Institut PPRIME, Poitiers - univ-orleans.fr...Introduction What is 4D-Vr?a rmalismFo ResultsConclusion 4D-Variational Data Assimilation using POD Reduced-Order Model Gilles TISSOT

Introduction What is 4D-Var? Formalism Results Conclusion

4D-Variational Data Assimilation using POD Reduced-OrderModel

Gilles TISSOT, Laurent CORDIER,Nicolas BENARD, Bernd R. NOACK

Institut PPRIME, Poitiers

GDR Contrôle Des Décollements

Orsay, 15-16 novembre 2012

1/21 G. TISSOT, Institut Pprime, GDR Contrôle Des Décollements


Context

Model based �ow control:

Flow

Controller

z(t)

y(t)

c(t)

w(t)

r(t)

Need for a dynamical model :

Representative of the �ow (at least input/ouput behavior).

Fast ⇒ Low order model.



Context

Model based �ow control:

Huge number of DoF ⇒ POD Reduced-Order Model

POD ROM coming from Galerkin projection imperfect

⇒ Data assimilation to improve the POD ROM

Principle: To combine di�erent sources of information to estimateat best (in an optimal way) the state of the system.

Imperfect observations (incomplete, noised)

Imperfect model (simpli�ed)

A priori knowledge of the state of the system

Approaches:

Stochastic method (ex: Kalman Filter)

Variational method: Minimisation of a cost functional J



Outline

1 What is 4D-Var?

2 FormalismStrong dynamical constraintWeak dynamical constraint

3 ResultsDNS DataDNS Data - Twin experimentsPIV Data



Data AssimilationWhat is Data Assimilation?

We have some observations Y.

X

t

Y




We know a model M. u model parameters

∂X (t)

∂t+ M(X (t), u) = 0 ; X (0) = X0

X

t

Y




We know a background solution (X b0, ub). (η, u) control parameters.

∂X (t)

∂t+ M(X (t), u) = 0 ; X (0) = X b

0 + η

X

t

Y

Background

solution

Xb

0




We know a background solution (X b0, ub). (η, u) control parameters.

∂X (t)

∂t+ M(X (t), u) = 0 ; X (0) = X b

0 + η

X

t

Y

X (t)=X(t ; X , u )bBackground

solution

Xb

0b




4D-Var: search for (ηa, ua) = argmin(J(η, u))

J(η, u) =1

2

∫ T

0

‖Y−H(X (t; η, u))‖2R−1dt+1

2‖η‖2B−1+

1

2‖u−ub‖2C−1

X

t

Y

X (t)=X(t ; X , u )

X0

bBackground

solution

Analysed

solution

Xb

0b

a




4D-Var: search for (ηa, ua) = argmin(J(η, u))

J(η, u) =1

2

∫ T

0

‖Y−H(X (t; η, u))‖2R−1dt+1

2‖η‖2B−1+

1

2‖u−ub‖2C−1

X

t

Y

X (t)=X(t ; X , u )

a

X (t)=X(t ; X , u )

X0

bBackground

solution

Analysed

solution

Xb

0b

a

a



Context

Objectives of Data Assimilation: Estimate



Context

Objectives of Data Assimilation: Forecast



Formalism

Strong dynamical constraintVariational Data Assimilation



FormalismStrong dynamical constraint (Papadakis, 2007)

Model:

∂X (t)

∂t+ M(X (t), u) = 0

X (0) = X0 + η

Cost functional:

J(η, u) =1

2

∫ T

0

‖Y−H(X (t; η, u))‖2R−1dt+1

2‖η‖2B−1+

1

2‖u−ub‖2C−1

H observation operator

R, B and C correlation matrix which represent how we trust inthe observations and the background solutions.



FormalismStrong dynamical constraint Optimality system

Find ∇J for minimisation: λ(t) Lagrange multiplier

Adjoint equation

−∂λ∂t

(t) +

(∂M∂X

)∗λ(t) =

(∂H∂X

)∗R−1 (H(X (t))− Y)

λ(T ) = 0

Optimality condition

∂J

∂η= λ(0) + B−1η

∂J

∂u= −

∫ T

0

(∂M∂u

)∗λ(t)dt + C−1(u − ub)



Formalism

Weak dynamical constraintVariational Data Assimilation



FormalismWeak dynamical constraint (Papadakis, 2007)

Model:

∂X (t)

∂t+ M(X (t)) = w(t) ; X (0) = X0 + η

Cost functional:

J(η,w(t)) =1

2

∫ T

0

‖Y−H(X (t))‖2R−1dt+1

2‖η‖2B−1+

1

2

∫ T

0

‖w(t)‖2W−1dt

Adjoint equation: Unchanged

Optimality condition:

∂J

∂η= λ(0) + B−1 ;

∂J

∂w(t)= λ(t) +W−1w(t)



ResultsDNS Data Strong constraint

DNS data from a cylinder wake (DNS code Icare - IMFT)

Re = 200

Nt = 200

2 periods of vortex shedding

6 POD modes used and represent 99% of the �ow energy

POD Decomposition:

v(x , t) = vm(x) +Nt∑i=1

aPi (t)Φi (x)




Observations Y: aPi (t) (D'Adamo, 2007)Model Mgal :

daRidt

(t) = Ci +

Ngal∑j=1

LijaRj (t) +

Ngal∑j=1

Ngal∑k=1

QijkaRj (t)aRk (t)

aR(0) = aP(0) + η

Background solution: taken as initial control parameters

η0 = 0

C 0

i = L0ij = Q0

ijk = 0

Covariance matrices:

R−1 = I, B−1=C−1 = σ2I, σ = 10−3




-2

0

2

-3 -2 -1 0 1 2 3 4

a2

a1

1

-0.6

-0.3

0

0.3

0.6

-3 -2 -1 0 1 2 3 4

a3

a1

1

-0.6

-0.3

0

0.3

0.6

-0.4 0 0.4

a4

a3

1

-1

-0.5

0

0.5

-0.6 -0.3 0 0.3 0.6

a6

a5

POD

4D-Var

1

Figure: 4D-Var for the DNS dataset.



ResultsDNS Data - Twin experiments Strong constraint

-2

0

2

-3 -2 -1 0 1 2 3 4

a2

a1

1

-0.6

-0.3

0

0.3

0.6

-3 -2 -1 0 1 2 3 4

a3

a1

1

-0.6

-0.3

0

0.3

0.6

-0.4 0 0.4

a4

a3

1

-1

-0.5

0

0.5

-0.6 -0.3 0 0.3 0.6

a6

a5

POD

4D-Var

1

Figure: 4D-Var twin experiments for the DNS dataset.



ResultsDNS Data - Twin experiments Strong constraint

e(t) =

√∑Ngal

i=1

(ai (t)− aTruei (t)

)2/ 1

T

∫ T0

∑Ngal

i=1aTruei (t)2dt

0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step

assimilation forecast< >< >

0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


perfect observations

noised observations

analysed solution

Figure 1:1

Figure: Time evolutions of errors. Comparison between the case of

perfect observations and the case of twin experiments.



ResultsPIV Data Strong and Weak constraint

PIV Data of a cylinder wake (N. Benard, Pprime)

Re = 40000

Nt = 128

10 periods of vortex shedding

16 POD modes used and represent 31% of the �ow energy

Background solution: taken as initial control parameters

Strong 4D-Var : Weak 4D-Var :

η0 = 0 Solution of the strong 4D-Var.C 0

i , L0

ij , Q0

ijk found by

Galerkin projection.

Covariance matrices:

Strong : σ = 1 ; Weak : σ = 10−3




-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

a2

a1

1

-0.4

-0.2

0

0.2

0.4

0.6

-1 -0.5 0 0.5 1

a3

a1

1

-1-0.8-0.6-0.4-0.2

00.20.40.60.81

1.2

0 20 40 60 80 100 120

a1

time step

1

-0.4-0.2

00.20.40.60.81

1.2

0 20 40 60 80 100 120

a3

time step

PODst 4D-Varw 4D-Var

1

Figure: 4D-Var for the PIV dataset. Strong and weak constraint.18/21 G. TISSOT, Institut Pprime, GDR Contrôle Des Décollements



Weak 4D-Var can relax the dynamical constraints and �t better theobservations.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80 100 120

e(t)

time step

4D-Var Strong

4D-Var Weak

Figure 1:1

Figure: Time evolution of the errors between observations and estimation.

But no forecast is possible ! (w(t) unde�ned)19/21 G. TISSOT, Institut Pprime, GDR Contrôle Des Décollements


Conclusion

Conclusion

Method coming from meteorology/oceanography, applied on�ow control.

4D-Var is a well established tool and the results depend on thequality of the inputs (model, observations, background).

The belief we have in each information has to be consistentwith its imperfections.

Summary

Two methods of 4D-Var were presented.

POD ROM improved by 4D-Var.

Tested by twin experiments, then on PIV data.



QUESTIONS???


FormalismPerfect model

Descent algorithm, we need ∇J:

Finite di�erences

X

t

X(t ; X +dŋ , u )

X0

0

X(t ; X , u+du )0

( ∂J∂η,δη) = J(η+εδη,u)−J(η,u)

ε

( ∂J∂u,δu) = J(η,u+εδu)−J(η,u)

ε

N + 1 temporal integrations

Adjoint method

X,

t

X(t ; X , u )X(0)

0

λ(t ; X(t),u)

λ

λ(0) λ(T)=0

∇J is a function of λ(t)

2 temporal integrations


ResultsDNS Data - Twin experiments Perfect Model

Twin experiments

De�ne a true state:Analysed solution of the previous 4D-Var

Create arti�cial observations:Noising the original observations by adding X/σi whereX ' N (0, σ2t ), σt = 0.2 and σi = 2[ i+1

2].

Compare the true state and the new analysed solution:

e(t) =

√∑Ngal

i=1

(ai (t)− aTruei (t)

)2/ 1

T

∫ T0

∑Ngal

i=1aTruei (t)2dt


ResultsDNS Data - Initial condition sensitivity Perfect Model

η is small (≈ 0.1% mode amplitude).Is it a sensitive control parameter?

Now we run the same simulation with η = 0 ...



-2

0

2

-3 -2 -1 0 1 2 3 4

a2

a1

1

-0.6

-0.3

0

0.3

0.6

-3 -2 -1 0 1 2 3 4

a3

a1

1

-0.6

-0.3

0

0.3

0.6

-0.4 0 0.4

a4

a3

1

-1.5

-1

-0.5

0

0.5

-0.6 -0.3 0 0.3 0.6

a6

a3

POD

4D-Var

1

Figure: 4D-Var with η = 0.25/21 G. TISSOT, Institut Pprime, GDR Contrôle Des Décollements


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


0.0001

0.001

0.01

0.1

1

10

100

0 50 100 150 200 250 300 350 400

e(t)

time step


η 6= 0η = 0

Figure 1:1

Figure: Time evolutions of errors. Comparison between the case of

perfect observations with η 6= 0 and η = 0.


Documents

Institut PPRIME, Poitiers - univ-orleans.fr...Introduction What is 4D-Vr?a rmalismFo ResultsConclusion 4D-Variational Data Assimilation using POD Reduced-Order Model Gilles TISSOT