Diffusion Maps as Invariant Functions of Dynamical Systems

7/28/2019 Diffusion Maps as Invariant Functions of Dynamical Systems

1/11

Mezi Research Group

SIAM Conference on Applications of Dynamical Systems

Snowbird, UT, May 2013

Marko Budii

Igor Mezi

DIFFUSION MAPS AS

INVARIANT FUNCTIONS OF

DYNAMICAL SYSTEMS

Funding:

ONR MURI

Ocean 3D+1


2/11

Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013 2

Invariant functions define barriers to dynamical transport

and simplify dynamics.

Classical mechanics:

Integrals of motion foliate the state space.

We can reduce system to each of the leaves of the

foliation and reduce order of the system.

Fluid dynamics:

Boundaries of (sub-)level sets of

invariant functions arebarriers to material transport.

Invariant functions: functions on the state space whose values are constant along trajectories.


3/11

xp = u(xp), xp(0) = p

(p, t) 7! xp(t)

f(p) := limT!1

1

T

ZT0

f(xp())d

: !

p! g(xp(t)) p! g p

p! f(xp(t)) p! f(p)

f(xp(t)) f(p)

Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013

Every invariant function is an

average of a scalar function along trajectories.

3

xp(t)p

PTrajectory:

Time-average:

Vector field Initial state

Time averages are invariant functions:

Different initial functions may not yield independent

time-averages.

Level sets of time-averages are invariant sets.


4/11

R


Ergodic Quotient is a static representation of

dynamical invariants in the space of averaged functions.

4

p 7!

264f1(p)

f2(p)...

375

Functions selected from a basis

xp = u(t, xp)

Entire trajectories are represented

by single points.

Axes are averagedfunctions.

f2(p)

f1(p)

f(p) := limT!1

1

T

ZT0

f(xp())d

Ergodic Quotient (EQ)

maps initial conditions to

infinite vectors

of time averages.

Ergodic Quotient

Any function with the domain in EQ

is again an invariant function.


5/11


Sobolev space topology on Ergodic Quotient

captures families of orbits as continuous segments.

5

State space

portrait

Reeb Graphs

(ideal EQ structure)

Fixed point onseparatrices preventsegments fromconnecting.

d(p1, p2)2 =

X

k2Zd

fk(p1) fk(p2)2

(1 + |k|2)s Acts as a low-pass filter:

de-emphasizes

small scale differences.

Wavevectors

Ergodic Quotientf2(p)

f1(p)

Desired topology:

Time-averages of Fourier harmonics are

spatial Fourier coefficients of

averaging distributions, supported on orbits.

P

fk(x) = ex

Required choice of observables:

Sobolev space norm does the trick:


6/11


Diffusion coordinates reduce ambient dimension,

while preserving intrinsic geometry of the EQ.

6

Ergodic Quotient

f2(p)

f1(p)

The ambient dimension is the number of

functions averaged (axes).

It is ideally large, to achieve

a high resolution.

It is known a priori.

The intrinsic dimension depends on the

complexity of dynamics, but can be

very small, even zero (ergodicity).

It is unknown a priori.

To disentangle the wire (EQ),sort the points by time it takes them to heat up.

Heat sources are placed onan entangled wire (EQ).Diffusion Maps:

[Coifman, Lafon,

ACHA, 2006]

Diffusion coordinates

pulled back onto the state space

are invariant functions,

sensitive to similarity in dynamics.

p

!

264

f1(p)

f2(p)...

375!

264

1((p))2((p))

..

.

375

Averaging DiffusionMaps

Practical matters:Bubacarr Bah,MSc Thesis,Oxford, 2008.
http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/http://eprints.maths.ox.ac.uk/740/


7/11

1


Coloring the state space by values of dominant

diffusion coordinates reveals large scale features.

7

21 3

Diffusion pseudocolors identify invariant sets

Different colors indicate there is no

material transport between regions.

Coordinates of higher order

distinguish between finer features.

State space portraitEQ in Diffusion Coordinates Coloring of the state space

1

2

Averaging

+Diffusion Maps


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R

z

0

2

0.2

0.3

0.1

0.1

Cluster 3

Cluster 2

Cluster 1

Cluster 0

2

1

1 0.5 0 0.5 11

0.8

0.6

0.4

0.2

0

0.2

0.4

0.

6

0.8

1


Clustering in diffusion coord. identifies large-scale invariant sets.

8

0.00

0.50

1.00

0

2

4

Cluster

x

z

y

3

1

0.00

0.50

1.00 0.00

0.50

1.00

6

5

Index

6

Index 4 Index 20.15

0.1

0.05

00.

05

0.1

0.08

0.04

0

0.

04

0.080.03

0.01

0.01

0.03

0.05

Vortices identified as clusters ofpoints in diff. coord. space.

z

R

Secondary vortex appears asa bifurcation in the primary segment

Steady ABC flow:

Time-periodic Hills vortex ring:

Time-periodic perturbation splits

the primary vertex core.

State space

State space

Ergodic Quotient

Ergodic Quotient


9/11Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013

Future: method resolves structures in finite-time, unsteady flows.

9by Drew Poje (CUNY)

HYCOM simulation of 2D Gulf Stream (along surface of equal potential density of water)

Averaging Time = 20 days

Averaging Time = 10 days

1 2 3

1 2 3

12

3

1Color:

12

3

1Color:


10/11Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013

To achieve automatic analysis,

there are several issues that need resolving:

10

Tuning the diffusion bandwidth.

Incorrect bandwidth can artificially

connect or disconnect regions.

Tuning is heuristic: Lafon (min. distance),

Lee (neighborhood size), Singer (linear sensitivity).

Detection of intrinsic dimensionand independent coordinates.

Cluster 3

Cluster 2

Cluster 1

Cluster 0

2

1

1 0.5 0 0.5 11

0.8

0.6

0.4

0.

2

0

0.2

0.4

0.6

0.8

1

Due to highly

non-convex structures,

K-means clustering

is not a good choice.

Alternative segmenting

would be preferable.

Diffusion eigenvalues depend on the tuning

Where is the

spectral gap?

k = 5

k = 4

k = 3

k = 2

0.3 0.35 0.4 0.45 0.51

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

k

Dependence of diffusion coordinates

Post-processing:

K-means, proximity graphs, etc.

What is

relevant for

dynamical

systems?

Ergodic quotient for Hills ring vortex


11/11Marko Budii: Diffusion Maps as Invariant Functions of Dynamical SystemsMay 23, 2013 11

Take away points:

Budii and Mezi, Geometry of the ergodic quotient reveals coherent structures in flows,Physica D, 241, (2012).

Budii,Ergodic Quotients in Analysis of Dynamical Systems, PhD Thesis,UC Santa Barbara, (2012).

Budii, Mohr, and Mezi,Applied Koopmanism, Chaos 22, (2012).

Ergodic Quotient (EQ) is a geometric representation

of the space of invariant functions.

Time-averaged functions provide computable axes for EQ.

Diffusion Maps provide efficient axes for EQ,

while preserving intrinsic geometry.

EQ was successfully used to extract invariant sets

in the state space.

Even in finite-time, method recovers significant structures.

Documents

Diffusion Maps as Invariant Functions of Dynamical Systems