CARI2020: Hend Ben Ameur & Nizar Kharrat & Mohamed Hedi RiahiCari2020 senegal

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A posteriori error estimation in an adaptive
multidimensional parameterization algorithm
Hend Ben Ameur & Nizar Kharrat & Mohamed H´edi Riahi
CARI’20
14-17 October Ecole Polytechnique de Thi`es, S´en´egal
Mohamed H´edi Riahi LAMSIN-ESPRIT 1 14-17 October 2020 1 / 10

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Direct and inverse problem in hydrogeology
Model:Equation of grandwater flow
S
∂Φ
∂t
− div(T Φ) = Q in Ω × (0, tf )
Φ = Φd on ΓD × (0, tf )
(−T Φ).n = qN on ΓN × (0, tf )
Φ((x, y), 0) = Φ0(x, y) in Ω
S storage coefficient, T hydraulic transmissivity, Φ piezometric head and Q distributed source
terms
Inverse problem of estimation of parameters
Direct problem: S, T=⇒Φ(S, T).
Inverse problem: dobs = Φmes=⇒S, T.
Misfit function:
J(S, T) =
1
2 i,j
(Φ((xi, yi), tj) − dobs
i,j )2
Mohamed H´edi Riahi LAMSIN-ESPRIT 2 14-17 October 2020 2 / 10

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Adaptative Parametrization
Parameterization with unknown in each mesh
cell
1 Too many parameters to estimate.
2 Number (dobs) of measures is reduced .
Reduce the number of parameters to estimate
1 Hypothesis: constant parameter by
zone.
2 (S, T) and interfaces.
Mohamed H´edi Riahi LAMSIN-ESPRIT 3 14-17 October 2020 3 / 10

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Adaptative Parametrization
Adaptive parameterization: Refinement indicators
idea of adaptive parameterization method
From one iteration to the next add only one degree of freedom.
H.Ben Ameur, G.Chavent, J.Jaffr´e,
Refinement and coarsening indicators for adaptive parametrization: application to the estimation of hydraulic
transmissivities, Inverse Problems., (2002), 775.
H.Ben Ameur, F.Cl´ement, P.Weis, G.Chavent,
The multidimensional refinement indicators algorithm for optimal parameterization, Journal of Inverse and Ill-Posed
Problems., (2008), 107–126.
Mohamed H´edi Riahi LAMSIN-ESPRIT 3 14-17 October 2020 3 / 10

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A posteriori error estimators
Goal: Improve the resolution of the inverse problem of parameter
estimation S et T (the adaptive parameterization algorithm).
Estimate (S, T) ⇔ Minimize J.
Solving the direct problem.
Solving the adjoint problem.
⇒ Better approximation of direct and adjoint problems.
Incorporating
Adaptive parametrization algorithm + Adaptive refinement algorithm.
H.Ben Ameur, N.Kharrat, Z.Mghazli,
Incorporating a posteriori error estimators in an adaptive parametrization algorithm, Journal of Inverse and Ill-Posed
Problems., (2015),.
Becker, R and Braack, M and Meidner, D and Rannacher, R and Vexler, B,
Adaptive finite element methods for PDE-constrained optimal control problems, Reactive flows, diffusion and transport.,
(2007), 177–205.
Mohamed H´edi Riahi LAMSIN-ESPRIT 4 14-17 October 2020 4 / 10

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Development of estimators
J(S, T, Φ) − J(Sh, Th, Φτh) = ηh,Z + Rh
J(S, T, Φ) − J(Sh, Th, Φτh) = J(S, T, Φ) − J(S, T, Φτ ) +
J(S, T, Φτ ) − J(Sc
h, Tc
h, Φc
τh) + J(Sc
h, Tc
h, Φc
τh) − J(Sh, Th, Φτh)
J(S, T, Φ) − J(S, T, Φτ ): the error due to time discretization.
J(S, T, Φτ ) − J(Sc
h, Tc
h, Φc
τh): the error due to space discretization.
J(Sc
h, Tc
h, Φc
τh) − J(Sh, Th, Φτh): the error due to parameter
discretization.
Mohamed H´edi Riahi LAMSIN-ESPRIT 5 14-17 October 2020 5 / 10

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optimization problems
Minimize J(S, T, Φ) subject to the continuous state equation , (S, T) ∈ Q2 (Pc)
with Q := P0((0, tf ), L∞(Ω)).
Minimize J(S, T, Φτ ) subject to the state equation to discretize in time with dG(r),
(S, T) ∈ Q2 (Pτ )
Minimize J(Sc
h, Tc
h, Φτh) subject to the state equation to discretize in space with cG(s) ,
(Sc
h, Tc
h) ∈ Z2 (Pc
τh)
Minimize J(Sh, Th, Φτh) subject to the state equation to discretize in space-time with
cG(s)dG(r) and discretization of parameters, (Sh, Th) ∈ Q2
h (Pτh)
with Qh = {σ ∈ Z; σ|In
∈ P0(In, P0(K)), ∀K ∈ T n
h , n = 1, · · · , Nτ }
Mohamed H´edi Riahi LAMSIN-ESPRIT 6 14-17 October 2020 6 / 10

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Computing of error estimators
Becker, R and Braack, M and Meidner, D and Rannacher, R and Vexler, B,
Adaptive finite element methods for PDE-constrained optimal control problems, Reactive flows, diffusion and transport.,
(2007), 177–205.
Theorem
Let ξ = (Φ, S, T, ψ), ξτ = (Φτ , S, T, ψτ ), ξc
τh = (Φτh, Sc
h, T c
h, ψτh) and ξτh = (Φτh, Sh, Th, ψτh) be
stationary points resp of L and L we have
L (ξ; δξ) = L (ξ; δξ) = 0 ∀δξ ∈ W, (1)
L (ξτ ; δξτ ) = 0 ∀δξτ ∈ Wr
τ , (2)
L (ξ
c
τh; δξ
c
τh) = 0 ∀δξc
τh ∈ W
c,r,s
τh
, (3)
L (ξτh; δξτh) = 0 ∀δξτh ∈ W
r,s
τh
. (4)
Then, there holds for the errors with respect to the cost functional J due to the dG(r)-time, cG(s)-space:
J(S, T, Φ) − J(S, T, Φτ ) =
1
2
L (ξτ ).(ξ − ˆξτ ) + Rτ , (5)
J(S, T, Φτ ) − J(S
c
h, T
c
h, Φ
c
τh) =
1
2
L (ξ
c
τh).(ξτ − ˆξ
c
τh) + R
c
h, (6)
J(S
c
h, T
c
h, Φ
c
τh) − J(Sh, Th, Φτh) =
1
2
L (ξτh).(ξ
c
τh − ˆξτh) + Rh. (7)
where ˆξτ = (
ˆ
Φτ , ˆS, ˆT , ˆψτ ), ˆξc
τh = (
ˆ
Φτh, ˆSc
h, ˆT c
h, ˆψτh) and ˆξτh = (
ˆ
Φτh, ˆSh, ˆTh, ˆψτh) are arbitrarily chosen and
Rτ , Rc
h and Rh have the same structure as given in Lemma 1 for L = L.
Mohamed H´edi Riahi LAMSIN-ESPRIT 7 14-17 October 2020 7 / 10

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Numerical tests
Φd = 1
Φd = 2
−T Φ.n = 0 −T Φ.n = 0
Figure: The synthetic aquifer
Adaptive parametrization algorithm.
Adaptive parametrization algorithm+ Global mesh refinement.
Adaptive parametrization algorithm +Adaptive refinement algorithm.
Mohamed H´edi Riahi LAMSIN-ESPRIT 8 14-17 October 2020 8 / 10

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Conclusions and current work
1 Develop refinement indicators.
2 Simultaneous Estimation of S and T Using the Adaptive
Parametrisation Algorithm.
3 Develop a posteriori error estimators .
4 Incorporating of the posterior error estimation in the adaptive
parametrisation algorithm.
5 Simultaneous Estimation of S and T using the combination of the
two algorithms.
6 To study the influence of the posterior indicators in the case where
the two coefficients do not have the same parameterization.
7 Study the time discretization error.
Mohamed H´edi Riahi LAMSIN-ESPRIT 9 14-17 October 2020 9 / 10

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THANK YOU FOR YOUR ATTENTION
Mohamed H´edi Riahi LAMSIN-ESPRIT 10 14-17 October 2020 10 / 10