spectrum!

is almost-periodic.

ConclusionsConclusions

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

-Useful for Lagrangian studies in fluid mechanics.-Useful for Lagrangian studies in fluid mechanics.

Invariant measures and time-averagesInvariant measures and time-averages

Example:Example: Probability histogramsProbability histograms!!

-1 0

κ i

2• But But poor for dynamicspoor for dynamics: :

IrrationalIrrational ’s’sproduce the same statisticsproduce the same statistics

Dynamical evolution of uncertainty: simple examplesDynamical evolution of uncertainty: simple examples

Types of uncertainty:Types of uncertainty:

•EpistemicEpistemic (reducible) (reducible)•Aleatory Aleatory (irreducible)(irreducible)•A-prioriA-priori (initial conditions, (initial conditions, parameters, model structure)parameters, model structure)•A-posterioriA-posteriori (chaotic dynamics, (chaotic dynamics, observation error)observation error)

Expanding maps: Expanding maps: x’=2xx’=2x

Uncertainty in CDF metric: ExamplesUncertainty in CDF metric: Examples

Uncertainty strongly dependent on distribution of initial conditions.Uncertainty strongly dependent on distribution of initial conditions.