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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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Modes librariesApproximation of bifurcation diagrams
Outline
1 Modes libraries
2 Approximation of bifurcation diagrams
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Modes librariesApproximation of bifurcation diagrams
Outline
1 Modes libraries
2 Approximation of bifurcation diagrams
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Modes librariesApproximation of bifurcation diagrams
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
Dr. Filippo Terragni Reduced Order Modeling Applications 4 / 2 0
M d lib i
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Modes librariesApproximation of bifurcation diagrams
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
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
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Modes libraries
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Modes librariesApproximation of bifurcation diagrams
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
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
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Modes libraries
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Modes librariesApproximation of bifurcation diagrams
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
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Modes libraries
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Modes librariesApproximation of bifurcation diagrams
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)
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Modes libraries
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Modes librariesApproximation of bifurcation diagrams
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.
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 2 0
Modes libraries
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Approximation of bifurcation diagrams
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.
Here, we set homogeneous Dirichlet boundary conditions
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 2 0
Modes libraries
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Approximation of bifurcation diagrams
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.
Here, we set homogeneous Dirichlet boundary conditions
Depending on the parameter values, different solutions appear
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 2 0
Modes librariesA i i f bif i di
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Approximation of bifurcation diagrams
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.
Here, we set homogeneous Dirichlet boundary conditions
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
Dr. Filippo Terragni Reduced Order Modeling Applications 6 / 2 0
Modes librariesA i ti f bif ti di
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Approximation of bifurcation diagrams
Example: dynamics in the CGLE
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Modes librariesApproximation of bifurcation diagrams
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Approximation of bifurcation diagrams
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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Modes librariesApproximation of bifurcation diagrams
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Approximation of bifurcation diagrams
Outline
1 Modes libraries
2 Approximation of bifurcation diagrams
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Modes librariesApproximation of bifurcation diagrams
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Approximation of bifurcation diagrams
Setting
Bifurcation phenomena are of paramount scientific interest andhave been the object of active research over the last decades.
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Modes librariesApproximation of bifurcation diagrams
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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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Modes librariesApproximation of bifurcation diagrams
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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)
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Modes librariesApproximation of bifurcation diagrams
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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)
equation (1) can be regarded as a nonlinear dynamical system
Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 20
Modes librariesApproximation of bifurcation diagrams
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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)
equation (1) can be regarded as a nonlinear dynamical system
is a real parameter associated with some physical property ofthe system
Dr. Filippo Terragni Reduced Order Modeling Applications 10 / 20
Modes librariesApproximation of bifurcation diagrams
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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)
equation (1) can be regarded as a nonlinear dynamical system
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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Modes librariesApproximation of bifurcation diagrams
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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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Modes librariesApproximation of bifurcation diagrams
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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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Modes librariesApproximation of bifurcation diagrams
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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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Modes librariesApproximation of bifurcation diagrams
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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)
2 a suitable way to go along the various values of (continuation)
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Modes librariesApproximation of bifurcation diagrams
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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)
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)
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Modes librariesApproximation of bifurcation diagrams
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
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Modes librariesApproximation of bifurcation diagrams
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
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Modes librariesApproximation of bifurcation diagrams
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 .
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Modes librariesApproximation of bifurcation diagrams
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)
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Modes librariesApproximation of bifurcation diagrams
Time integration via POD based ROMs
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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)
A standard numerical method may need huge computationalresources (slow)
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Modes librariesApproximation of bifurcation diagrams
Time integration via POD based ROMs
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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)
A standard numerical method may need huge computationalresources (slow)
If the given system is dissipative, we can successfully integrateone POD-based ROM for each value of (fast)
Dr. Filippo Terragni Reduced Order Modeling Applications 15 / 20
Modes librariesApproximation of bifurcation diagrams
Time integration via POD based ROMs
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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)
A standard numerical method may need huge computationalresources (slow)
If the given system is dissipative, we can successfully integrateone POD-based ROM for each value of (fast)
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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Modes librariesApproximation of bifurcation diagrams
Time integration via POD-based ROMs
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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)
A standard numerical method may need huge computationalresources (slow)
If the given system is dissipative, we can successfully integrateone POD-based ROM for each value of (fast)
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
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Modes librariesApproximation of bifurcation diagrams
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
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Modes librariesApproximation of bifurcation diagrams
Again remember the CGLE
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Again remember the CGLE
The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
Dr. Filippo Terragni Reduced Order Modeling Applications 17 / 20
Modes librariesApproximation of bifurcation diagrams
Again remember the CGLE
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Again remember the CGLE
The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
Symmetries are x 1 x , u u eic
Dr. Filippo Terragni Reduced Order Modeling Applications 17 / 20
Modes librariesApproximation of bifurcation diagrams
Again remember the CGLE
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Again remember the CGLE
The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
Symmetries are x 1 x , u u eic
(linear growth) is the bifurcation parameter
Dr. Filippo Terragni Reduced Order Modeling Applications 17 / 20
Modes librariesApproximation of bifurcation diagrams
Again remember the CGLE
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g
The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
Symmetries are x 1 x , u u eic
(linear growth) is the bifurcation parameter
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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Modes librariesApproximation of bifurcation diagrams
Again remember the CGLE
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g
The 1D complex Ginzburg-Landau equation (CGLE) is
tu = (1 + i)2
xxu + u (1 + i)|u|2u , with xu = 0 at x = 0, 1
where u is a complex variable and (,,) are real parameters.
Symmetries are x 1 x , u u eic
(linear growth) is the bifurcation parameter
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
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Modes librariesApproximation of bifurcation diagrams
Example (Terragni & Vega, Physica D 241, 2012)
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p ( )
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Modes librariesApproximation of bifurcation diagrams
Example (Terragni & Vega, Physica D 241, 2012)
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Modes librariesApproximation of bifurcation diagrams
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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