Context
1/18
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Model Reduction using Dynamic Mode Decomposition
Gilles TISSOT, Laurent CORDIER,
Nicolas BENARD, Bernd R. NOACK
Institut PPRIME, Poitiers
3
rd
GDR Symposium Flow Separation Control
November 7-8, 2013 Lille
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Context
Model based ow control:
w(t)
c(t)
z(t)
Flow
y(t)
Controller
r(t)
Need for a dynamical model :
Representative of the ow (at least input/ouput behavior).
Fast
2/18
⇒
Low-order model.
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Context
3/18
Reduced-Order Modeling:
Decompose the solution as
u (x , t ) =
Na
X
i =1
Φi (x )νi (t ).
Keep the most important modes as a basis.
Galerkin projection of the equations on the subspace
⇒
ROM.
Which basis?
Stability global modes :
linearized system.
Balanced truncation : optimal in input/output.
POD : snapshot based, optimal in energy content.
DMD: snapshot based method. Represents the system dynamics.
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Outline
4/18
1
DMD algorithm
2
Determine reduced basis
3
Reduced-order modelling
4
Results
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
DMD algorithm
Conclusion
(Schmid 2010 )
Let the matrix of the snapshots be
N = (v , . . . , v )
1
N
V1
N snapshots,
N ∈ RNx ×N
V1
Hypothesis 1
∃A ∈ RNx ×Nx ,
linear operator, such as
v k + 1 = Av k ,
∀k ∈ [1, N − 1]
Objective:
Determine the eigenvectors/values of A, without knowing A.
Hypothesis 2
{v 1 , . . . , v N −1 }
are linearly independent.
v N = c1 v 1 + · · · + cN −1 v N −1 + r
5/18
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
DMD algorithm
(Schmid 2010 )
Combining Hyp. 1 and Hyp. 2
N −1 = V N −1 C + r e T
AV1
N −1
1
Similarity transformation
with C the Companion matrix:
0
C
B
B
=B
@
...
...
0
0
c2
.
.
.
..
.
.
.
.
.
.
.
0
...
1
0
1
1
c1
c
c
C
C
C
A
i
with
i = λi y i
v n = V1N −1 c ⇒ c =
stant.
6/18
N −1
V1
+
1
.
vn
N−
then
Φi = V1N −1 y i
N−
can be found by pseudo-inverse of V1
1
Reconstruction using
Eigen-elements of A
If C y
Conclusion
AΦi
≈ λi Φi ,
dened up to a con-
Companion matrix properties:
vk =
NX
−1
i =1
Φi λki −1
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Determine reduced basis
How to perform a truncation?
vk =
NX
−1
i =1
Φi λki −1
N
−1
modes with linear dynamics behavior.
DMD modes selection:
Choice depends on the objective.
Frequency / Growth rate:
Mode amplitude:
kΦi k2
ωi =
arg(λi )
∆t
; σi =
log(|λi |)
∆t
Energy contribution:
Ei
=
1
T
Z T
t /∆t 2 dt = kΦ k2 e 2σi T − 1
Φi λi
i
2σi T
0
Hard because of non-orthogonality.
7/18
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Determine reduced basis
8/18
Optimized DMD: (Chen
et al. 2012 )
Recompute modes as:
vk =
p
X
i =1
Φ̂i λ̂ki −1 + r (tk )
Find the best
(Φ̂i , λ̂i )
with
such that
p
N −1
Γ=
N
X
k =1
kr (tk )k2
minimal.
Minimize the residual under the linear dynamics constraint
Computationally expensive.
⇒
Analytical gradient computation.
Other variants:
Low-rank and sparse DMD (Jovanovi¢
et al. 2012 )
Optimal Mode Decomposition (OMD) (Goulart
Chronos-Koopman analysis (Cammilleri
et al. 2012 )
et al. 2013 )
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Reduced-order modelling
9/18
Linear time propagator:
vk =
p
X
i =1
Φ̂i λ̂ki −1
Kinematic
linear recursion.
Galerkin projection:
Project Navier-Stokes equations onto the reduced basis
p
p X
p
X
X
dν
Qijk νj νk .
Gij j = Ci + Lij νj +
dt
j =1
j =1 k =1
j =1
p
X
Φ̂i
Imperfect
due
to
truncation errors.
Combine both by 4D-Var: (Papadakis 2007 )
η = νi (0) − νi0 and u = (Gij , Ci , Lij , Qijk )T minimizing
N p 2 σ
σu
1 XX
η
νi (tk ; η, u )) − λ̂ki −1 +
kηk2 +
ku − u b k2
J (η, u ) =
Find
2
(0, u b )
k =1 i =1
2
background state (Galerkin model) weighted by
2
ση , σu .
G. TISSOT, Institut PPRIME, 3
rd
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Data
10/18
Data: PIV Data of a cylinder wake Re
= 13000.
Classical DMD:
Nt
= 1000.
25 periods of vortex shedding.
i
p=7 modes selected with E criterion.
Optimized DMD:
Nt
= 256.
6 periods of vortex shedding.
p=7 Optimized DMD modes.
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
DMD Spectrum
DMD eigenvalues:
Frequencies/growth rates:
1
0.1
0.05
0
0
-0.05
σj
Im(λj )
0.5
-0.5
-0.25
-0.3
-1
-1
-0.5
0
Re(λj )
0.5
-0.1
-0.15
-0.2
1
Modes amplitude:
-20
-10
0
ωj
10
20
Energy contribution:
St
1 1:
Figure
= 0 .2
St
= 0.2
0.1
11/18
Ej
1
q
f
kΦ
jk
1
1
-20
-10
0
10
20
0.1
-20
-10
0
10 rd
G. TISSOT, Institut PPRIME, 3
20
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Modes
12/18
u
u
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
4
3
Modes with higher
2
y
Classical DMD
1
energy contribution.
0
-3
-2
-1
x
0
Re(Φ27 )
3
2
1
0
1
-3
u
2
as initial condition.
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
1 1:
Figure
1
0
-3
-2
-1
x
-1
x
0
1
0
1
Re(Φ̂2 )
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1 1:
Figure
4
3
y
3
Selected DMD modes
y
Optimized DMD
-2
u
Re(Φ̂1 )
4
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
4
y
Re(Φ1 )
2
1
0
-3
-2
-1
x
0
rd
G. TISSOT, Institut PPRIME, 3
1
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Modes
12/18
u
u
Re(Φ79 )
Re(Φ29 )
0.6
4
Classical DMD
3
0.2
3
Modes with higher
2
0
2
1
-0.2
1
0
-0.4
0
energy contribution.
y
0.4
y
4
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.6
-3
-2
-1
x
0
1
-3
u
1 1:
Figure
0.04
4
0.02
3
Selected DMD modes
2
0
2
y
3
y
0
1
Re(Φ̂6 )
0.06
Optimized DMD
as initial condition.
-1
x
u
Re(Φ̂4 )
4
-2
1
-0.02
1
0
-0.04
0
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
1 1:
Figure
-0.06
-3
-2
-1
x
0
1
-3
-2
-1
x
0
1
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
13/18
snapshot reconstruction
Original snapshots:
Classical DMD:
2
2
1.5
4
1.5
4
1.5
3
1
3
1
3
1
2
0.5
2
0.5
2
0.5
1
0
1
0
0
-0.5
0
-0.5
-3
-2
-1
x
0
-3
-2
-1
x
0
1 1:
Figure
-1
0
-2
-1
x
0
1
L2 -norm error:
-1
x
0
1
1
1 1:
Figure
-0.5
1
-1
0
0.5
1 1:
Figure
3
0
2
-1.5
-3
-2
4
y
y
-0.5
-1
-3
0.5
3
0
1
0
1
4
0.5
2
0
-0.5
1
1
3
1
-1
1
4
y
4
-1
v
Optimized DMD:
2
y
u
y
th
y
5
0
2
-0.5
1
-1
0
-1.5
-3
-2
-1
x
0
45.6%
1
-1.5
-3
-2
-1
x
15.6%
0
1
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Reduced-Order Model
4D-Var solved in Optimized DMD space:
Observations:
Linear time propagator
Nt
= 256
λ̂ik −1 .
snapshots.
Background solution: taken as initial control parameters
η0 = 0
0
Gij0 , Ci0 , L0
ij , Qijk found by Galerkin projection.
Covariance matrices:
ση = σu = 1.
14/18
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Reduced-Order Model
1.5
1.2
1
1
0.5
0.8 Assimilation
0.6
Forecast
Projection
λ̂k−1
j
4D-Var
0.2
0
20
40
60
t
80
100
-1
-1.5
120
2
40
60
t
80
100
120
0
-0.5
80
100
120
0.5
0
-0.5
-1
-1
-1.5
-1.5
0
20
40
60
t
1 1:
Figure
1 1:
Figure
1
0.5
Re(ν6 )
Re(ν4 )
20
1.5
1 1:
Figure
1
-2
0
2
1.5
15/18
0
-0.5
0.4
0
Re(ν3 )
Re(ν1 )
1.4
80
100
120
-2
0
20
40
60
t
1 1:
Figure
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Results
Reduced-Order Model
16/18
erel
Relative reconstruction error: erel
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
20
40
=
ku [1,...,p] (x , t ) − u (x , t )k
ku (x , t )k
60
t
Projection
λ̂k−1
4D-Var
80
100
120
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
Conclusion
17/18
DMD extracts dynamically relevant features.
Selection of modes dicult by classical DMD.
Optimized DMD leads to a good reduced basis.
Linear time propagator and Galerkin model combined by
4D-Var.
Tested on PIV measurements of turbulent cylinder wake.
Promising technique in model reduction for ow control.
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium
Context
Outline
DMD algorithm
Determine reduced basis
Reduced-order modelling
Results
Conclusion
QUESTIONS???
18/18
rd
G. TISSOT, Institut PPRIME, 3
GDR Symposium