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A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible fluid flow
Journal of Computational and Applied Mathematics 270 (2014) 522–530
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A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible fluid flow N. Akkari a,b,∗ , A. Hamdouni a,∗∗ , E. Liberge a , M. Jazar b a
University of La Rochelle, LaSIE-Laboratory of the Engineering Sciences for the Environment, Avenue Michel Crépeau 17042 La Rochelle Cedex 1, France b Lebanese University, LaMA-Laboratory of Mathematics and Applications, B.P 826 Tripoli, Lebanon
article
info
Article history: Received 4 July 2013 Received in revised form 14 November 2013 Keywords: ROM–POD Sensitivity Parametric evolution POD modes number Error estimates Navier–Stokes equations
abstract In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier–Stokes equations for a 2D incompressible fluid flow. We present a numerical test of our theoretical result, for an unsteady fluid flow in a channel around a circular cylinder. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and main results 1.1. Statement of the problem Boundary problems intervene widely in applications in fluid mechanics such as stability study, inverse problems, and optimization problems. There is also the optimal control problem by variation of the fluid flows characteristic parameters [1–4]. In fact, this problem is reduced to solve the Navier–Stokes equations for different parametric values and to minimize a given functional with respect to these values, so that one can reach a target situation. But the resolution cost of this optimization problem at each Reynolds number is very high from two points of view: PC storage capacity and time for computations. To solve this problem in practice, we reduce the degrees of freedom of the Navier–Stokes system. A reduced order model is then used to predict the behavior of fluid flows at different parametric values. In particular, we are interested in the models reduction by projection, the POD technique (Proper Orthogonal Decomposition) [5]. Given X a Hilbert space, and (u(t ))t a family in X living on a time interval (0, T ) and integrable in
∗ Corresponding author at: University of La Rochelle, LaSIE-Laboratory of the Engineering Sciences for the Environment, Avenue Michel Crépeau 17042 La Rochelle Cedex 1, France. ∗∗ Corresponding author. E-mail addresses: [email protected] (N. Akkari), [email protected] (A. Hamdouni), [email protected] (E. Liberge), [email protected] (M. Jazar). 0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.11.025
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the sense of L2 (0, T ; X ), it is about to construct a projection subspace spanned by a basis Φ and which minimizes the error obtained a posteriori by the orthogonal projection of u on a subspace of X of finite dimension N:
u − Π Φ u22
L (0,T ;X )
N
2 ≤ u − ΠNΨ uL2 (0,T ;X ) ,
∀N , and ∀Ψ an orthonormal Hilbert basis of X .
This minimization problem is equivalent to the eigenvalues problem of the compact, selfadjoint and positive operator
R, defined as follows: R Φ = µΦ , where
R:
X
ϕ
→
X
Rϕ =
→
1 T
T
(u(t ), ϕ)X u(t )dt . 0
One can verify that the optimal error by POD is equal to the remainder of the POD-eigenvalues series of R: 1 T
u − Π Φ u22 N
L (0,T ;X )
=
+∞
µn .
n =N +1
The problem that appears naturally is to find a way to control the parametric evolution in a reduced order model by POD. The previous works concerning this problem are numerical algorithms that are supposed to build a reduced order model by POD, adapted to changes of the characteristics of a fluid flow. Amsallem et al. [6–8], build a geometric interpolation algorithm of POD basis at different parametric values. There are also other techniques based on the iterative algorithm of Greedy [9,10], that enriches a POD basis computed at a given parametric value, by evaluating the errors obtained a posteriori by the orthogonal projection on this basis of solutions associated with other parametric values [11–17]. Besides these works, the prediction of a validity domain of a reduced order model by a POD associated with only one parameter, is becoming crucial. This could lead from the one hand to interpolate reduced order models available each one in a confidence region around a characteristic parameter, and from the other hand to initialize the Greedy algorithm. Terragni and Vega [18] presented an argument to use the POD approximation at a given parametric value to construct the bifurcation diagram of dissipative systems, which is not the aim of the present paper. We are interested then, in mathematical contributions concerning the study of the sensitivity of a reduced order model by a reference POD basis. More precisely, the mathematical formulation of the problem is given as follows: 1.2. Mathematical formulation of the problem Let us consider a problem describing the evolution of a parametric solution uλ (λ ∈ R+∗ ) in a Hilbert space X . V denotes a subspace of X and Vh a subspace of V of dimension M:
∂ uλ = Aλ (uλ (t )) ∂t u (0) = u . λ 0
(1)
From now on, we are interested in the ordinary differential equation of dimension M obtained by a Galerkin projection of the initial complete problem (1) on Vh :
h ∂ uλ , v h = A (uh (t )), v h ∀v h ∈ V λ λ h ∂t h h h h uλ (0), v = u0 , v ∀v ∈ Vh .
(2)
A solution uhλ0 of this equation associated with a parameter λ0 is computed once and for all. A POD basis Φ λ0 =
λ Φn 0
n=1,...,M
of dimension M in X , associated with uhλ0 on a time interval (0, T ), will lead to construct a ROM–POD
describing the evolution of an approximation uˆ λ,λ0 of uhλ , in a subspace of dimension N very small compared to M. We λ the POD eigenvalues sequence associated to the POD basis. denote µn 0 n=1,...,M
∀t ∈ (0, T ) and ∀x ∈ X : uˆ λ,λ0 (t , x) =
N n=1
0 aλ,λ (t )Φnλ0 (x), n
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λ,λ0
where ∀n = 1, . . . , N , an
(t ) is the solution of:
λ,λ0 dan
= Aλ (ˆuλ,λ0 (t )), Φnλ0 λ,λ0 an (0) = u0 , Φnλ0 .
(3)
dt
The question which arises naturally is the following: To what extent uˆ λ,λ0 remains an accurate approximation to uhλ ? Then, the main problem is to give a mathematical criteria in order to determine a confidence region for a reference POD basis. It appears also the control problem of the dimension of the reduced order model, so we can improve its performance by increasing the retaining POD modes while constructing the ROM, without any loss of the reduction concept. It is worth noting that, F. Terragni, E. Valero and J.M. Vega developed in [19] a numerical adaptive method to accelerate time dependent numerical solvers of PDEs, by the use of error estimates to predict the required number of POD modes in order to preserve stability of a reduced order model by POD in connection with high-order modes truncation. Also, this is not the problem treated in this paper. 1.3. Main result A result we are proposing for a particular class of evolution problems, is a priori estimate of the squared error in the space L2 (0, T ; X ), obtained by a Galerkin approach for uhλ by a reference POD basis Φ λ0 :
λ
Formal result 1. There exist two decreasing sequences: f1 0 (N )
N =1,...,M
λ
and f2 0 (N )
N =1,...,M
: such that,
h Bλ |λ − λ0 |α λ λ u − uˆ λ,λ 22 ≤ f1 0 (N ) + f2 0 (N ) 0 . λ 0 L (0,T ;X ) λ0
(4)
2
The estimate (4) shows that the parametric ROM–POD error uhλ − uˆ λ,λ0 L2 (0,T ;X ) , is controlled by a first term fλ10 (N ) which will estimate the squared POD-Galerkin error relative to the parameter λ0 , and a second term which is a function of the alpha-th power of the distance between λ and λ0 , multiplied by an fλ20 (N ) which tend to zero when N becomes close to M. This result establishes an a priori estimate of the decrease rate of the squared ROM–POD error, especially when the two parameters λ and λ0 are distant. This rate depends on the one of the sequence fλ20 (N ). Moreover, for a fixed POD modes number N, this result shows the dependency of the ROM–POD error with respect to the distance between the parameters.
1.4. Paper organization In what follows, we give the proof elements of our principal result. In Section 2, we prove Theorem 1 with α = 2 for a quasi-linear parabolic equation. In Section 3, we show that the same proof applies in particular to the Navier–Stokes equations for a 2D incompressible fluid flow. The parameter of these equations will be the Reynolds number described by the variation of the viscosity λ ∈ R+∗ , for given length and velocity of reference. In Section 4, we present a numerical test of our theoretical estimate (4) for a fixed POD modes number N and by varying only the viscosity λ. We do this for an unsteady fluid flow in a channel around a circular cylinder. The numerical simulations are done using code-Saturne. The Navier–Stokes equations are solved by the finite volume method. In Section 5, we conclude by giving some perspectives of this work. 2. Proof of the main result for a quasi-linear parabolic equation We place our problem in the case of a semi-discrete quasi-linear parabolic equation with a dissipative term: We denote X = [L2 (Ω )]d , V = [H 1 (Ω )]d where Ω is a bounded open set, connected and Lipschitz of Rd , and Vh is a subspace of V of dimension M. f is given in L2loc ([0, +∞[, X ). This equation is given by its weak formulation as follows:
h ∂ uλ , v h + b uh (t ), uh (t ), v h = −λa uh (t ), v h + F (v h ) ∀v h ∈ V t h λ λ λ ∂t X uh (0), v h = u , v h ∀v h ∈ Vh , 0 λ X X
(5)
where b, a and Ft are respectively a trilinear form, a bilinear coercive one and a linear one given by the following expressions: – ∀v1 , v2 ∈ V , a (v1 , v2 ) = (∇v1 , ∇v2 )[L2 (Ω )]d×d . – ∀v1 , v2 , v3 ∈ V , b (v1 , v2 , v3 ) = (A(v1 ).B (∇v2 ), v3 )X , where A and B are respectively linear in v1 and ∇v2 . – ∀v ∈ V , Ft (v) = (f (t ), v)X .
(.) denotes the scalar product in the corresponding space. We precise some additional notations, that could be useful to our proof.
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– Cp denotes the constant relative to the coercivity of the bilinear form a, which is nothing but the square of a Poincaré constant. – Ca , Cb are respectively the two norms of the forms a and b on (Vh , ∥.∥X ) and K = uhλ0 . L∞ (0,T ;X )
2.1. The study of the squared POD-Galerkin error relative to a fixed parameter λ0 We denote uˆ λ0 = uˆ λ,λ0 , for λ = λ0 . We prove a result on the decrease rate of the squared POD-Galerkin error relative to the parameter λ0 :
2 h uλ0 − uˆ λ0 2
L (0,T ;X )
.
We show first the following a priori estimate: Proposition 1. ∀t ∈ (0, T ) and ∀ε > 0, one has:
M 2 λ2 C 2 N h µλn 0 . ΠΦ λ0 uλ0 − uˆ λ0 (t ) ≤ 2Cb K + 0 a T exp ε − 2λ0 Cp + 6Cb K t X ε n=N +1
(6)
λ
Proof. We replace v h by Φn 0 in Eq. (5) (written for the parameter λ0 ) and we use the commutativity between the projector operator Π and the time derivative dtd , then we can deduce the following equality:
1 d N 2 ΠΦ λ0 uhλ0 − uˆ λ0 = −b uhλ0 − uˆ λ0 (t ), uhλ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X 2 dt
− b uˆ λ0 (t ), uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) − λ0 a uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) . We use the properties of the two forms a and b, and we apply Young’s inequality, then we get the following three inequalities:
−b
uhλ0 − uˆ λ0 (t ), uhλ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t )
≤ Cb K
2 1 2 1 h uλ0 − ΠΦN λ0 uhλ0 (t ) + ΠΦN λ0 uhλ0 − uˆ λ0 (t ) 2
2
X
X
2 + Cb K ΠΦN λ0 uhλ0 − uˆ λ0 (t ) , −b uˆ λ0 (t ), uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X 2 1 2 2 1 h N h N h ≤ Cb K uλ0 − ΠΦ λ0 uλ0 (t ) + ΠΦ λ0 uλ0 − uˆ λ0 (t ) + Cb K ΠΦN λ0 uhλ0 − uˆ λ0 (t ) , 2
2
X
X
X
and ∀ϵ > 0,
−λ0 a
uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t )
≤
2 λ20 Ca2 h uλ0 − ΠΦN λ0 uhλ0 (t ) X 2ε 2 2 ε N h + ΠΦ λ0 uλ0 − uˆ λ0 (t ) − λ0 Cp ΠΦN λ0 uhλ0 − uˆ λ0 (t ) . 2
X
X
Therefore,
2 2 d N ΠΦ λ0 uhλ0 − uˆ λ0 ≤ 6Cb K + ε − 2λ0 Cp ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X X dt 2 λ20 Ca2 h + 2Cb K + uλ0 − ΠΦN λ0 uhλ0 (t ) . ε
X
We integrate over an interval (0, t ) ⊂ (0, T ) and we apply the Gronwall lemma, then we get:
M 2 λ20 Ca2 N h T µλn 0 . ΠΦ λ0 uλ0 − uˆ λ0 (t ) ≤ exp 6Cb K + ε − 2λ0 Cp t 2Cb K + X ε n =N +1 This ends the proof. By choosing ε small, the preceding estimate (6) shows the competition between the quasi-linear term and the one corresponding to the coercivity of the bilinear form a. By this estimate, the only way to reduce the POD-Galerkin error
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N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
is to choose N close to M. Then, a solution to this problem could be by doing a priori estimate of the two constants Cb and Cp . This could lead to choose an artificial viscosity verifying a certain condition, so that the term in the exponential becomes negative (6). Then, under this hypothesis we prove the following proposition: Proposition 2. If we choose an artificial viscosity λsm that verifies the condition:
λsm >
6Cb K + ε 2Cp
− λ0 ,
(7)
then the squared POD-Galerkin error for the parameter λ0 will be controlled as follows:
M h λ2sm Ca2 u − uˆ λ 22 ≤ 2T 1 + 2C K + µλn 0 . b λ0 0 L (0,T ;X ) ε n =N +1
(8)
λ f1 0 ( N )
Proof.
2 h h N h u − uˆ λ 22 ≤ 2 u − Π u λ λ0 λ0 0 L (0,T ;X ) Φ 0 λ0 2
L (T,X )
2
L (T,X )
.
λ
M
– uhλ0 − Π N λ0 uhλ0 =T Φ L2 (0,T ;X )
2 + 2 ΠΦN λ0 uhλ0 − uˆ λ0 2
n =N +1
µn 0 .
2 – Thanks to (7) and to the previous proposition, Π N λ0 uhλ0 − uˆ λ0 Φ 2
L (0,T ;X )
≤ 2Cb K +
λ20 Ca2 ε
T
M
n=N +1
λ
µn 0 .
This ends the proof. 2.2. Control of ∥uhλ − uhλ0 ∥2L2 (0,T ;X ) We prove a result on the parametric sensitivity of the solutions uhλ of Eq. (5). Proposition 3. We denote β a positive real number close to zero. Then, the mapping defined by:
β
2Cp
, +∞
λ
−→
L2 (0, T ; X )
→
uhλ
is locally Lipschitz. Moreover, at a given neighborhood of λ0 , we have the following inequality:
h |λ − λ0 |2 u − uh 22 . ≤ B λ λ λ0 L (0,T ;X ) 0 λ0 Bλ0 :=
TCa2 K λ0 4Cb β
[exp(4Cb KT ) − 1],
Proof. We denote w h (t ) = uhλ − uhλ0 (t ), which verifies the following weak formulation:
d dt
wh , v
X
+ b w h (t ), uhλ (t ), v + b uhλ0 , wh (t ), v + λa w h (t ), v + (λ − λ0 ) a(uhλ0 (t ), v) = 0.
We replace v by w h (t ) in the preceding equation, then we get: 1 d 2 dt
w h 2 = −b wh (t ), uh (t ), wh (t ) − b uh (t ), wh (t ), wh (t ) λ λ0 X − λa(w h (t ), wh (t )) − (λ − λ0 ) a(uhλ0 (t ), wh (t )).
We use the properties of the two forms a and b, and we apply Young’s inequality, then we get the following three inequalities:
2 −b w h (t ), uhλ (t ), wh (t ) ≤ Cb K w h (t )X , 2 −b uhλ0 (t ), wh (t ), wh (t ) ≤ Cb K w h (t )X ,
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and,
2 γ 2 C 2K 2 −λa(w h (t ), wh (t )) − (λ − λ0 ) a(uhλ0 (t ), wh (t )) ≤ −λCp w h (t )X + w h (t )X + |λ − λ0 |2 a . 2 2γ Choosing γ = β , we can prove that −2λCp + γ < 0. Therefore, 2 2 w h 2 ≤ 4Cb K w h (t )2 + Ca K |λ − λ0 |2 . X X dt γ
d
We integrate over an interval (0, t ) ⊂ (0, T ) and we apply the Gronwall lemma, then we get: 2 h 2 w (t ) ≤ TC 2 K 2 λ0 exp(4Cb Kt ) |λ − λ0 | . a X γ λ0
λ
2.3. Completion of the proof of Theorem 1: choice of the sequence (f2 0 (N ))N =1,...,M and a priori estimate of its terms λ
Choice of the sequence (f2 0 (N ))N =1,...,M and an a priori estimate of its terms the following proposition, that will be a key point in order to give a priori estimate of the terms of the sequence We prove λ
f2 0 ( N )
N =1,...,M
.
For convenience, we suppose that d = 2 and Ω = (0, 1) × (0, 1), without any loss of generality: Proposition 4. Let (Φn )n=1,...,M be an orthonormal basis of (Vh , ∥.∥X ). We denote Φn = Φn1 , Φn2
x2
fn1 (x1 , x2 ) =
0
Then,
M
n =1
x1
Φn1 (y1 , y2 )dy1 dy2 and fn2 (x1 , x2 ) =
0
∥ ∥ < fn 2X
x2
0
x1
T
. We define fn = fn1 , fn2
T
:
Φn2 (y1 , y2 )dy1 dy2 .
0
1 . 2
Proof. For i = 1, 2: fni
(x1 , x2 ) =
Φni
, 1[0,x1 ]×[0,x2 ]
, X
2 then fni X =
1
1
i Φ , 1[0,x n
0
0
2 dx1 dx2 .
1 ]×[0,x2 ] X
2 M 2 1 1 Therefore, n=1 fni X < 0 0 1[0,x1 ]×[0,x2 ] X dx1 dx2 , which conclude to the result.
Completion of the proof of Theorem 1
h 2 2 u − uˆ λ,λ 22 ≤ 2 uhλ0 − uˆ λ0 L2 (0,T ;X ) + 2 uhλ − uhλ0 − ΠΦN λ0 uhλ − uhλ0 2 λ 0 L (0,T ;X ) L (0,T ;X ) 2 N h + 2 ΠΦ λ0 uλ − uhλ0 − uˆ λ,λ0 − uˆ λ0 2 . L (0,T ;X )
2
– The squared POD-Galerkin error uhλ0 − uˆ λ0
L2 (0,T ;X )
λ
is estimated by f1 0 (N ), based on Proposition 2.
2 2 h λ0 h − ΠΦN λ0 uhλ − uhλ0 2 = M . n=N +1 uλ − uλ0 , Φn L (0,T ;X ) X L2 (0,T ;R) 2 2 M h λ0 h – The Galerkin error Π N λ0 uhλ − uhλ0 − uˆ λ,λ0 − uˆ λ0 is controlled by u − u , Φ ; n λ λ0 n=N +1 Φ L2 (0,T ;X ) X L2 (0,T ;R) This is shown easily by reducing the semi-discrete model describing the evolution of uhλ − uhλ0 (t ).
– uhλ − uhλ0
Therefore, the parametric squared POD-Galerkin error is essentially controlled by the remainder λ
M
n=N +1
∥(uhλ −
uhλ0 , Φn 0 )X ∥2L2 (0,T ;R) . Then, based on the previous Proposition 4, a way to study the decrease rate of this remainder will λ
be by considering the remainder of the primitives sum of the reference POD modes Φn 0 . This is shown simply by applying successively the Green formula and the Cauchy–Schwarz inequality to each one of the orthogonal projection coefficients λ
(uhλ − uhλ0 )(t ), Φn 0
. Therefore,
X
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N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530 M h 2 uλ − uhλ0 , Φnλ0 X 2
L (T,R)
n=N +1
λ
where f2 0 (N ) =
M
n =N +1
is controlled by
Bλ0 |λ − λ0 |2
λ0
λ
f2 0 (N ),
2 λ0 fn . X
This ends the proof of Theorem 1. Heuristic result: precision on the decrease rate of the squared ROM–POD error with respect to the solutions regularity A precision on the decrease rate of the parametric squared ROM–POD error could be obtained with respect to the solutions regularity. In fact, we prove the following a priori estimate: Proposition 5. Under the following restrictive condition: uhλ − uhλ0 ∈ L2 (0, T ; [H m (Ω )]2 ), an a priori estimate of the squared ROM–POD error is given by:
h Bλ |λ − λ0 |2 λ u − uˆ λ,λ 22 ≤ f1 0 ( N ) + c ( N ) 0 λ 0 L (0,T ;X ) λ0 where c (N ) is of the order of
1 , Nm
(9)
when N becomes close to M.
Proof. By applying successively the Green formula and the Cauchy–Schwarz inequality and by repeating this step m-times
λ
to each (uhλ − uhλ0 )(t ), Φn 0
, we prove that
M
n =N +1
X
2 h λ uλ − uhλ0 , Φn 0 2 X
the m-iterated primitives sum of the reference POD modes, which conclude to the result.
L (0,T ;R)
is finally controlled by the remainder of
3. Application of Theorem 1 to the Navier–Stokes equations for a 2D incompressible fluid flow We place now our problem in the particular case of the semi-discrete Navier–Stokes equations for a 2D incompressible fluid flow. Then, we denote X = [L2 (Ω )]2 , V = v ∈ [H01 (Ω )]2 , div v = 0 where Ω is an open bounded set of R2 , which is regular and connected. Vh is a subspace of V of dimension M. Let u0 be given in V and f be given in L2loc ([0, +∞[, X ). These equations are given by their weak formulation:
h ∂ uλ , v h + b uh (t ), uh (t ), v h = −λa uh (t ), v h + F (v h ) ∀v h ∈ V t h λ λ λ ∂t X h h h h uλ (0), v X = u0 , v X ∀v ∈ Vh .
(10)
– ∀v1 , v2 ∈ V , a (v1 , v2 ) = (∇v1 , ∇v2 )[L2 (Ω )]4 . – ∀v1 , v2 , v3 ∈ V , b (v1 , v2 , v3 ) = (div (v1 ⊗ v2 ), v3 )X . – ∀v ∈ V , Ft (v) = (f (t ), v)X . We recall that the parameter of these equations is the Reynolds number described by the variation of the viscosity Uh . λ We mention also, that under the above hypothesis given on u0 and f , one can prove the existence and uniqueness of a global solution for the complete Navier–Stokes equations, which verify:
λ ∈ R+∗ , for given length and velocity of reference: Re =
uλ ∈ C 0 ([0, +∞[, V ) ∩ L2loc [0, +∞[, [H 2 (Ω )]2 ∩ V
,
duλ dt
∈ L2loc ([0, +∞[, X ),
[20].
It is obvious that for this particular case of the incompressible Navier–Stokes equations with homogeneous boundary conditions, the procedure we did in the previous section could be repeated exactly in the same way as for the quasi-linear equation. Therefore, the a priori estimate (4) of Theorem 1 and the heuristic estimate (9) of Proposition 5 are valid. 4. Numerical test 4.1. Flow configuration In this section, we present a numerical test to our theoretical estimate (4), for a fixed POD modes number N and by varying only the viscosity λ. Our study configuration is for an unsteady fluid flow in a channel, around a circular cylinder (see Fig. 1). The inlet condition is a flat fluid flow. We impose an outlet condition, and symmetry conditions on the two parallel walls (Γ1 and Γ2 ) of the channel.
N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
529
Fig. 1. Flow configuration.
Fig. 2. Evolution of ln uhλ − uˆ λ,λ0 L2 (T,X ) with respect to ln(|λ − λ0 |) (dotted line) and theoretical error (green line).
4.2. Numerical results
For a fixed POD modes number N, we plot the logarithm of the parametric ROM–POD error uhλ − uˆ λ,λ0 L2 (0,T ;X ) , as a function of log(|λ − λ0 |). Indeed, the reference POD basis is associated with a viscosity λ0 which corresponds to a Reynolds number equal to 125. And, we varied the viscosity λ in order to have computations associated with Reynolds numbers varying in the interval [70, 180]. We obtain the dotted line shown in Fig. 2. In the same figure, we plot the logarithm of the error obtained in theory. We get the green line. We retrieve then, the linear dependency of the ROM–POD error with respect to viscosity evolution. 5. Conclusion and perspectives Conclusion In this work, we have established a mathematical result on the sensitivity of a reduced order model of a quasi-linear parabolic equation, by a reference POD basis. We have proved the linear dependency of the ROM–POD error with respect to the parametric evolution. We have obtained also a priori estimate of the decrease rate of the squared ROM–POD error with respect to the POD modes number N. This rate depends on the one of the remainder of the primitives series of the reference POD basis vectors. An accuracy on this rate is strongly related to the solutions regularity: for a regularity of type uhλ − uhλ0 ∈ L2 (0, T ; H m ), it is a decrease rate of which the asymptotic behavior is similar to N1m . The applicability of these results was shown in the case of the Navier–Stokes equations for a 2D incompressible fluid flow. A numerical test was done for an unsteady fluid flow in a channel around a circular cylinder, in order to validate the linear dependency of the ROM–POD 1 . error with respect to Re Perspectives There are many perspectives of our previous work concerned about studying the sensitivity of a reduced order model by a reference POD basis. The first one is to apply our result in order to give validations of interpolation techniques (using the Lie group theory) of reduced models available each one in a confidence region around a reference parameter. Furthermore, we hope to obtain sharper estimates of the terms corresponding to Theorem 1, without supposing supplementary conditions on the solutions regularity: an idea will be to introduce the Meyer norm of a POD basis.
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Besides all this work, we plan to improve our theoretical result by formulating our problem in a different way. So, we propose to study the sensitivity of a reduced order model by a POD associated this time with a reference solution uhλ0 and ∂ uh
∂ uh
with its parametric derivative ∂λλ (λ0 ) (we suppose ∂λλ (λ0 ) ∈ L2 (0, T ; X )). Then, besides the previous works concerned only about numerical computations of the parametric sensitivity of a solution [21,22], it will be important to give mathematical contributions in order to be able to determine a confidence region of a reduced order model by an expanded POD basis. References [1] C. Allery, C. Béghein, A. Hamdouni, Applying proper orthogonal decomposition to the computation of particle dispersion in a two-dimensional ventilated cavity, Commun. Nonlinear Sci. Numer. Simul. 10 (8) (2005) 907–920. [2] C. Allery, C. Béghein, A. Hamdouni, Investigation of particle dispersion by a ROM POD approach, Internat. Appl. Mech. 44 (1) (2008) 133–142. [3] K. Kunisch, S. Volkwein, Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (2) (1999) 345–371. [4] H.V. Ly, H.T. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor, in: Center for Research in Scientific Computation, North Carolina State University, 1998. [5] J.L. Lumley, The Structure of inhomogeneous turbulent flows, in: A.M. Yaglom, V.I. Tatarski (Eds.), Atmospheric Turbulence and Radio Wave Propagation, 1967, pp. 166–178. [6] D. Amsallem, J. Cortial, K. Carlberg, C. Farhat, A method for interpolation on manifolds structural dynamics reduced-order models, Internat. J. Numer. Methods Eng. 80 (2009) 1241–1258. [7] D. Amsallem, C. Farhat, An interpolation method for adapting reduced order models and application to aeroelasticity, Amer. Inst. Aeronaut. Astronaut. 46 (7) (2008) 1803–1813. [8] D. Amsallem, J. Cortial, C. Farhat, On-demand CFD-based aeroelastic predictions using a database of reduced-order bases and models, in: 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 5–8 January 2009, AIAA Paper 2009-800. [9] G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a Posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng. 15 (3) (2008) 229–275. [10] M.A. Grepl, A.T. Patera, A Posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, M2AN Math. Model. Numer. Anal. 39 (1) (2005) 157–181. [11] A. Manzoni, A. Quarteroni, G. Rozza, Computational reduction for parametrized PDEs: strategies and applications, Milan J. Math. 80 (2) (2012) 283–309. [12] N. Nguyen, G. Rozza, P. Huynh, A. Patera, Reduced basis approximation and a Posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation, in: Large-Scale Inverse Problems and Quantification of Uncertainty, vol. 8, 2010, pp. 151–178. [13] J. Eftang, A. Patera, E. Ronquist, An hp certified reduced basis method for parametrized elliptic partial differential equations, SIAM J. Sci. Comput. 32 (6) (2010) 3170–3200. [14] T. Lassila, G. Rozza, Parametric free-form shape design with PDE models and reduced basis method, Comput. Methods Appl. Mech. Eng. 199 (2010) 1583–1592. [15] B. Haasdonk, Convergence rates of the POD-Greedy method, M2AN Math. Model. Numer. Anal. 47 (2013) 859–873. [16] A. Quarteroni, G. Rozza, A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations in industrial applications, J. Math. Ind. 1 (3) (2011). [17] M. Bergmann, C.-H. Bruneau, A. Iollo, Enablers for robust POD models, J. Comput. Phys. 228 (2) (2009) 516–538. [18] Filippo Terragni, José M. Vega, On the use of POD-based ROMs to analyse bifurcations in some dissipative systems, Physica D 241 (2012) 1393–1405. [19] Filippo Terragni, Eusebio Valero, José M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, SIAM J. Sci. Comput. 33 (6) (2011) 3538–3561. [20] F. Boyer, P. Fabrie, Eléments d’Analyse pour l’Étude de Quelques Modèles d’Écoulements de Fluides Visqueux Incompressibles, Springer, 2006. [21] A. Hay, I. Akhtar, J.T. Borggaard, On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions, Internat. J. Numer. Methods Fluids 68 (2012) 122–134. [22] A. Hay, J. Borggaard, D. Pelletier, Improved low-order modelling from sensitivity analysis of the proper orthogonal decomposition, J. Fluid Mech. 629 (2009) 41–72.
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Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible fluid flow N. Akkari a,b,∗ , A. Hamdouni a,∗∗ , E. Liberge a , M. Jazar b a
University of La Rochelle, LaSIE-Laboratory of the Engineering Sciences for the Environment, Avenue Michel Crépeau 17042 La Rochelle Cedex 1, France b Lebanese University, LaMA-Laboratory of Mathematics and Applications, B.P 826 Tripoli, Lebanon
article
info
Article history: Received 4 July 2013 Received in revised form 14 November 2013 Keywords: ROM–POD Sensitivity Parametric evolution POD modes number Error estimates Navier–Stokes equations
abstract In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier–Stokes equations for a 2D incompressible fluid flow. We present a numerical test of our theoretical result, for an unsteady fluid flow in a channel around a circular cylinder. © 2013 Elsevier B.V. All rights reserved.
1. Introduction and main results 1.1. Statement of the problem Boundary problems intervene widely in applications in fluid mechanics such as stability study, inverse problems, and optimization problems. There is also the optimal control problem by variation of the fluid flows characteristic parameters [1–4]. In fact, this problem is reduced to solve the Navier–Stokes equations for different parametric values and to minimize a given functional with respect to these values, so that one can reach a target situation. But the resolution cost of this optimization problem at each Reynolds number is very high from two points of view: PC storage capacity and time for computations. To solve this problem in practice, we reduce the degrees of freedom of the Navier–Stokes system. A reduced order model is then used to predict the behavior of fluid flows at different parametric values. In particular, we are interested in the models reduction by projection, the POD technique (Proper Orthogonal Decomposition) [5]. Given X a Hilbert space, and (u(t ))t a family in X living on a time interval (0, T ) and integrable in
∗ Corresponding author at: University of La Rochelle, LaSIE-Laboratory of the Engineering Sciences for the Environment, Avenue Michel Crépeau 17042 La Rochelle Cedex 1, France. ∗∗ Corresponding author. E-mail addresses: [email protected] (N. Akkari), [email protected] (A. Hamdouni), [email protected] (E. Liberge), [email protected] (M. Jazar). 0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.11.025
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523
the sense of L2 (0, T ; X ), it is about to construct a projection subspace spanned by a basis Φ and which minimizes the error obtained a posteriori by the orthogonal projection of u on a subspace of X of finite dimension N:
u − Π Φ u22
L (0,T ;X )
N
2 ≤ u − ΠNΨ uL2 (0,T ;X ) ,
∀N , and ∀Ψ an orthonormal Hilbert basis of X .
This minimization problem is equivalent to the eigenvalues problem of the compact, selfadjoint and positive operator
R, defined as follows: R Φ = µΦ , where
R:
X
ϕ
→
X
Rϕ =
→
1 T
T
(u(t ), ϕ)X u(t )dt . 0
One can verify that the optimal error by POD is equal to the remainder of the POD-eigenvalues series of R: 1 T
u − Π Φ u22 N
L (0,T ;X )
=
+∞
µn .
n =N +1
The problem that appears naturally is to find a way to control the parametric evolution in a reduced order model by POD. The previous works concerning this problem are numerical algorithms that are supposed to build a reduced order model by POD, adapted to changes of the characteristics of a fluid flow. Amsallem et al. [6–8], build a geometric interpolation algorithm of POD basis at different parametric values. There are also other techniques based on the iterative algorithm of Greedy [9,10], that enriches a POD basis computed at a given parametric value, by evaluating the errors obtained a posteriori by the orthogonal projection on this basis of solutions associated with other parametric values [11–17]. Besides these works, the prediction of a validity domain of a reduced order model by a POD associated with only one parameter, is becoming crucial. This could lead from the one hand to interpolate reduced order models available each one in a confidence region around a characteristic parameter, and from the other hand to initialize the Greedy algorithm. Terragni and Vega [18] presented an argument to use the POD approximation at a given parametric value to construct the bifurcation diagram of dissipative systems, which is not the aim of the present paper. We are interested then, in mathematical contributions concerning the study of the sensitivity of a reduced order model by a reference POD basis. More precisely, the mathematical formulation of the problem is given as follows: 1.2. Mathematical formulation of the problem Let us consider a problem describing the evolution of a parametric solution uλ (λ ∈ R+∗ ) in a Hilbert space X . V denotes a subspace of X and Vh a subspace of V of dimension M:
∂ uλ = Aλ (uλ (t )) ∂t u (0) = u . λ 0
(1)
From now on, we are interested in the ordinary differential equation of dimension M obtained by a Galerkin projection of the initial complete problem (1) on Vh :
h ∂ uλ , v h = A (uh (t )), v h ∀v h ∈ V λ λ h ∂t h h h h uλ (0), v = u0 , v ∀v ∈ Vh .
(2)
A solution uhλ0 of this equation associated with a parameter λ0 is computed once and for all. A POD basis Φ λ0 =
λ Φn 0
n=1,...,M
of dimension M in X , associated with uhλ0 on a time interval (0, T ), will lead to construct a ROM–POD
describing the evolution of an approximation uˆ λ,λ0 of uhλ , in a subspace of dimension N very small compared to M. We λ the POD eigenvalues sequence associated to the POD basis. denote µn 0 n=1,...,M
∀t ∈ (0, T ) and ∀x ∈ X : uˆ λ,λ0 (t , x) =
N n=1
0 aλ,λ (t )Φnλ0 (x), n
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N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
λ,λ0
where ∀n = 1, . . . , N , an
(t ) is the solution of:
λ,λ0 dan
= Aλ (ˆuλ,λ0 (t )), Φnλ0 λ,λ0 an (0) = u0 , Φnλ0 .
(3)
dt
The question which arises naturally is the following: To what extent uˆ λ,λ0 remains an accurate approximation to uhλ ? Then, the main problem is to give a mathematical criteria in order to determine a confidence region for a reference POD basis. It appears also the control problem of the dimension of the reduced order model, so we can improve its performance by increasing the retaining POD modes while constructing the ROM, without any loss of the reduction concept. It is worth noting that, F. Terragni, E. Valero and J.M. Vega developed in [19] a numerical adaptive method to accelerate time dependent numerical solvers of PDEs, by the use of error estimates to predict the required number of POD modes in order to preserve stability of a reduced order model by POD in connection with high-order modes truncation. Also, this is not the problem treated in this paper. 1.3. Main result A result we are proposing for a particular class of evolution problems, is a priori estimate of the squared error in the space L2 (0, T ; X ), obtained by a Galerkin approach for uhλ by a reference POD basis Φ λ0 :
λ
Formal result 1. There exist two decreasing sequences: f1 0 (N )
N =1,...,M
λ
and f2 0 (N )
N =1,...,M
: such that,
h Bλ |λ − λ0 |α λ λ u − uˆ λ,λ 22 ≤ f1 0 (N ) + f2 0 (N ) 0 . λ 0 L (0,T ;X ) λ0
(4)
2
The estimate (4) shows that the parametric ROM–POD error uhλ − uˆ λ,λ0 L2 (0,T ;X ) , is controlled by a first term fλ10 (N ) which will estimate the squared POD-Galerkin error relative to the parameter λ0 , and a second term which is a function of the alpha-th power of the distance between λ and λ0 , multiplied by an fλ20 (N ) which tend to zero when N becomes close to M. This result establishes an a priori estimate of the decrease rate of the squared ROM–POD error, especially when the two parameters λ and λ0 are distant. This rate depends on the one of the sequence fλ20 (N ). Moreover, for a fixed POD modes number N, this result shows the dependency of the ROM–POD error with respect to the distance between the parameters.
1.4. Paper organization In what follows, we give the proof elements of our principal result. In Section 2, we prove Theorem 1 with α = 2 for a quasi-linear parabolic equation. In Section 3, we show that the same proof applies in particular to the Navier–Stokes equations for a 2D incompressible fluid flow. The parameter of these equations will be the Reynolds number described by the variation of the viscosity λ ∈ R+∗ , for given length and velocity of reference. In Section 4, we present a numerical test of our theoretical estimate (4) for a fixed POD modes number N and by varying only the viscosity λ. We do this for an unsteady fluid flow in a channel around a circular cylinder. The numerical simulations are done using code-Saturne. The Navier–Stokes equations are solved by the finite volume method. In Section 5, we conclude by giving some perspectives of this work. 2. Proof of the main result for a quasi-linear parabolic equation We place our problem in the case of a semi-discrete quasi-linear parabolic equation with a dissipative term: We denote X = [L2 (Ω )]d , V = [H 1 (Ω )]d where Ω is a bounded open set, connected and Lipschitz of Rd , and Vh is a subspace of V of dimension M. f is given in L2loc ([0, +∞[, X ). This equation is given by its weak formulation as follows:
h ∂ uλ , v h + b uh (t ), uh (t ), v h = −λa uh (t ), v h + F (v h ) ∀v h ∈ V t h λ λ λ ∂t X uh (0), v h = u , v h ∀v h ∈ Vh , 0 λ X X
(5)
where b, a and Ft are respectively a trilinear form, a bilinear coercive one and a linear one given by the following expressions: – ∀v1 , v2 ∈ V , a (v1 , v2 ) = (∇v1 , ∇v2 )[L2 (Ω )]d×d . – ∀v1 , v2 , v3 ∈ V , b (v1 , v2 , v3 ) = (A(v1 ).B (∇v2 ), v3 )X , where A and B are respectively linear in v1 and ∇v2 . – ∀v ∈ V , Ft (v) = (f (t ), v)X .
(.) denotes the scalar product in the corresponding space. We precise some additional notations, that could be useful to our proof.
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525
– Cp denotes the constant relative to the coercivity of the bilinear form a, which is nothing but the square of a Poincaré constant. – Ca , Cb are respectively the two norms of the forms a and b on (Vh , ∥.∥X ) and K = uhλ0 . L∞ (0,T ;X )
2.1. The study of the squared POD-Galerkin error relative to a fixed parameter λ0 We denote uˆ λ0 = uˆ λ,λ0 , for λ = λ0 . We prove a result on the decrease rate of the squared POD-Galerkin error relative to the parameter λ0 :
2 h uλ0 − uˆ λ0 2
L (0,T ;X )
.
We show first the following a priori estimate: Proposition 1. ∀t ∈ (0, T ) and ∀ε > 0, one has:
M 2 λ2 C 2 N h µλn 0 . ΠΦ λ0 uλ0 − uˆ λ0 (t ) ≤ 2Cb K + 0 a T exp ε − 2λ0 Cp + 6Cb K t X ε n=N +1
(6)
λ
Proof. We replace v h by Φn 0 in Eq. (5) (written for the parameter λ0 ) and we use the commutativity between the projector operator Π and the time derivative dtd , then we can deduce the following equality:
1 d N 2 ΠΦ λ0 uhλ0 − uˆ λ0 = −b uhλ0 − uˆ λ0 (t ), uhλ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X 2 dt
− b uˆ λ0 (t ), uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) − λ0 a uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) . We use the properties of the two forms a and b, and we apply Young’s inequality, then we get the following three inequalities:
−b
uhλ0 − uˆ λ0 (t ), uhλ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t )
≤ Cb K
2 1 2 1 h uλ0 − ΠΦN λ0 uhλ0 (t ) + ΠΦN λ0 uhλ0 − uˆ λ0 (t ) 2
2
X
X
2 + Cb K ΠΦN λ0 uhλ0 − uˆ λ0 (t ) , −b uˆ λ0 (t ), uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X 2 1 2 2 1 h N h N h ≤ Cb K uλ0 − ΠΦ λ0 uλ0 (t ) + ΠΦ λ0 uλ0 − uˆ λ0 (t ) + Cb K ΠΦN λ0 uhλ0 − uˆ λ0 (t ) , 2
2
X
X
X
and ∀ϵ > 0,
−λ0 a
uhλ0 − uˆ λ0 (t ), ΠΦN λ0 uhλ0 − uˆ λ0 (t )
≤
2 λ20 Ca2 h uλ0 − ΠΦN λ0 uhλ0 (t ) X 2ε 2 2 ε N h + ΠΦ λ0 uλ0 − uˆ λ0 (t ) − λ0 Cp ΠΦN λ0 uhλ0 − uˆ λ0 (t ) . 2
X
X
Therefore,
2 2 d N ΠΦ λ0 uhλ0 − uˆ λ0 ≤ 6Cb K + ε − 2λ0 Cp ΠΦN λ0 uhλ0 − uˆ λ0 (t ) X X dt 2 λ20 Ca2 h + 2Cb K + uλ0 − ΠΦN λ0 uhλ0 (t ) . ε
X
We integrate over an interval (0, t ) ⊂ (0, T ) and we apply the Gronwall lemma, then we get:
M 2 λ20 Ca2 N h T µλn 0 . ΠΦ λ0 uλ0 − uˆ λ0 (t ) ≤ exp 6Cb K + ε − 2λ0 Cp t 2Cb K + X ε n =N +1 This ends the proof. By choosing ε small, the preceding estimate (6) shows the competition between the quasi-linear term and the one corresponding to the coercivity of the bilinear form a. By this estimate, the only way to reduce the POD-Galerkin error
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N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
is to choose N close to M. Then, a solution to this problem could be by doing a priori estimate of the two constants Cb and Cp . This could lead to choose an artificial viscosity verifying a certain condition, so that the term in the exponential becomes negative (6). Then, under this hypothesis we prove the following proposition: Proposition 2. If we choose an artificial viscosity λsm that verifies the condition:
λsm >
6Cb K + ε 2Cp
− λ0 ,
(7)
then the squared POD-Galerkin error for the parameter λ0 will be controlled as follows:
M h λ2sm Ca2 u − uˆ λ 22 ≤ 2T 1 + 2C K + µλn 0 . b λ0 0 L (0,T ;X ) ε n =N +1
(8)
λ f1 0 ( N )
Proof.
2 h h N h u − uˆ λ 22 ≤ 2 u − Π u λ λ0 λ0 0 L (0,T ;X ) Φ 0 λ0 2
L (T,X )
2
L (T,X )
.
λ
M
– uhλ0 − Π N λ0 uhλ0 =T Φ L2 (0,T ;X )
2 + 2 ΠΦN λ0 uhλ0 − uˆ λ0 2
n =N +1
µn 0 .
2 – Thanks to (7) and to the previous proposition, Π N λ0 uhλ0 − uˆ λ0 Φ 2
L (0,T ;X )
≤ 2Cb K +
λ20 Ca2 ε
T
M
n=N +1
λ
µn 0 .
This ends the proof. 2.2. Control of ∥uhλ − uhλ0 ∥2L2 (0,T ;X ) We prove a result on the parametric sensitivity of the solutions uhλ of Eq. (5). Proposition 3. We denote β a positive real number close to zero. Then, the mapping defined by:
β
2Cp
, +∞
λ
−→
L2 (0, T ; X )
→
uhλ
is locally Lipschitz. Moreover, at a given neighborhood of λ0 , we have the following inequality:
h |λ − λ0 |2 u − uh 22 . ≤ B λ λ λ0 L (0,T ;X ) 0 λ0 Bλ0 :=
TCa2 K λ0 4Cb β
[exp(4Cb KT ) − 1],
Proof. We denote w h (t ) = uhλ − uhλ0 (t ), which verifies the following weak formulation:
d dt
wh , v
X
+ b w h (t ), uhλ (t ), v + b uhλ0 , wh (t ), v + λa w h (t ), v + (λ − λ0 ) a(uhλ0 (t ), v) = 0.
We replace v by w h (t ) in the preceding equation, then we get: 1 d 2 dt
w h 2 = −b wh (t ), uh (t ), wh (t ) − b uh (t ), wh (t ), wh (t ) λ λ0 X − λa(w h (t ), wh (t )) − (λ − λ0 ) a(uhλ0 (t ), wh (t )).
We use the properties of the two forms a and b, and we apply Young’s inequality, then we get the following three inequalities:
2 −b w h (t ), uhλ (t ), wh (t ) ≤ Cb K w h (t )X , 2 −b uhλ0 (t ), wh (t ), wh (t ) ≤ Cb K w h (t )X ,
N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
527
and,
2 γ 2 C 2K 2 −λa(w h (t ), wh (t )) − (λ − λ0 ) a(uhλ0 (t ), wh (t )) ≤ −λCp w h (t )X + w h (t )X + |λ − λ0 |2 a . 2 2γ Choosing γ = β , we can prove that −2λCp + γ < 0. Therefore, 2 2 w h 2 ≤ 4Cb K w h (t )2 + Ca K |λ − λ0 |2 . X X dt γ
d
We integrate over an interval (0, t ) ⊂ (0, T ) and we apply the Gronwall lemma, then we get: 2 h 2 w (t ) ≤ TC 2 K 2 λ0 exp(4Cb Kt ) |λ − λ0 | . a X γ λ0
λ
2.3. Completion of the proof of Theorem 1: choice of the sequence (f2 0 (N ))N =1,...,M and a priori estimate of its terms λ
Choice of the sequence (f2 0 (N ))N =1,...,M and an a priori estimate of its terms the following proposition, that will be a key point in order to give a priori estimate of the terms of the sequence We prove λ
f2 0 ( N )
N =1,...,M
.
For convenience, we suppose that d = 2 and Ω = (0, 1) × (0, 1), without any loss of generality: Proposition 4. Let (Φn )n=1,...,M be an orthonormal basis of (Vh , ∥.∥X ). We denote Φn = Φn1 , Φn2
x2
fn1 (x1 , x2 ) =
0
Then,
M
n =1
x1
Φn1 (y1 , y2 )dy1 dy2 and fn2 (x1 , x2 ) =
0
∥ ∥ < fn 2X
x2
0
x1
T
. We define fn = fn1 , fn2
T
:
Φn2 (y1 , y2 )dy1 dy2 .
0
1 . 2
Proof. For i = 1, 2: fni
(x1 , x2 ) =
Φni
, 1[0,x1 ]×[0,x2 ]
, X
2 then fni X =
1
1
i Φ , 1[0,x n
0
0
2 dx1 dx2 .
1 ]×[0,x2 ] X
2 M 2 1 1 Therefore, n=1 fni X < 0 0 1[0,x1 ]×[0,x2 ] X dx1 dx2 , which conclude to the result.
Completion of the proof of Theorem 1
h 2 2 u − uˆ λ,λ 22 ≤ 2 uhλ0 − uˆ λ0 L2 (0,T ;X ) + 2 uhλ − uhλ0 − ΠΦN λ0 uhλ − uhλ0 2 λ 0 L (0,T ;X ) L (0,T ;X ) 2 N h + 2 ΠΦ λ0 uλ − uhλ0 − uˆ λ,λ0 − uˆ λ0 2 . L (0,T ;X )
2
– The squared POD-Galerkin error uhλ0 − uˆ λ0
L2 (0,T ;X )
λ
is estimated by f1 0 (N ), based on Proposition 2.
2 2 h λ0 h − ΠΦN λ0 uhλ − uhλ0 2 = M . n=N +1 uλ − uλ0 , Φn L (0,T ;X ) X L2 (0,T ;R) 2 2 M h λ0 h – The Galerkin error Π N λ0 uhλ − uhλ0 − uˆ λ,λ0 − uˆ λ0 is controlled by u − u , Φ ; n λ λ0 n=N +1 Φ L2 (0,T ;X ) X L2 (0,T ;R) This is shown easily by reducing the semi-discrete model describing the evolution of uhλ − uhλ0 (t ).
– uhλ − uhλ0
Therefore, the parametric squared POD-Galerkin error is essentially controlled by the remainder λ
M
n=N +1
∥(uhλ −
uhλ0 , Φn 0 )X ∥2L2 (0,T ;R) . Then, based on the previous Proposition 4, a way to study the decrease rate of this remainder will λ
be by considering the remainder of the primitives sum of the reference POD modes Φn 0 . This is shown simply by applying successively the Green formula and the Cauchy–Schwarz inequality to each one of the orthogonal projection coefficients λ
(uhλ − uhλ0 )(t ), Φn 0
. Therefore,
X
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N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530 M h 2 uλ − uhλ0 , Φnλ0 X 2
L (T,R)
n=N +1
λ
where f2 0 (N ) =
M
n =N +1
is controlled by
Bλ0 |λ − λ0 |2
λ0
λ
f2 0 (N ),
2 λ0 fn . X
This ends the proof of Theorem 1. Heuristic result: precision on the decrease rate of the squared ROM–POD error with respect to the solutions regularity A precision on the decrease rate of the parametric squared ROM–POD error could be obtained with respect to the solutions regularity. In fact, we prove the following a priori estimate: Proposition 5. Under the following restrictive condition: uhλ − uhλ0 ∈ L2 (0, T ; [H m (Ω )]2 ), an a priori estimate of the squared ROM–POD error is given by:
h Bλ |λ − λ0 |2 λ u − uˆ λ,λ 22 ≤ f1 0 ( N ) + c ( N ) 0 λ 0 L (0,T ;X ) λ0 where c (N ) is of the order of
1 , Nm
(9)
when N becomes close to M.
Proof. By applying successively the Green formula and the Cauchy–Schwarz inequality and by repeating this step m-times
λ
to each (uhλ − uhλ0 )(t ), Φn 0
, we prove that
M
n =N +1
X
2 h λ uλ − uhλ0 , Φn 0 2 X
the m-iterated primitives sum of the reference POD modes, which conclude to the result.
L (0,T ;R)
is finally controlled by the remainder of
3. Application of Theorem 1 to the Navier–Stokes equations for a 2D incompressible fluid flow We place now our problem in the particular case of the semi-discrete Navier–Stokes equations for a 2D incompressible fluid flow. Then, we denote X = [L2 (Ω )]2 , V = v ∈ [H01 (Ω )]2 , div v = 0 where Ω is an open bounded set of R2 , which is regular and connected. Vh is a subspace of V of dimension M. Let u0 be given in V and f be given in L2loc ([0, +∞[, X ). These equations are given by their weak formulation:
h ∂ uλ , v h + b uh (t ), uh (t ), v h = −λa uh (t ), v h + F (v h ) ∀v h ∈ V t h λ λ λ ∂t X h h h h uλ (0), v X = u0 , v X ∀v ∈ Vh .
(10)
– ∀v1 , v2 ∈ V , a (v1 , v2 ) = (∇v1 , ∇v2 )[L2 (Ω )]4 . – ∀v1 , v2 , v3 ∈ V , b (v1 , v2 , v3 ) = (div (v1 ⊗ v2 ), v3 )X . – ∀v ∈ V , Ft (v) = (f (t ), v)X . We recall that the parameter of these equations is the Reynolds number described by the variation of the viscosity Uh . λ We mention also, that under the above hypothesis given on u0 and f , one can prove the existence and uniqueness of a global solution for the complete Navier–Stokes equations, which verify:
λ ∈ R+∗ , for given length and velocity of reference: Re =
uλ ∈ C 0 ([0, +∞[, V ) ∩ L2loc [0, +∞[, [H 2 (Ω )]2 ∩ V
,
duλ dt
∈ L2loc ([0, +∞[, X ),
[20].
It is obvious that for this particular case of the incompressible Navier–Stokes equations with homogeneous boundary conditions, the procedure we did in the previous section could be repeated exactly in the same way as for the quasi-linear equation. Therefore, the a priori estimate (4) of Theorem 1 and the heuristic estimate (9) of Proposition 5 are valid. 4. Numerical test 4.1. Flow configuration In this section, we present a numerical test to our theoretical estimate (4), for a fixed POD modes number N and by varying only the viscosity λ. Our study configuration is for an unsteady fluid flow in a channel, around a circular cylinder (see Fig. 1). The inlet condition is a flat fluid flow. We impose an outlet condition, and symmetry conditions on the two parallel walls (Γ1 and Γ2 ) of the channel.
N. Akkari et al. / Journal of Computational and Applied Mathematics 270 (2014) 522–530
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Fig. 1. Flow configuration.
Fig. 2. Evolution of ln uhλ − uˆ λ,λ0 L2 (T,X ) with respect to ln(|λ − λ0 |) (dotted line) and theoretical error (green line).
4.2. Numerical results
For a fixed POD modes number N, we plot the logarithm of the parametric ROM–POD error uhλ − uˆ λ,λ0 L2 (0,T ;X ) , as a function of log(|λ − λ0 |). Indeed, the reference POD basis is associated with a viscosity λ0 which corresponds to a Reynolds number equal to 125. And, we varied the viscosity λ in order to have computations associated with Reynolds numbers varying in the interval [70, 180]. We obtain the dotted line shown in Fig. 2. In the same figure, we plot the logarithm of the error obtained in theory. We get the green line. We retrieve then, the linear dependency of the ROM–POD error with respect to viscosity evolution. 5. Conclusion and perspectives Conclusion In this work, we have established a mathematical result on the sensitivity of a reduced order model of a quasi-linear parabolic equation, by a reference POD basis. We have proved the linear dependency of the ROM–POD error with respect to the parametric evolution. We have obtained also a priori estimate of the decrease rate of the squared ROM–POD error with respect to the POD modes number N. This rate depends on the one of the remainder of the primitives series of the reference POD basis vectors. An accuracy on this rate is strongly related to the solutions regularity: for a regularity of type uhλ − uhλ0 ∈ L2 (0, T ; H m ), it is a decrease rate of which the asymptotic behavior is similar to N1m . The applicability of these results was shown in the case of the Navier–Stokes equations for a 2D incompressible fluid flow. A numerical test was done for an unsteady fluid flow in a channel around a circular cylinder, in order to validate the linear dependency of the ROM–POD 1 . error with respect to Re Perspectives There are many perspectives of our previous work concerned about studying the sensitivity of a reduced order model by a reference POD basis. The first one is to apply our result in order to give validations of interpolation techniques (using the Lie group theory) of reduced models available each one in a confidence region around a reference parameter. Furthermore, we hope to obtain sharper estimates of the terms corresponding to Theorem 1, without supposing supplementary conditions on the solutions regularity: an idea will be to introduce the Meyer norm of a POD basis.
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Besides all this work, we plan to improve our theoretical result by formulating our problem in a different way. So, we propose to study the sensitivity of a reduced order model by a POD associated this time with a reference solution uhλ0 and ∂ uh
∂ uh
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