Introduction Non standard approach Recovery of the final state Computations
Some results on data assimilation problems and
Tychonov regularization
Jean-Pierre Puel
Laboratoire de Mathématiques de Versailles,
Université de Versailles Saint-Quentin,
45 avenue des Etats Unis, 78035 Versailles Cedex, France
[email protected]
CIMPA 2009, Pointe à Pitre, January 2009
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Computations
Outline
1
Introduction
Motivation
Mathematical problem
Variational data assimilation
2
Non standard approach
Carleman estimate
Some functonal analysis
Relations with Tychonov regularization
3
Recovery of the final state
Adjoint controllability problem
Approximation by optimal control
4
Computations
Linear quasi-geostrophic ocean model
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
Data Assimilation : Very important problem in
Weather prediction
Climatology
Oceanology
All environment sciences
Corresponds to an enormous amount of computing time, for
example 60% of computing time in meteorology.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
For example : weather prediction.
To compute evolution of air masses, pressure, temperature, etc. on
an annulus around the planet up to an altitude of 51 km.
Model : enriched every day but rather satisfactory at the moment.
Computers : more and more powerful ... reasonable possibilities.
Missing : knowledge of the state variables to-day or yesterday
(initial conditions) on the whole spatial domain !! in order to
predict the evolution.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
For simplicity of presentation, all the presentation will be given on
the heat equation.
Analogous results and proofs for
Diffusion-convection equations
Linearized Navier-Stokes equations
Linearized Boussinesq
More generally : coupled (linearized) systems of
diffusion-convection and Navier-Stokes
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
All operators considered here are linear.
Up to now : linear method.
Hope of extension to some nonlinear systems...
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
Mathematical problem : Heat equation
∂y
− ∆y = f on Ω × (0, T ),
∂t
y = 0 on ∂Ω × (0, T ),
y (0) = y0 in Ω.
y0 is unknown, but we know measurements of the solution on a
subdomain ω × (0, T0 ), T0 < T .
h = yω×(0,T0 ) .
For example
0 = yesterday . T0 = to-day. T = 2T0 = to-morrow.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
A simple translation by ỹ solution (which can be computed) of
∂ỹ
− ∆ỹ = f on Ω × (0, T ),
∂t
ỹ = 0 on ∂Ω × (0, T ),
ỹ (0) = 0 in Ω
reduces the problem to the case where f = 0 called (HE).
∂y
− ∆y = 0 on Ω × (0, T ),
∂t
y = 0 on ∂Ω × (0, T ),
(2)
y (0) = y0 in Ω.
(3)
Jean-Pierre Puel
(1)
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
Question :
How to recover the value of the state variable y (t)
at a time t, 0 ≤ t ≤ T0 in order to compute y
on the time interval (T0 , T )?
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
Classical method : Variational data assimilation.
In use at the moment in several meteorological centers (Reading,
Meteo-France...)
Based on optimal control ideas.
Try to recover y (0) = y0 , then run the system from t = 0 to
t = T.
y0 : control variable. Solve the equation for y0 given → y (y0 ).
Define
Z
Z
1 T0
J0 (y0 ) =
|y (y0 ) − h|2 dxdt
2 0
ω
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
To find ȳ0 such that
J0 (ȳ0 ) = min J0 (y0 )
y0
(OC )
This problem does not have a solution (ill-posed problem).
If we take a minimizing sequence y0n , then we only know that
T0
Z
0
Z
|y (y0n )|2 dxdt ≤ C .
ω
This gives no estimate on y0n (in any known functional space)!!
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Motivation
Computations
Mathematical problem Variational data assimilatio
Remedy :
Tychonov regularization.
For α > 0 (Tychonov regularization parameter) define
1
Jα (y0 ) =
2
T0
Z
0
Z
|y (y0 ) − h|2 dxdt +
ω
α
||y0 ||2
2
Now find yα such that
Jα (yα ) = min Jα (y0 )
y0
(TROC )
This problem has a unique solution yα (classical methods)
This problem is known to be unstable when α → 0 ....but seems to
work in practice !
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
In fact :
Global Carleman estimates (difficult) plus energy estimates
(standard) provide a weighted estimate on y .
We have the following estimate:
|y (T0 )|2L2 (Ω)
Z
T0
Z
e
+
0
Ω
−2sη
Z
2
T0
Z
|∇y | dxdt +
0
Z T0 Z
≤C
|y |2 dxdt
0
e −2sη |y |2 dxdt
Ω
ω
where s > 0 and η is a weight such that
η(x, t) > 0 in Ω × (0, T0 ), η → +∞ when t → 0,
∀δ > 0, η(x, t) ≤ Mδ on Ω × (δ, T0 ).
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Remarks :
This estimate does not make any reference to the initial value y0 !!
It gives an estimate in classical Sobolev spaces L2 (δ, T0 ; H01 (Ω)) for
every δ > 0 and an estimate on |y (T0 )|L2 (Ω) .
It implies the unique continuation property which is classical for
this example, but less classical for more complex systems.
This estimate is the main mathematical difficulty to solve when
trying to apply the method to your favorite evolution system.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Consider the space
V = {y , y is a (regular) solution of (HE), y0 ∈ L2 (Ω)}
and define on this space
||y ||2V =
Z
0
T0
Z
|y |2 dxdt.
ω
Then ||y ||V is a prehilbertian norm on V because of unique
continuation property.
Now define V as the completion of V.
Then V is a Hilbert space for the scalar product associated to the
norm ||y ||V .
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Because of Carleman estimate we have
∀δ > 0, V ⊂ L2 (δ, T0 ; H01 (Ω)).
Essentially we have
V = {y ∈ L2e −2sη (0, T0 ; H01 (Ω)), y is a solution of (HE(1)-(2)),
Z T0 Z
|y |2 dxdt < +∞}.
0
ω
An element of V may not have any value at time t = 0 !!
But its value at each time t > 0, in particular at time t = T0
makes perfect sense in L2 (Ω) for example.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
We still have the Carleman estimate for elements of V
|y (T0 )|2L2 (Ω)
Z
T0
Z
e
+
0
−2sη
Ω
Z
2
T0
|∇y | dxdt +
0
Z T0 Z
≤C
|y |2 dxdt
0
Z
e −2sη |y |2 dxdt
Ω
ω
For y ∈ V we define
Z
T0
Z
J(y ) =
0
|y − h|2 dxdt.
ω
Notice that for y ∈ V, J and J0 have the same value but the
argument is different : J0 depends on the initial value y0 while J
depends on the trajectory y ∈ V .
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
We now consider the minimization problem (MP):
To find ŷ ∈ V such that
J(ŷ ) = min J(y ).
y ∈V
(MP)
We immediately obtain the following result.
Theorem
There exists a unique solution ŷ ∈ V to the previous minimization
problem.
If we have two datas h1 and h2 the corresponding solutions ŷ1 and
ŷ2 satisfy
Z T0 Z
2
||ŷ1 − ŷ2 ||V ≤
|h1 − h2 |2 dxdt.
0
Jean-Pierre Puel
ω
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Remark : This result implies in particular the stability (in the
V -norm) of the solution with respect to perturbations in the
“measurements”.
A priori we know that for every δ > 0
ŷ ∈ C ([δ, T0 ]; L2 (Ω)) ∩ L2 (δ, T0 ; H01 (Ω)).
Without any further hypothesis, we cannot say anything about the
convergence of Tychonov regularization. But if we assume some
additional regularity on the solution ŷ we can obtain a convergence
result.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Theorem
Let us assume that ŷ ∈ C ([0, T0 ]; L2 (Ω)) with ŷ (0) = ŷ0 . Then we
have
J(ŷ ) = J0 (ŷ0 ).
Moreover, when α → 0, the solution yα of the Tychonov
regularized problem converges to ŷ0 strongly in L2 (Ω). If y α is the
solution of (HE) corresponding to yα we have
Z
0
T0
Z
|ŷ − y α |2 dxdt ≤ α|ŷ0 |L2 (Ω) .
ω
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Proof (classical).
We have
Jα (yα ) ≤ Jα (ŷ0 )
and
J(ŷ ) ≤ J(y α ).
Therefore
Z
Z
1 T0
α
|y α − h|2 dxdt + |yα |2L2 (Ω)
2 0
2
ω
Z
Z
1 T0
α
≤
|ŷ − h|2 dxdt + |ŷ0 |2L2 (Ω)
2 0
2
ω
Z T0 Z
α
1
|y α − h|2 dxdt + |ŷ0 |2L2 (Ω) .
≤
2 0
2
ω
As a consequence we have
∀α > 0, |yα |2L2 (Ω) ≤ |ŷ0 |2L2 (Ω) .
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
After extraction of a subsequence (still denoted by yα ), we can
suppose that
yα * ỹ0 in L2 (Ω) weakly
and if ỹ denotes the solution of (HE) with ỹ (0) = ỹ0 we have
y α → ỹ in L2 (0, T0 ; L2 (Ω))
and also in various topologies.
Now we have
α
|yα |2L2 (Ω) → 0, when α → 0
2
so that necessarily, when α → 0,
1
2
Z
0
T0
Z
1
|y − h| dxdt →
2
ω
α
2
Jean-Pierre Puel
Z
0
T0
Z
|ŷ − h|2 dxdt.
ω
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Therefore we must have
Z
Z
Z
Z
1 T0
1 T0
|ŷ − h|2 dxdt =
|ỹ − h|2 dxdt.
2 0
2
ω
ω
0
From uniqueness in the minimization problem, we see that
necessarily we have
ŷ = ỹ in Ω × (0, T0 ),
so that
ŷ0 = ỹ0 .
We now know that
yα * ŷ0 in L2 (Ω) weakly
and
|yα |2L2 (Ω) ≤ |ŷ0 |2L2 (Ω) .
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
This implies strong convergence in L2 (Ω) of yα towards ŷ0 .
As ŷ is the minimizer of J we see from the Euler-Lagrange
equation associated to this minimization problem that
Z
T0
Z
∀z ∈ V ,
(ŷ − h).(ŷ − z)dxdt = 0.
0
(4)
ω
Now a simple calculation gives
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
Z
T0
Z
|ŷ − y α |2 dxdt + α|ŷ0 − yα |2L2 (Ω) =
0
ω
Z T0 Z
Z T0 Z
α
2
2
|y − h| dxdt + α|yα |L2 (Ω) +
|ŷ − h|2 dxdt +
0
ω
0
ω
Z T0 Z
2
α|ŷ0 |L2 (Ω) − 2
(ŷ − h).(y α − h)dxdt − 2α(ŷ0 , yα )L2 (Ω) ≤
0
ω
Z T0 Z
Z T0 Z
2
|ŷ − h|2 dxdt − 2
(ŷ − h).(y α − h)dxdt +
0
ω
0
ω
2α|ŷ0 |2L2 (Ω) − 2α(ŷ0 , yα )L2 (Ω) ≤
Z T0 Z
2
(ŷ − h).(ŷ − y α )dxdt + 2α(ŷ0 , ŷ0 − yα )L2 (Ω) .
0
ω
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
But as y α ∈ V we obtain
Z
0
T0
Z
ω
|ŷ − y α |2 dxdt + α|ŷ0 − yα |2L2 (Ω) ≤ 2α(ŷ0 , ŷ0 − yα )L2 (Ω) .
This gives immediately the convergence result.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Carleman
Computations
estimate Some functonal analysis Relations with Ty
If in addition ŷ0 is in the range of the controlled adjoint equation
we can obtain a better convergence rate.
Theorem
If there exists w ∈ L2 (0, T0 ; L2 (ω)) such that the solution q of
∂q
− ∆q = w .χω in Ω × (0, T0 ),
∂t
q = 0, on Γ × (0, T0 ),
−
q(T0 ) = 0, in Ω,
satisfies
q(0) = ŷ0 ,
then we have
|ŷ0 − yα |2L2 (Ω) ≤ C α,
Z
0
Jean-Pierre Puel
T0
Z
|ŷ − y α |2 dxdt ≤ C α2 .
ω
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
Recovery of the final state
We have already seen that, due to the fact that ŷ ∈ V ,
|ŷ (T0 )|2L2 (Ω)
T0
Z
Z
≤C
0
|ŷ |2 dxdt.
ω
This gives a stability result for the recovery of ŷ (T0 ) from
“measurements” of ŷ on ω × (0, T0 ).
On the other hand, from the Euler-Lagrange equation associated
to the minimization problem (MP) we have
Z
T0
Z
(h − ŷ )zdxdt = 0, ∀z ∈ V .
0
(EL)
ω
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
Let us now consider the following exact controllability problem
(null controllability) for the adjoint equation.
If ψ ∈ L2 (Ω) we know (null controllability results) that there exists
v = v (ψ) ∈ L2 (0, T0 ; L2 (ω)) such that the solution ϕ of
∂ϕ
− ∆ϕ = v .χω , in Ω × (0, T0 ),
∂t
ϕ = 0, on Γ × (0, T0 ),
−
ϕ(T0 ) = ψ in Ω,
satisfies
ϕ(0) = 0.
Moreover we can take v (ψ) of minimal norm in L2 (0, T0 ; L2 (ω))
and we then have
|v (ψ)|L2 (0,T0 ;L2 (ω)) ≤ C |ψ|L2 (Ω)
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
We also know (by a duality argument) that v (ψ) is obtained as the
restriction of z on ω × (0, T0 ) where z is the minimum (in V ) of
the functional
Z
Z
Z
1 T0
2
|z| dxdt +
z(T0 )ψdx.
K (z) =
2 0
ω
Ω
As z ∈ V we have from (EL)
T0
Z
Z
T0
Z
Z
hzdxdt =
0
so that
Z
T0
ŷ zdxdt
ω
Z
0
Z
T0
ω
Z
hv (ψ)dxdt =
0
ω
ŷ v (ψ)dxdt.
0
Jean-Pierre Puel
ω
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
Now multiplying the equation for ϕ by ŷ , because ϕ(0) = 0, we
obtain (this can be shown rigorously)
Z
Z
T0
Z
Z
ψŷ (T0 )dx = −
Ω
T0
Z
v (ψ)ŷ dxdt = −
0
ω
v (ψ)hdxdt.
0
ω
The control v (ψ) can be “computed” for every ψ ∈ L2 (Ω). The
measurement h is given. Therefore, at least in theory, this enables
us to recover the coefficient
Z
ψŷ (T0 )dx
Ω
for every ψ in a Hilbert basis of L2 (Ω), and therefore ŷ (T0 ), at the
price of solving a null controlability problem for each element of a
Hilbert basis.....
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
Approximation by optimal control
For β > 0 we can consider an approximation by an optimal control
problem.
Define
Z
Z
Z
1
1 T0
2
Kβ (v ) =
|v |2 dxdt,
|ϕ(0)| dx +
2β Ω
2 0
ω
Look for vβ ∈ L2 (0, T0 ; L2 (ω)) such that
Kβ (vβ ) =
min
v ∈L2 (0,T0 ;L2 (ω))
Kβ (v )
For every β > 0 this problem has a unique solution vβ which can
be characterized by an optimality system using an adjoint state
(classical optimal control).
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
It is easy to show that when β → 0,
vβ → v (ψ) in L2 (0, T0 ; L2 (ω))
Z
T0
Z
−
Z
T0
Z
Z
vβ hdxdt → −
0
ω
v (ψ)hdxdt =
0
ω
ψŷ (T0 )dx
Ω
Therefore we can compute an approximation of the desired
coefficient at the price of solving an optimal control problem (for
each ψ).
Rate of convergence : requires a regularity hypothesis on the
adjoint state associated to the exact controllability problem.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Adjoint
Computations
controllability problem Approximation by optimal cont
As
Z
q∈V →
ψq(T0 )dx
Ω
is a linear continuous map, from Riesz Theorem there exists p ∈ V
such that
Z T0 Z
Z
∀q ∈ V ,
pqdxdt =
ψq(T0 )dx.
0
ω
Ω
If we suppose that in addition
p ∈ C ([0, T0 ]; L2 (Ω))
then we obtain
1
|vβ − v (ψ)|L2 (0,T0 ;L2 (ω)) ≤ 2β 2 |p(0)|L2 (Ω) ,
Z
Z T0 Z
1
| ŷ (T0 )ψdx +
h.vβ dxdt| ≤ C β 2 |p(0)|L2 (Ω) ,
Ω
0
Jean-Pierre Puel
ω
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
It would be important to use a reduced basis such as POD... in
order to reduce the number of optimal control problems (or exact
controllability problems) to solve. Work of P.Hepperger
(Diplomarbeit, T.U. München) which seems very promising in this
direction.
Computations with a classical basis done in Garcia-Osses-Puel for a
large scale ocean model (reasonable results).
Computations with a spectral basis on the same problem
(Garcia-Osses-Puel).
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
Linear quasi-geostrophic ocean model :

1

ut − AH ∆u + γu + (f0 + βx2 ) k ∧ u + ∇p = T



ρ0

div u = 0 in Ω × (0, T ),

 u = 0 on Γ × (0, T ),



u(0) = u0 in Ω,
in Ω × (0, T ),
u(x, t) and p(x, t), respectively, denote the velocity and the
pressure of the fluid at (x, t) = (x1 , x2 , t) ∈ IR2 × IR+ .
AH represents the horizontal eddy viscosity coefficient, γ is the
bottom friction coefficient, ρ0 is the fluid density, T is the wind
stress, and (f0 + βx2 )k ∧ u is the Coriolis term, with
k ∧ u = (−u2 , u1 ). We have used β-plane approximation, with
β = 2Ω0 R −1 cos θ̃0 , where Ω0 and R are the angular velocity and
radius of the Earth, respectively, and θ̃0 a reference latitude.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
Formulation in terms of the stream function ψ(x, t). Since
div u = 0, u = 0 on Γ × (0, T ), and Ω is a connected subset of IR ,
we can introduce the stream function ψ(x, t)









∂
∂ψ
(∆ψ) − m ∆2 ψ + s ∆ψ +
= −curl T
∂t
∂x1
∂ψ
= 0 on Γ × (0, T ),
ψ=
∂n
∆ψ(0) = −curl u0 = ∆ψ0 in Ω,
Ro
in Ω × (0, T ),
(
where the coefficients Ro , s and m are the non-dimensional
Rossby, Stommel and Munk numbers, respectively:
Ro =
U
,
βL2
m =
Jean-Pierre Puel
AH
,
βL3
s =
γ
.
βL
(6)
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
U denotes a typical horizontal velocity, L is a representative
horizontal length scale of ocean circulation. Ω = [0, 1] × [0, 1] and
T0 = 0.05. For the Rossby, Munk and Stommel numbers, we
consider :
Ro = 1.5 × 10−3 ,
m = 1 × 10−4 ,
s = 5 × 10−3 ,
which correspond to
γ = 1 × 10−7 s−1 , AH = 2 × 103 m2 s−1 , L = 106 m, T = 1year ,
U = 0.03m s−1 , β = 2 × 10−11 m−1 s−1 , D0 = 800 m.
For the wind stress, we use
1
x2
T = (τ1 , τ2 ) = exp(π t) − cos(π ), 0 .
π
L
2
(7)
The following series of test problems have been done with
∆t = T0 /50.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
0.4
30vp obs4
0.35
0.3
|z(0)|21,Ω
0.25
0.2
0.15
α=0.025
0.1
0.05
0
0
5
10
15
20
25
30 x1
2
||h||L2(L2(O))
2
Control norm h
2
L (L2 (O))
versus the term |z(0)|21,Ω for different values
of the penalty parameter α. The observatory O for this test is
[0, 1] × [0.3, 0.7]. We have chosen for our tests α = 0.025.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
ψN
rec,h
ψN
h
ψN
rec,h
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
0.8
1
Contour lines of stream function at T0 using real physical parameters.
Exact (left), recovered using 11 eigenvalues (center) and recovered
using 64 eigenvalues (right). The area between the dotted lines
corresponds to the observatory O. The regularizing parameter is
α = 0.025.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
1
0.15
0.8
0.1
0.05
0.6
0
0.4
−0.05
0.2
0
0
−0.1
−0.15
0.2
Jean-Pierre Puel
0.4
0.6
0.8
1
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
1
0.1
0.8
0.05
0.6
0
0.4
−0.05
0.2
−0.1
0
0
0.2
0.4
0.6
0.8
1
Relative error in percentage between the recovered stream function
using 11 eigenvalues (above) and 64 eigenvalues (below) and the exact
solution. The observatory is O = [0, 1] × [0.3, 0.7]. The regularizing
parameter is α = 0.025.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
0.085
0.08
Error
0.075
0.07
0.065
0.06
0.055
10
20
30
40
50
Number of eigenvalues
60
70
Relative error versus the number of eigenvalues taken for the
projection in the assimilation method (bold line). The error level
obtained using a canonical finite element basis for the same mesh is
indicated by an horizontal dashed line. The observatory O for this
test is [0, 1] × [0.3, 0.7]. We have chosen for our tests 64 eigenvalues.
The regularizing parameter is α = 0.025.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
For the following experiment, we consider different observatory
n
sizes and some perturbation in the observations ψobs,h
given by:
n
n
ψ̂obs,h
= ψobs,h
+ A sin
kπx
L
sin
mπy L
sin(wt),
(8)
n
where A is the noise amplitude (0.5 ∗ max(ψobs,h
)) and k, m, w
were taken big enough. In Table ??, we present the relative errors
in L2 (Ω) and H 1 (Ω) for the final recovered stream function in both
case, with noise and in absence of it in the observatory set, using
64 eigenvalues. We obtain small values for each case.
Jean-Pierre Puel
Some results on data assimilation problems and Tychonov regul
Introduction Non standard approach Recovery of the final state Linear
Computations
quasi-geostrophic ocean model
Relative errors in L2 (Ω) and H 1 (Ω) for the final recovered stream function with
noise
noise in the observatory set (ψrec,h
) and in the absence of noise (ψrec,h ) versus
the observatory size O.
O
˛
˛
˛
˛ N
˛ψh −ψrec,h ˛
0,Ω
˛
˛
˛ N˛
˛ψh ˛
N −ψ
|ψh
rec,h |1,Ω
|ψ N |1,Ω
h
˛
˛
˛ N
noise ˛˛
˛ψh −ψrec,h
0,Ω
˛
˛
˛ N˛
˛ψh ˛
N −ψ noise |
|ψh
rec,h 1,Ω
|ψ N |1,Ω
h
[0, L] × [0.4L, 0.6L]
[0, L] × [0.3L, 0.7L]
[0, L] × [0.2L, 0.8L]
[0, L] × [0.1L, 0.9L]
[0, L] × [0, L]
0.1043
0.0634
0.0540
0.0460
0.0454
0.2561
0.2260
0.2137
0.2069
0.2063
0.1054
0.0668
0.0562
0.0471
0.0465
0.2582
0.2284
0.2159
0.2082
0.2077
0,Ω
Jean-Pierre Puel
0,Ω
Some results on data assimilation problems and Tychonov regul