M. Budišić: Time-Averages of Observables

8/6/2019 M. Budii: Time-Averages of Observables

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Mezi Research Group

SIAM Conference on Applications of Dynamical Systems

Snowbird, May 2011

Marko [email protected]

TIME-AVERAGES OF OBSERVABLES

The minisymposium that follows focuses on time averaging in analysis and design of dynamical systems. This presentation aims to introducethe recent advances in analyzing dynamical systems by averaging functions, or observables, along the trajectories of the system.

First of all, let me briefly say what I will NOT be talking about. This work should not be confused with averaging of the model where the orderof the model is reduced by substituting dynamics evolving on fast time scales by their averaged effect.

The time-averages of observables, on the other hand, involve averaging observables, or measurements, along trajectories of the system.Common names used are: time-averages, trajectory-averages, dynamical averages, Cesro means, ergodic averages, lagrangian averages,cumulative moving averages, statistics...


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Monday, May 23, 2011 Marko Budii: Time Averages of Observables

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Time-averages are robust under time-discretizationand round-off error even when trajectory is not.

[Sigurgeirsson, Stuart, 2001]

Motivation:

Time-averages provide macroscopic descriptions ofgeometry of dynamics: Lyapunov exponents, residence

times, power spectrum,...

[Cvitanovi et al., Chaos Book]

Trajectories are not robust descriptions of dynamics inchaotic systems, due to exponential sensitivity to errors.

Problem:

Use time-averages to extend the library of robust indicators of

dynamical behaviors for analysis and design.

In chaotic systems, trajectories are non-robust. Even the smallest perturbations, common in numerical simulations, result in completelydifferent trajectories. In Chaos Book, Cvitanovi argues that trajectories are wrong objects to look at in such cases, and that the best we cando is to analyze features of the set in state space that the trajectory explores. Despite trajectories being sensitive to perturbations,simulations and numerics show that averaging over them still results in reliable time-averages, as noticed by Sigurgeirsson and Stuart.

Time averages provide a macroscopic description of features of the set a trajectory is exploring in the state space. Many common dynamicalindicators (Lyapunov coefficients, residence times, power spectra...) can be formulated as time averages of particular observables alongtrajectories. This body of work aims to systematically extend the library of those indicators by a careful choice of observables that areaveraged.


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f(x) = limT

1

T

T

0

f[t(x)]dt

M

t :MM

f :M C

Flow map

Observable

Compact manifold

Birkhoff: Limits exist a.e.

wrt any measure preserved byflow.

Dynamics are modeled by an flow map on a COMPACT manifold. Observables are scalar functions on that manifold. In most of the cases,we will focus on continuous observables. Tilde above a function will indicate that we are taking a time average ofthe function along the trajectory. Of course, there exist analogous formulations for maps and PDEs.

Since the definition of the time averages includes taking infinite-time limits, it is a valid question to ask whether these limits exist or not. A largeclass of dynamical systems preserve SOME measure. Birkhoffs theorem asserts that time averages converge almost anywhere with respectto any measure preserved. A common examples are area-preserving systems, e.g., advection in divergence-free flows and Hamiltoniansystems. While Birkhoffs result provides some comfort, it might still include cases where the exceptional set, i.e., the set where the time-averages are ill-defined, includes a big portion of the state space.However, there is a known structure that is a good example of non-existence, an attracting heteroclinic loop. As trajectories are attracted to it,they linger for longer and longer periods of time close to each fixed point in the loop, which prevents the time average from settling down. Mostother configurations in state space have well defined time averages and they can be computed by numerical simulations.


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Trajectories slow down more and more

as they pass equilibria, preventing

convergence.

Non-existence: Attracting heteroclinic loop

(x) = limT

1

T

T

0

f[t(x)]dt

M

t :MM

f :M C

Flow map

Observable

Compact manifold

Birkhoff: Limits exist a.e.

wrt any measure preserved byflow.

Dynamics are modeled by an flow map on a COMPACT manifold. Observables are scalar functions on that manifold. In most of the cases,we will focus on continuous observables. Tilde above a function will indicate that we are taking a time average ofthe function along the trajectory. Of course, there exist analogous formulations for maps and PDEs.

Since the definition of the time averages includes taking infinite-time limits, it is a valid question to ask whether these limits exist or not. A largeclass of dynamical systems preserve SOME measure. Birkhoffs theorem asserts that time averages converge almost anywhere with respectto any measure preserved. A common examples are area-preserving systems, e.g., advection in divergence-free flows and Hamiltoniansystems. While Birkhoffs result provides some comfort, it might still include cases where the exceptional set, i.e., the set where the time-averages are ill-defined, includes a big portion of the state space.However, there is a known structure that is a good example of non-existence, an attracting heteroclinic loop. As trajectories are attracted to it,they linger for longer and longer periods of time close to each fixed point in the loop, which prevents the time average from settling down. Mostother configurations in state space have well defined time averages and they can be computed by numerical simulations.


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Orbits in regular zones Orbit in mixing zone Sticky orbit

images: [Levnaji, Mezi, 2010]

Convergence of time-averages

Coping with slow convergence:

cycle expansions

comp. benefits for adaptive sampling

finite-time analysis

Difference in averages at 0.6105 and 1.0105 steps

Intermittency: trajectories get temporarily

trapped around marginally stable periodic

islands.

Standard map convergence error (log)

[Cvitanovi et al., Chaos Book]

Even when time averages are well defined, the phenomenon of trajectories lingering close to marginally stable for extended period of timeand then moving away can (theoretically) cause arbitrarily slow convergence rates.

This is demonstrated on an example of a Chirikov standard map a Hamiltonian iterated map with a state space that contains bothregular and mixing zones that come in contact in a fractal region. Theoretically, the convergence can be arbitrarily slow even when thetime averages of observables exist. The graphs on the top show log-log plots difference between finite time averages and infinite timeaverages. Convergence of time averages in regular zones is dominated by geometric slope of -1. In mixing zones, convergence isdominated by slope of -1/2. In zones where intermittency occurs, convergence is characterized by intermittent periods of large errors astrajectories get trapped around marginally stable objects. Such zones are known to occur in KAM systems when tori break down.

Predrag Cvitanovic championed an analytical method of cycle expansions to compute time-averages in intermittent zones. He proposedthat, instead of averaging an observables along chaotic trajectories, we could obtain the result by averaging along periodic orbits that areembedded within the chaotic zones. This is a successful strategy, but it involves determining where unstable periodic orbits lie, which is not

an easy task. Therefore cycle expansions do not provide a general-enough replacement for direct averaging.

Mixing zones and intermittent zones are sufficiently explored by a single trajectory, due to local ergodicity with respect to measuressupported on sets containing such zones. If our goal is to compute time averages on a sufficiently dense set of points in the state space, wemight use this fact in an adaptive sampling of state space. While waiting for the time average to converge, the trajectory explores a fat set instate space, which does not have to be sampled by additional initial conditions later, as all points on the trajectory will have the same time-average and can be identified with the initial condition of the trajectory. Therefore, even if we have to wait for a long period of time, if our goalis coverage of state space, we get additional computational benefit for waiting for time averages to converge. This benefit is especiallyimportant when the space is high-dimensional

Finally, we might just have to accept that finite-time averages are the best that we can do. I will revisit this topic briefly at the end of thispresentation, but understanding of how to harness finite-time averages is one of the burning issues that users of time averages will have toface.


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Trajectory-based

numerical consistency

Statistics-based

numerical consistency?

Computing averages for continuous systems

When dynamical system is an iterative map, we have no issues of computing it on a computer. When simulating ODEs and PDEs, we usediscretization schemes to approximate continuous trajectories. Most discretization schemes are validated by providing error estimatescompared to trajectories of continuous systems. The work of Paul Tupper, Jonathan Mattingly, Andrew Stuart, Xiaoming Wang and othersdemonstrates that trajectory-consistency is not always enough to ensure that time-averages will be correctly computed when systemsare high-dimensional, or have a special structure that needs to be conserved.*** slide transition ***A common observation is that a numerical scheme shouldnt introduce additional attractors into dynamics. Both works by Tupper andWang show that some commonly used schemes fail to satisfy this requirement. Tupper analyzed molecular dynamics (high-DOFHamiltonian) integrators that conserve energy. If they are not symplectic, they might stabilize periodic orbits that are otherwise unstable forcontinuous systems. If the continuous system was ergodic, such stable periodic islands might destroy the ergodicity of the numerical schemeresulting in incorrectly computed time-averages.

Wang showed that in PDE simulations of heat convection, time-averaged indicators, such as Nusselt number, can be incorrectly computed

if the scheme introduces attractors that arent there for continuous system. This occurs in some commonly used numerical schemes. Histheoretical result about precompactness of the set of attractors generated by numerical schemes is essentially the same as Tuppers, implyingthat the culprit for non-consistency of time averages is the possible set of attractors that numerical methods might introduce, even at small stepsizes.

In conclusion, while the work of Sigurgeirsson and Stuart provides hope that time-averages can be computed reliably for maps, we still need totake care in using integrators for continuous systems, paying special attention to systems with symplectic structure and infinite-dimensionalsystems.


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Trajectory-based

numerical consistency

Statistics-based

numerical consistency?

Computing averages for continuous systems

[P. Tupper, 2007] Numerical ergodicity of Hamiltonian systems

1. ... is ergodic, e.g., has no artificial limit cycles.

2. ... conserves energy.

3. ... conserves phase space volume (symplectic).

Time averages are correct if numerical scheme:

[X. Wang, 2010] Simulation of cts. infinite-dimensional dissipative systems

1. ... numerical global attractors for small

time steps form a pre-compact set.

2. ... numerical scheme is finite-time

uniformly convergent.

Time averages are correct if system is uniformly cts. and:Proof-of-concept

numerical scheme forcomputation of Nusselt

no. in a convection

model.

Symplectic schemes perform betterthan energy-conserving ones.

When dynamical system is an iterative map, we have no issues of computing it on a computer. When simulating ODEs and PDEs, we usediscretization schemes to approximate continuous trajectories. Most discretization schemes are validated by providing error estimatescompared to trajectories of continuous systems. The work of Paul Tupper, Jonathan Mattingly, Andrew Stuart, Xiaoming Wang and othersdemonstrates that trajectory-consistency is not always enough to ensure that time-averages will be correctly computed when systemsare high-dimensional, or have a special structure that needs to be conserved.*** slide transition ***A common observation is that a numerical scheme shouldnt introduce additional attractors into dynamics. Both works by Tupper andWang show that some commonly used schemes fail to satisfy this requirement. Tupper analyzed molecular dynamics (high-DOFHamiltonian) integrators that conserve energy. If they are not symplectic, they might stabilize periodic orbits that are otherwise unstable forcontinuous systems. If the continuous system was ergodic, such stable periodic islands might destroy the ergodicity of the numerical schemeresulting in incorrectly computed time-averages.

Wang showed that in PDE simulations of heat convection, time-averaged indicators, such as Nusselt number, can be incorrectly computed

if the scheme introduces attractors that arent there for continuous system. This occurs in some commonly used numerical schemes. Histheoretical result about precompactness of the set of attractors generated by numerical schemes is essentially the same as Tuppers, implyingthat the culprit for non-consistency of time averages is the possible set of attractors that numerical methods might introduce, even at small stepsizes.

In conclusion, while the work of Sigurgeirsson and Stuart provides hope that time-averages can be computed reliably for maps, we still need totake care in using integrators for continuous systems, paying special attention to systems with symplectic structure and infinite-dimensionalsystems.


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ft f Time-averages are invariants of motion.

Level-sets of time-averaged observables partition the

state space into invariant sets.

f(x) f(x)t(x)Trajectories An observableTime averaged

observable

M = T2

The first important property of time-averages is their invariance with respect to dynamics.It is common knowledge that invariants of motion,such as energy and momenta in some mechanical systems, play a strong role in understanding the dynamics of systems. However, as saidbefore, we can compute time-averages for most dynamical systems. What role do time-averages play there?

Dynamical system I will be using throughout this presentation is a Chirikov standard map. It is a Hamiltonian system evolving on a 2-torus,with a parameter epsilon that takes it from a completely regular discrete shear, to global chaos. Lets visualize time-averages of an observablethat alternates signs on a checkerboard pattern. On the rightmost image I am coloring the level sets of the time-averaged observable.

As a consequence of invariance, level-sets of time-averaged observables are invariant sets. Therefore, by looking at level-sets of an averagedobservable, we can immediately establish between which regions of state space there is no transport. Just from looking at therightmost figure, we can immediately see that there is no transport between regions colored light-blue and red.


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f(x) g(x)


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Intersections of invariant partitions are again invariant partitions.

Intersecting increases the resolution of information.

Repeating the procedure for linearly independent observables

limits to theergodic partition collection of ergodic sets.

[Levnaji, Mezi, 2010]

Transport?

Transport?

[Susuki, Mezi, 2009]

There are two questions that are left by looking at a single averaged observable:1. Is there a transport within any region of uniform color?2. Does choice of the observables matter?

Different observables will result in different level-sets of time-averaged observables. Each of these level-set partitions will be an invariantpartition. Single invariant partition does not answer whether there is transport inside of individual level sets.*** slide transition ***However, by intersecting invariant partitions generated from different observables we can increase the resolution of information and get finerinformation about the level sets. Continuing this process for linearly independent observables refines the invariant partition more and more,limiting to the ergodic partition. In it, every set is an ergodic set. Within any ergodic set, any two points are connected by transport. In thissense, ergodic partition extends the notion of foliation of the state space into invariant tori for completely integrable dynamical systems.

Therefore, combining information from different time-averages can provide very detailed information about global dynamical features of the

state space. Yoshihiko Susukis paper is about applications of ergodic partition visualization to analysis of continuous power systems.


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[f g . . . ](x)


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Intersections of invariant partitions are again invariant partitions.

Intersecting increases the resolution of information.

Repeating the procedure for linearly independent observables

limits to theergodic partition collection of ergodic sets.

[Levnaji, Mezi, 2010][Susuki, Mezi, 2009]

[f g](x)

There are two questions that are left by looking at a single averaged observable:1. Is there a transport within any region of uniform color?2. Does choice of the observables matter?

Different observables will result in different level-sets of time-averaged observables. Each of these level-set partitions will be an invariantpartition. Single invariant partition does not answer whether there is transport inside of individual level sets.*** slide transition ***However, by intersecting invariant partitions generated from different observables we can increase the resolution of information and get finerinformation about the level sets. Continuing this process for linearly independent observables refines the invariant partition more and more,limiting to the ergodic partition. In it, every set is an ergodic set. Within any ergodic set, any two points are connected by transport. In thissense, ergodic partition extends the notion of foliation of the state space into invariant tori for completely integrable dynamical systems.

Therefore, combining information from different time-averages can provide very detailed information about global dynamical features of the

state space. Yoshihiko Susukis paper is about applications of ergodic partition visualization to analysis of continuous power systems.


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State space portrait

f1(x), f2(x), . . . l

Ergodic quotient

x t(x)

x

f(x) =M

fdx

Choice oftopology on EQ defines coherent structures,

e.g., discrete metric topology corresponds to ergodic partition.

Ergodic Quotient (EQ) and empirical measures

Averaging

f from cts. basis

EQ is the set of weak-representatives of

empirical measures for all initial cond.

Empirical measures represent time

averaging through spatial integration.

t(x) M

Going from state space portrait to time-averages quotients the state space by ergodic partition we lose the ability to resolve features inside ofergodic sets. That is OK, because we only need one initial condition to understand any ergodic set since the dynamics restricted to anyergodic set is a system ergodic with respect to a measure supported on the ergodic set. What do we gain? We gain the ability to comparedynamics between different parts of the systems in a robust way.

Averaging is a linear, bounded functional on the space of continuous functions that can be represented by delta-like invariant measures,supported along trajectories. Empirical measures are weak-limits (in infinite time) of such invariant measures. Averaging a basis of continuousfunctions produces a weak representation of empirical measures. Ergodic quotient is then the set of weak-representatives of empiricalmeasures.

The coherent structures that we can identify in state space are directly related to topologies imposed on the ergodic quotient.What do I mean by that? Example: assume we have sourced a lot of initial conditions on the state space and computed their time averages. Adiscrete metric topology is the one where objects are at distance zero only to themselves, and infinity to everybody else. Taking an infinite

intersection of level sets of observables achieves the same thing - any element of the limiting partition contains initial conditions which had thesame averages for all functions. Therefore, the ergodic partition, mentioned on the previous slide, is equivalent to ergodic quotient viewedthrough the lens of discrete metric topology.

But we can consider other, continuous, topologies.


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Negative Sobolev space normscompare residence times of

trajectories. [Mathew, Mezi, 2010]

Easily computed on the ergodic

quotient low pass filter!

Intrinsic coordinates recovered by

diffusion modes on data.

Elliptic Duffing Gyres[MB, Mezi, 2009]

Coherent structures: similarity in residence times

f1(x), f2(x), . . .

l

Ergodic quotient


t(x) M

Endowing the empirical measure space with a negative-index Sobolev space norm topology (proposed by G. Mathew) results defines aparticular type of coherent features. In that topology, trajectories are deemed to be a part of the same coherent structure if their residencetimes sets in the state space are similar. Moreover, by selecting a proper index of the Sobolev space norm we can make sure that differencesacross larger spatial scales matter more than differences over small scales. These types of norms amount to using Fourier basis asobservables, and comparing the distance between trajectories is comparing the energy of the sequences with appropriate discounting of thehigher wave numbers.

In ergodic quotient, that lives in a high-dimensional space, certain state space features will be mapped to low-dimensional structures,even if the state space itself is fairly high-dimensiona.1. The first image depicts an elliptic well that maps to a string in the ergodic quotient. This signifies that ergodic sets in an elliptic zone(periodic orbits in two-dimensional state space) can be parametrized by a single conserved quantity.2. The second image shows a Duffing-type of behavior, or a double-well, where the inside libration orbits each map to separate strings whichend up joining when we cross the lip of wells into the revolution orbits.

3. Conversely, convection cells do not have revolution orbits that would join two strings together. Instead they have hyperbolic fixed points intheir corners that ensure that the time averages stay apart.

These low-dimensional structures inhabit an infinitely-dimensional sequence space. Using a single averaged observable, we are coloringthe state space according to projections of such features to a one-dimensional space. In my own research, I extract the intrinsiccoordinates of such structures and use them to visualize features of the state space. This provides us with low-dimensional embeddings ofergodic quotient that maximize the detail shown.*** slide transition ***If we know which trajectories mapped to certain features in EQ, we can assign colors to different features of EQ, and use these colors to colorthe state space portrait in order to visualize coherent structures.*** slide transition ***For example, take the Arnold-Beltrami-Childress flow: steady state flow characterized by prominent vortices.We visualized the vorticesextracting features in the ergodic quotient and using them to color trajectories. The major elliptic region was colored by a gradient, while side

vortices were picked out as disconnected components and each given its own color. However, we have done that by studying only results ofsimulation, which could be potentially done in a model-free setting, e.g., for experimental data, but that is consistent with physical properties,such as vorticity of the velocity field.


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Primary/secondary vortices of an ABC flow

Negative Sobolev space normscompare residence times of

trajectories. [Mathew, Mezi, 2010]

Easily computed on the ergodic

quotient low pass filter!

Intrinsic coordinates recovered by

diffusion modes on data.

Elliptic Duffing Gyres[MB, Mezi, 2009]

Coherent structures: similarity in residence times

f1(x), f2(x), . . .

l

Ergodic quotient


t(x) M

Endowing the empirical measure space with a negative-index Sobolev space norm topology (proposed by G. Mathew) results defines aparticular type of coherent features. In that topology, trajectories are deemed to be a part of the same coherent structure if their residencetimes sets in the state space are similar. Moreover, by selecting a proper index of the Sobolev space norm we can make sure that differencesacross larger spatial scales matter more than differences over small scales. These types of norms amount to using Fourier basis asobservables, and comparing the distance between trajectories is comparing the energy of the sequences with appropriate discounting of thehigher wave numbers.

In ergodic quotient, that lives in a high-dimensional space, certain state space features will be mapped to low-dimensional structures,even if the state space itself is fairly high-dimensiona.1. The first image depicts an elliptic well that maps to a string in the ergodic quotient. This signifies that ergodic sets in an elliptic zone(periodic orbits in two-dimensional state space) can be parametrized by a single conserved quantity.2. The second image shows a Duffing-type of behavior, or a double-well, where the inside libration orbits each map to separate strings whichend up joining when we cross the lip of wells into the revolution orbits.

3. Conversely, convection cells do not have revolution orbits that would join two strings together. Instead they have hyperbolic fixed points intheir corners that ensure that the time averages stay apart.

These low-dimensional structures inhabit an infinitely-dimensional sequence space. Using a single averaged observable, we are coloringthe state space according to projections of such features to a one-dimensional space. In my own research, I extract the intrinsiccoordinates of such structures and use them to visualize features of the state space. This provides us with low-dimensional embeddings ofergodic quotient that maximize the detail shown.*** slide transition ***If we know which trajectories mapped to certain features in EQ, we can assign colors to different features of EQ, and use these colors to colorthe state space portrait in order to visualize coherent structures.*** slide transition ***For example, take the Arnold-Beltrami-Childress flow: steady state flow characterized by prominent vortices.We visualized the vorticesextracting features in the ergodic quotient and using them to color trajectories. The major elliptic region was colored by a gradient, while side

vortices were picked out as disconnected components and each given its own color. However, we have done that by studying only results ofsimulation, which could be potentially done in a model-free setting, e.g., for experimental data, but that is consistent with physical properties,such as vorticity of the velocity field.


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Quality of mixing and samplingErgodicity defect

Replaces ergodic/non-ergodic

with continuous measure ofarea-ergodicity.

-

Fourier basis as observables.

Applied to design of mixing, optimal

sampling of a distribution.

[Scott et al., 2010] [Mathew, Mezi, 2010]

Haar wavelets as observables.

Continuous Dynamical Indicators

The application of analysis of topology of empirical measures goes beyond visualization of features. Sherry Scott and George Mathew haveboth applied this concept to convert binary descriptions ergodic/non-ergodic and mixing/not-mixing into continuous quantificationsof ergodicity and mixing. Continuous quantifications can then be used as design parameters, for example to drive the system to ergodicitywith respect to a particular measure.

The goal will often determine the appropriate choice of observables. Sherry Scott looked at a multiscale measure of ergodicity by takingHaars wavelets as observables, which allow precise tuning of scale and position of features we are interested in.

George Mathew, on the other hand, applied this concept to design of trajectories. In his work, he needed to plan trajectories for searchers,e.g., unmanned vehicles (represented as dots), such that they collaboratively sample a particular known prior distribution on the state space.By comparing weak representations of empirical measures of searchers and the weak representation of the target distribution, hedesigned optimal controllers that drove searchers to quickly sample the desired distributions. The same idea was applied to design ofmixing trajectories.


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!1 !0.5 0 0.5 1!1

!0.5

0

0.5

1

(a)

Re{j}

Im{j}

f f

Uf= f T

Uf= f

Uf = ei2 f


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Focus on dynamics ofon the attractor, systems with recurrent dynamics.

Koopman Operator linear, unitary on attractor

captures full nonlinear dynamics

U =

n

k=1

ei2kP

k

T+ Uc

Time-averaging projects

onto eigenspace at 1

(invariant functions).

Complex eigenspacesreveal periodicity of

dynamics.

Koopman spectrum of a jet in the crossflow. [Rowley et al., 2009]

[Mezi, Banaszuk, 2004.]

T :MM, T

So far I have talked about the relationship between state space features and the time-averaged observables. What is the relationship betweenobservable and its time-average? At every step of the averaging process, we are composing an observable with the evolution of thesystem. Operator that corresponds to that operation is the Koopman operator. It is a linear, and it is dual to the Perron-Frobenius operatorthat evolves densities of initial conditions. On the attractor, Koopman operator is unitary with respect to the L2 function space over anyinvariant measure on the attractor.

Let me stress that the operator itself is LINEAR in observables but that it captures full dynamics of a NONLINEAR dynamical system.The unitarity of the operator enables us to study its spectrum in an effort to understand its behavior. The image shows a numerically computedspectrum of a Koopman operator for a fluid flow.

Time-averaging projects the observable onto the eigenspace at 1 for the Koopan operator, which contains all invariant functions for dynamics.The complex eigenspaces reveal periodicity in dynamics of the system corresponding to the frequency of the eigenvalue associated with theeigenspace. We can access the projection to periodic eigenspaces by harmonic averages, which are introduced on the next slide.

Koopman operator is an instantaneous object; its definition contains only a finite time advancement of dynamics. The infinite-timeaveraging is just a TOOL through which we analyze the eigenspace at 1 the space of invariant functions. If we want to access othereigenspaces, we can modify simple time averages to form harmonic averages.


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f(x) = limN

1

N

N1

n=0

ei2nf[Tn(x)]

Uf= ei2 f

[Mezi, Banaszuk, 2004.]

Harmonic averages

f1 g1 h1 . . .

f2 g2 h2 . . .

f3 g3 h3 . . ....

.

.

....

. . .Discrete

Fourier

Transform

3

Harmonic

Quotientat

R. Mohr, I Mezi: Applications to analysis of computer networks (in progress)

Harmonic averages are formed from time averages by adding a complex term that rotates with frequency of the eigenvalue we are interestedin. Harmonic averages span the complex eigenspaces in the same fashion that time averaged spanning. Harmonic averaging projects anobservable to the complex eigenspace.

For any initial condition studies, there are two different parameters to choose: an observable and the frequency of averaging. While this mightseem daunting, fixing any of those parameters results in familiar objects:1. Fixing an observable, and computing all frequencies, results in a DFT of an observable.2. Fixing a frequency, and computing averages of all continuous observables results in harmonic quotients: analogues to ergodic quotientsdiscussed earlier, except oscillating at a certain frequency.Ryan Mohr, also at UC Santa Barbara, is working on applications of these methods to analysis of computer TCP networks.*** slide transition ***To demonstrate what harmonic averages look like, Ill turn again to the standard map. Looking at harmonic averages at period 5, we see thataverages over features that arent periodic with period 5 are equal to zero. The level sets at non-zero values correspond to periodic sets which

return to themseves after 5 time steps. As harmonic averages are complex functions, we can use both real and imaginary part to distinguishthe phase within these sets.


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Koopman Operator is an instantaneous object. Do we really need theinfinite-time formalism to analyze it?

finite-time Lyapunov exponents [Haller, 2010] and more...

finite-time turbulence quantifiers [Foias, Jolly, 2005]

time-averaged velocity [Poje, Haller, Mezi, 1999], [Mezi et al. 2010]

1

T

T

0

f[t(x)]dt = lim

1

0

f[(t mod T)(x)]dt

Finite time averaging is equivalent to periodizing the dynamics.

Finite-time averages

As mentioned, Koopman operator is instantaneous object, in case of iterated maps, it composes observable with a single step dynamics.This gives hope that analyzing it through infinite time averages is not necessary.

There have been several research efforts to convert infinite time indicators to finite time.

As a glimpse of one path to understand finite time averages, this identity shows that finite-time averages are equivalent to infinite timeaverages for modified flow map. The modification consists of periodizing dynamics of the original time map with the period equal to the timeover which finite-time average was computed.

While this might seem artificial and useless at first, this approach is readily used in signal processing, when Discrete Fourier Transform isused to analyze and filter measured signals that are not necessarily periodic. However Koopman operator frames this approach within thecontext of full nonlinear dynamics.


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Wealth of information is waiting to be managed.

Fully nonlinear data can be analyzed by linear tools.

Choosing observables: can we do better than the entire function

basis? Compressed sensing? Empirical measure reconstruction?

Finite-time averages are here to stay.

Infinite averaging is just a tool to access the instantaneous

Koopman op. What can finite-time averages tell us about it?

Physical data, nonrecurrent systems require finite-time averages.

Numerical methods should be designed with time-averaging in mind.

Trajectory-based consistency does not always imply consistency of

time-averages for Hamiltonian and infinite-dimensional systems.

Conclusions

These are the main take-aways from this talk:

1. Numerical integrators that were built around trajectories of ODEs might not be adequate for computation of statistics. On the other hand,since time averages show increased robustness to noise and numerical errors, as compared to trajectories, we might be able to buildaveraging-integrators that would be more efficient at computing time averages of trajectories, while ensuring the statistical consistency.

2. Moving the full nonlinear data to the linear world results in a wealth of information stored in averages of observables. At this point, unlesswe are physically motivated to choose a particular observable, we store the information in a large number of observables. A more selectiveapproach to choosing observable could enable us to reduce computational requirements of the method. It might be possible to use ideasfrom compressed sensing to do this. A step further would involve full empirical measure reconstruction, effectively viewing the ergodic quotientas input data to the inverse problem. Due to singular nature of a lot of empirical measures, with respect to Lebesgue, makes it difficult to usecurrent inverse problem solutions without encountering large reconstruction artifacts.

3. A requirement for viable methods is the ability to handle experimental data, and truly aperiodic data. It seems that to do this, we haveto abandon the infinity and embrace finite-time averages. Understanding how finite-time features relate to properties of the system is animportant step in acceptance of time-averaging as a standard tool in analysis and design.


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Main references

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M. Budišić: Time-Averages of Observables

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