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Modes librariesApproximation of bifurcation diagrams

Reduced Order Modeling Applications

Applications to bifurcation problems

ECMI Summer School 2013

Leganes, July 18 Dr. Filippo Terragni

Dr. Filippo Terragni Reduced Order Modeling Applications 1 / 2 0
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Outline

1 Modes libraries

2 Approximation of bifurcation diagrams

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Outline

1 Modes libraries


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An empirical property of POD modes

The major computational cost in a POD-based method is

associated with the snapshots calculationDecreasing the number of necessary snapshots is equivalent toreducing the CPU effort


M d lib i
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Consider a problem where some parameters are present.

The POD basis depends weakly on the problem parameters

POD modes computed for some values of the parameters maybe good to describe the solutions for other, different values also


Modes libraries
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Consider a problem where some parameters are present.

The POD basis depends weakly on the problem parameters

POD modes computed for some values of the parameters maybe good to describe the solutions for other, different values also

This is not that surprising: Fourier modes generically work We could store POD modes coming from various simulations

and create useful databases (libraries) of modes


Modes libraries
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Modes libraries

A modes library is simply a set of POD modes

This can be computed in various ways

applying POD to a set of generic functions (e.g., Fourier modesor other orthogonal polynomials)

storing the final POD basis used in a generic run of the adaptive

ROM described in the previous session

mixing two (or more) sets of different modes (after weighting)and finally applying POD


Modes libraries
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Modes libraries

A modes library is simply a set of POD modes

This can be computed in various ways

applying POD to a set of generic functions (e.g., Fourier modesor other orthogonal polynomials)

storing the final POD basis used in a generic run of the adaptive

ROM described in the previous session

mixing two (or more) sets of different modes (after weighting)and finally applying POD

The obtained modes, suitably weighted, can be used to

1 construct a ROM to approximate the solutions of the problem

2 start up the adaptive method described in the previous session(as old modes)

for some, generic parameter values (in a certain range)


Modes libraries
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Remember the CGLE

The 1D complex Ginzburg-Landau equation (CGLE) is

tu = (1 + i)2xxu + u (1 + i)|u|2u , with u = 0 at x = 0, 1

where u is a complex variable and (,,) are real parameters.


Modes libraries
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Approximation of bifurcation diagrams

Remember the CGLE




Here, we set homogeneous Dirichlet boundary conditions


Modes libraries
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Remember the CGLE





Depending on the parameter values, different solutions appear


Modes librariesA i i f bif i di
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Remember the CGLE





Depending on the parameter values, different solutions appear

For < 1 and larger than a critical value, the system mayexhibit complex behaviors (e.g., chaotic dynamics) at large time


Modes librariesA i ti f bif ti di
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Example: dynamics in the CGLE


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Example: dynamics in the CGLE

The number of necessary snapshots is drastically reduced (CPU effort also)

Mixing different modes is satisfactory (more directions are spanned)

Modes from simple dynamics provide good results (even in complex cases)


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Outline

1 Modes libraries



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Setting

Bifurcation phenomena are of paramount scientific interest andhave been the object of active research over the last decades.


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pp b g s

Setting

Bifurcation phenomena are of paramount scientific interest andhave been the object of active research over the last decades.

For instance, nonlinearity promotes instabilities bifurcations thatcan be either dangerous (e.g., flutter) or beneficial (e.g., promotingfavorable transversal convection in microcooling devices)

Computation can be fairly heavy POD-based ROMs may be useful


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pp g

Setting

Consider the general (parabolic) problem

tq = Lq + f(q, t , ) (1)

with suitable boundary and initial conditions, where q is defined ona bounded domain, L is a linear operator, f is a nonlinear operator.

some additional, convenient assumptions can be added to justify what will

be introduced later (omitted)

Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 20

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Setting


tq = Lq + f(q, t , ) (1)




equation (1) can be regarded as a nonlinear dynamical system


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Setting


tq = Lq + f(q, t , ) (1)





is a real parameter associated with some physical property ofthe system


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Setting


tq = Lq + f(q, t , ) (1)





is a real parameter associated with some physical property ofthe system

plays the role of a bifurcation parameter, namely changingits value will alter the topological features of the solutions of (1)


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What is a bifurcation?

A bifurcation is a qualitative change produced in the phase portraitof the system for a certain value of the bifurcation parameter

( the phase portrait is defined as the set of all orbits curves parametrized by t

associated with the solutions of (1) for all possible initial conditions, which yields

a global qualitative picture of the dynamics )


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What is a bifurcation?

A bifurcation is a qualitative change produced in the phase portraitof the system for a certain value of the bifurcation parameter

( the phase portrait is defined as the set of all orbits curves parametrized by t

associated with the solutions of (1) for all possible initial conditions, which yields

a global qualitative picture of the dynamics )

We would like to study these qualitative changes in the dynamics(e.g., steady, periodic, quasi-periodic, or chaotic) in the range 0 < 1

A bifurcation diagram shows the large-time values of a quantityassociated with the solutions as a function of


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Bifurcation diagrams

Constructing a bifurcation diagram requiresthree ingredients

1 a convenient quantity to plot (in order to clearly appreciatechanges in the solutions)


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2 a suitable way to go along the various values of (continuation)


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2 a suitable way to go along the various values of (continuation)

3 an efficient method to time integrate the problem for each valueof in the range 0 < 1 (in order to approach stable states)



Th P i
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The Poincare map

For the system (1), consider the Poincare hypersurface

H(q) := q, Lq + f(q, t , ) 12

ddt

q2 = 0 ,

which contains

all steady solutions

at least two points of each periodic solution

at least two points of any time oscillation of q for other morecomplex solutions



Th P i
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The Poincare map

For the system (1), consider the Poincare hypersurface

H(q) := q, Lq + f(q, t , ) 12

ddt

q2 = 0 ,

which contains

all steady solutions

at least two points of each periodic solution at least two points of any time oscillation of q for other morecomplex solutions

Thus, intersections of the solutions with H are associated with localmaxima (and minima) of the Poincare map t q2 .

We can plot q at those time instants in 0 tA < t tBwhere the Poincare map exhibits local maxima a

a The time interval 0 < t tA is disregarded since it contains the transientbehaviors in which the solutions approach the asymptotic states



C ti ti
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Continuation

The bifurcation parameter span 0 < 1 is discretized by .

At the first value of , we choose a generic (nonsymmetric) initialcondition at t = 0 and integrate the problem in 0 < t tB .

For subsequent (increasing) values of , the initial condition at t = 0

is the final state (at t = tB) for the previous value of .



Ti i t ti i POD b d ROM
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Time integration via POD-based ROMs

Constructing a bifurcation diagram requires solving the problem

many times (for each ) in a large time span (to discard transients)



Time integration via POD based ROMs
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A standard numerical method may need huge computationalresources (slow)



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If the given system is dissipative, we can successfully integrateone POD-based ROM for each value of (fast)



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Since the POD modes depend weakly on the problem parameters,we can successfully integrate only one POD-based ROM for allvalues of (very fast)



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Since the POD modes depend weakly on the problem parameters,we can successfully integrate only one POD-based ROM for allvalues of (very fast)

let us apply the last procedure



A simple method
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A simple method

Terragni & Vega, Physica D 241 (2012)

1 Choose a generic initial condition, a non-small time span, and a

fixed value of ; then, run a time dependent numerical solver tocalculate the associated orbit q(t, )

2 Select N time instants and apply POD to the set of snapshotsq(t1, ), . . . , q(tN, )

3 Construct the GS (depending on ) based on the n mostenergetic POD modes and compute its bifurcation diagram

4 Validate results repeating the procedure with more POD modes



Again remember the CGLE
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tu = (1 + i)2

xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1




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tu = (1 + i)2



Symmetries are x 1 x , u u eic



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tu = (1 + i)2




(linear growth) is the bifurcation parameter



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g


tu = (1 + i)2





Thanks to the Neumann boundary conditions, we may havesimple solutions of the form u(x, t) = eit u0 , where u0 can beconstant, dependent on x only, or time periodic also



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g


tu = (1 + i)2





Thanks to the Neumann boundary conditions, we may havesimple solutions of the form u(x, t) = eit u0 , where u0 can beconstant, dependent on x only, or time periodic also

For < 1 and larger than a critical value, the system mayexhibit complex behaviors (e.g., chaotic dynamics) at large time



Example (Terragni & Vega, Physica D 241, 2012)
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p ( )



Example (Terragni & Vega, Physica D 241, 2012)
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Some references
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1 M. L. Rapun, F. Terragni & J. M. VegaMixing snapshots and fast time integration of PDEs

IV International Conference on Computational Methods for CoupledProblems in Science and Engineering, COUPLED PROBLEMS 2011,article 246 (2011), pp. 112

2 F. Terragni & J. M. VegaOn the use of POD-based ROMs to analyze bifurcations in some dissipative

systems

Physica D 241 (2012), pp. 13931405

3 E. L. Allgower & K. GeorgIntroduction to Numerical Continuation Methods

SIAM Classics in Applied Mathematics 45, 2003

4 J. D. CrawfordIntroduction to bifurcation theory

Rev. Mod. Phys. 63 (1991), pp. 9911037

5 I. S. Aranson & L. KramerThe world of the complex Ginzburg-Landau equation

Rev. Mod. Phys. 74 (2002), pp. 99143

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5 Extensions