HAL Id: hal-03278113
https://hal.archives-ouvertes.fr/hal-03278113
Preprint submitted on 5 Jul 2021
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
BOUNDARY VALUE PROBLEMS INITIAL CONDITION IDENTIFICATION BY A WAVELET
GALERKIN METHOD
Kaïs Ammari, Souleymane Kadri Harouna
To cite this version:
Kaïs Ammari, Souleymane Kadri Harouna. BOUNDARY VALUE PROBLEMS INITIAL CONDI-
TION IDENTIFICATION BY A WAVELET GALERKIN METHOD. 2021. �hal-03278113�
IDENTIFICATION BY A WAVELET GALERKIN METHOD
KA¨IS AMMARI AND SOULEYMANE KADRI HAROUNA
Abstract. A new numerical method for the identification of initial conditions in wave propagation or heat conduction problems is constructed. The method is based on the wavelet basis Galerkin discretization of the model equations, without time discretization as in classical approches. A singular value decom- position is applied to the resulting discrete system to get explicitly its solution, and then to reconstruct the initial condition from measurements. Numerical results on toy solutions show the efficiency and the accuracy of the method.
Contents
1. Introduction 1
2. Wavelet basis and multiscale decomposition 2
2.1. Deslauriers-Dubuc Interpolating Wavelet 4
3. Wave equation 4
3.1. Wavelet Galerkin method for the wave equation 5 3.2. Wave equation Initial conditions identification 7
4. Heat equation 9
4.1. Wavelet Galerkin method for the heat equation 9
4.2. Initial temperature reconstruction 10
5. Numerical results 11
6. Conclusion 13
References 13
1. Introduction
The reconstruction and identification of initial conditions for systems involving parabolic or hyperbolic partial differential equation is an inverse problem exten- sively investigated in the recent decades [2, 11]. This question often arises in the non-destructive testing of materials, the development of the transient elastography technique, river pollution, population dynamics [2, 9], etc. In this work, we study the initial condition reconstruction from a numerical point of view.
The adopted strategy consists first to use a Galerkin spatial discretization of the model equation to transform it in an ordinary differential equation system of finite dimension. The problem lies then in the reconstruction of the initial condition of the resulting system. Moreover, for the wave and heat equations, these systems are
2010Mathematics Subject Classification. 35R30, 35K05.
Key words and phrases. control, identification, wavelet numerical discretization.
1
linear. Thus, their solutions are explicitly obtained if we know the exponential of the mass and the stiffness matrices of the used basis in the Galerkin method.
Due to their localization property both in space and frequency, wavelet bases are a good choice for the spatial discretization of partial differential equations. Indeed, the two scale relation satisfied by the wavelet generator allows to compute the mass and stiffness matrices coefficient without using numerical integration quadrature formula and this is done by solving an eigenvalue problem [1]. Furthermore, the spectrum of these matrices and their explicit optimal preconditioners are known [3].
Moreover, wavelet based method has the advantage of improving the discretization accuracy just by increasing the number of wavelet generator vanishing moment and physical boundary condition can be incorporated in the basis construction easily, see [10, 13].
Once the discretization matrices are computed, a singular value decomposition (SDV) is applied [14], and a change of basis transforms the first system into a diagonal one, for which we know the analytical expression of its solution. This avoids use of iterative procedure with constraints on the time step as in the common approaches [12]. From the solution analytical formula, to reconstruct the initial data, it is only necessary to observed the solution wavelet coefficients at a choosen time. This amounts to knowing the value of this solution at the spatial discretization grid points. These measurements are supposed to be provided in advance and then classical algorithms are used to project it onto the wavelet space [13].
Despite of the wavelet basis numerical projection error of the method, the mag- nitude of the numerical error on the reconstructed initial condition is the same as the numerical error of the Galerkin discretization on the exact solution. This error has polynomial decay that depends on the data smoothness and the wavelet basis number of vanishing moments. In general, we have an error in O(N−s) whereN denotes the number of grid points (the number of degrees of freedom) and s > 0 is at most the wavelet basissmoothness. Then, the accuracy and efficiency of the proposed method have to be compared the performance of the algorithm proposed in [6], where a time sampling is used instead of spatial discretization and with an initial temperature reconstruction error also aroundO(N−s).
The outline of this paper is as follows. In Section 2, we briefly introduce the wavelet basis construction and then we specify their numerical properties useful to prove the proposed method error estimates. Section 3 and Section 4 are dedicated to the initiale condition identification in the case of the wave equation and the heat equation, respectively. After proving the a priori error estimates of the wavelet- based Galerkin method, we show the error bound for our reconstruction algorithm.
Section 5 shows numerical results that support the theoretical results.
2. Wavelet basis and multiscale decomposition
In this section we briefly introduce the wavelet basis using the multiresolution analysis formalism as in the reference books [5]. A multiresolution analysis ofL2(R) is a sequence of closed nested subspaces{Vj}j∈Z that satisfy:
(i) Vj⊂Vj+1,∩j∈ZVj ={0} and∪j∈ZVj=L2(R).
(ii) ∀f ∈L2(R) : f(x)∈Vj⇔f(2x)∈Vj+1 andf(x)∈V0⇔f(x−k)∈V0. (iii) There exists a functionφsuch that{φ(.−k)}k∈Z is a Riesz basis ofV0. The functionφis called scaling function and the refinement relation (ii) leads to:
(1) φ(x) =X
k∈Z
hk
√
2φ(2x−k), hk ∈R.
The mask {hk}k∈Z is called scaling function filter and its support coincides with the support ofφ[4]. The parameterj defines the resolution. Definition (i) allows to write:
(2) Vj+1=Vj⊕Wj,
whereWj is just a topological complement, which is not unique. The wavelet basis is defined as a basis ofWj and using (ii), the wavelet generatorψ∈W0 is defined by its two-scale relation inV1:
(3) ψ(x) =X
k∈Z
gk
√
2φ(2x−k), gk∈R.
Similarly, the mask{gk}k∈Z is called wavelet filter. Settingφj,k= 2j/2φ(2jx−k) andψj,k= 2j/2ψ(2jx−k), we have:
Vj= span{φj,k : k∈Z} and Wj = span{ψj,k : k∈Z}.
Orthogonal wavelet basis consists in definingWj as the orthogonal complement of Vj:
(4) Vj+1=Vj⊕Wj, Wj =Vj+1∩(Vj)⊥. Generally, the spaceWj is defined as:
(5) Vj+1=Vj⊕Wj, Wj =Vj+1∩( ˜Vj)⊥.
where{V˜j}j∈Z is another multiresolution analysis ofL2(R) with scaling function ˜φ and wavelet ˜ψ. In this case, (Vj,V˜j)j∈Zis referred as a biorthogonal multiresolution analysis ofL2(R) [4, 5].
The multiscale projection of a functionf ∈L2(R) onto Vj and Wj are defined respectively by:
(6) Pj(f) =X
k∈Z
⟨f,φ˜j,k⟩φj,k and Qj(f) =X
k∈Z
⟨f,ψ˜j,k⟩ψj,k.
From (4) and (5), we haveQj(f) =Pj+1(f)− Pj(f) and the multiscale decompo- sition off ∈L2(R) reads:
f =Pj(f) +X
ℓ≥j
Qℓ(f).
Moreover, forregularscaling function generators, iff ∈Hs(R), we have the follow- ing Jackson and Bernstein inequalities [4, 5]:
(7)
∥Pj(f)−f∥L2(0,1)≤C2−js∥f∥Hs(0,1) and ∥Pj(f)∥Hs(0,1)≤C2js∥Pj(f)∥L2(0,1), s >0.
The wavelet basis construction extends easily to higher dimension using tensor product of the one-dimensional wavelet basis [5]. In practice we use the construction on the interval [0,1] or the hypercube [0,1]d, d >1 satisfying physical boundary conditions [10, 13]. Then, compactly support generators are used and accordingly
a minimum resolution jmin > 0 is set so as not to have function support outside [0,1] due to the dilatation factor, see [13] and reference therein.
Wavelet bases have many numerical properties that make them useful in practice.
For example, interpolation properties can be imposed in the basis construction [5, 7, 8]. In particular, this problem occurs in the initialization step of the fast wavelet transform algorithm and if not, we have to use quadrature formula to get the solution scaling function coefficients [10, 13]. That is, to compute the solution projection ontoVj.
2.1. Deslauriers-Dubuc Interpolating Wavelet. Deslauriers and Dubuc [7]
and Dubuc [8] introduced an interpolation scheme that constructs a function on R from its values {f(k)}k∈Z. The basic characteristics of interpolating wavelets basis require that the scaling function generatorφsatisfies the following condition [7]:
(8) φ(k) =
1, k= 0, 0, k̸= 0.
Relation (8) is useful in terms of the approximation, its allows to represent any functionf as a linear combination of the basis functions at levelj:
(9) f(x) =X
k∈Z
ckφj,k(x).
Evaluating the function at a dyadic grid pointx= 2−jmyields:
(10) f(2−jm) =X
k∈Z
ck2j/2φ(m−k) = 2j/2cm. The two scale relation satisfied byφis given by
(11) φ(x) =X
k∈Z
φ(k/2)√
2φ(2x−k)
which means that the filter coefficients{hk}k∈Zare equal to the half valuesφ(k/2).
The main advantage of an interpolating basis is that the multiscale projectionPj(f) is known from the values off at grid points. Then, using the Deslauriers-Dubuc wavelet is an issue that allows us to consider the measurements data as scaling function coefficients in the reconstruction procedure. However, in general these coefficients are approximated using quadrature formulas and this is what we will do in the following without loss of generality.
3. Wave equation
In this section we apply the wavelet based finite element method to recover the initial conditions in the wave propagation problem. For the sake of easy understand- ing, the approach is presented in the one spatial dimension case. The generalization to higher dimensions is straightforward.
The initial-boundary value problem for the one-dimensional waves propagation, both traveling and standing, is expressed as:
(12)
∂t2u(t, x) =ν∂x2u(t, x),
u(0, x) =u0(x) and ∂tu(0, x) =u1(x),
wherex∈[0,1] andt∈[0, T]. To complete (12), the wave is supposed to be fixed at boundaries which implies homogeneous Dirichlet boundary condition: u(t,0) = u(t,1) = 0, see [12].
To solve numerically the problem (12), Fourier’s basis is very practical because it allows to compute explicitly the solution. However, the Fourier basis is not com- pactly supported and the fast Fourier transform algorithm exists only for periodic function. In this paper, to compute the numerical solution of (12), we will use a wavelet based Galerkin method in space and we won’t need a time discretization as in the common approaches [12].
3.1. Wavelet Galerkin method for the wave equation. The wave propaga- tion model (12) is a linear hyperbolic equation. In the literature, there exists a lot of work on Galerkin finite element method for the discretization of hyperbolic equations, see [12] and references therein. We recall in this section some well known results [12] in the particular case of wavelet based method which are necessary to show the accuracy and the convergence of our method. We assume that the problem initial data and the wavelet basis have the necessary requiredregularity to satisfy the different estimates and then we start by presenting briefly the wavelet based Galerkin method.
According to the boundary conditions, the considered approximation spaces (Vj)j≥jmin provide a multi-resolution analysis of H01(0,1). The wavelet based Galerkin method applied to (12) consists in looking for its solutionuj∈Vj and in the following discrete form:
(13) uj(t, x) =
Nj
X
k=1
⟨u,ψ˜j,k⟩ψj,k(x) =
Nj
X
k=1
dj,k(t)ψj,k(x).
Then, integration by part and the boundary condition lead to:
(14)
Nj
X
k=1
d′′j,k(t)
Z 1 0
ψj,k(x)ψj,m(x)dx+νdj,k(t) Z 1
0
ψ′j,k(x)ψ′j,m(x)dx
= 0, form= 1, . . . , Nj. Thus, the coefficients [dj,k] solve the following linear system:
(15) Aj
d′′j,k(t)
+Rj[dj,k(t)] = 0,
where Aj and Rj denote respectively the mass matrix and the stiffness matrix of the considered wavelet basis:
(16) [Aj]k,m= Z 1
0
ψj,k(x)ψj,m(x)dx and [Rj]k,m=ν Z 1
0
ψ′j,k(x)ψ′j,m(x)dx.
These matrices are symmetric and positive, in practice one can explicitly com- pute their elements [1] and optimal diagonal preconditioner exits for these matrices [3].
The main characteristic of the Galerkin finite element method is that a priori error estimate exists which allows to prove its convergence. Precisely, in the case of wavelet based method we have the following proposition:
Proposition 1. Let uj and u be solutions of (14) and (12), respectively. If the initial conditions u0(x) and u1(x) and the wavelet basis are regular enough, then
we have:
(17) ∥uj−u∥L2(0,1)≤C2−js, for allj≥jmin ands >0.
Proof. Let ej(t, x) = u(t, x)−uj(t, x) be the discretization error and we assume that the two solutions areregular enoughto allow the following estimations:
∥ej∥L2(0,1)=∥u−uj∥L2(0,1)≤ ∥Pj(u)−uj∥L2(0,1)+∥Pj(u)−u∥L2(0,1). (18)
Then, from the Jackson inequality (7):
∥Pj(u)−u∥L2(0,1)≤C2−js∥u∥Hs(0,1),
we see that the second term in the right part of (18) is bounded byC2−js, if the solution u ∈ Hs(0,1). For the first term, we will use classical a posteriori error estimates for hyperbolic equations [12]. Indeed, let θj = uj− Pj(u) and taking
∂t[uj− Pj(u)] =∂tθj as test function in (12) and (14), we obtain:
⟨∂t2u−ν∂2xu, ∂tθj⟩=⟨∂t2uj−ν∂x2uj, ∂tθj⟩, and this rewrites in:
⟨∂t2[u− Pj(u)]−ν∂x2[u− Pj(u)], ∂tθj⟩=⟨∂t2θj−ν∂x2θj, ∂tθj⟩.
Integration by part gives:
1 2
d dt
∥∂tθj∥2L2(0,1)+ν∥∂xθj∥2L2(0,1)
=⟨∂t2[u−Pj(u)], ∂tθj⟩+ν⟨∂x[u−Pj(u)], ∂x∂tθj⟩ (19) ≤ ∥∂t2[u−Pj(u)]∥L2(0,1)∥∂tθj∥L2(0,1)+ν∥∂x[u−Pj(u)]∥L2(0,1)∥∂x∂tθj∥L2(0,1). After integration int, we obtain
∥∂(20)tθj(t)∥2L2(0,1)+ν∥∂xθj(t)∥2L2(0,1) ≤ ∥∂tθj(0)∥2L2(0,1)+ν∥∂xθj(0)∥2L2(0,1)
+ 2
Z t 0
∥∂2t[u− Pj(u)]∥L2(0,1)∥∂tθj∥L2(0,1)
+ 2
Z t 0
ν∥∂x[u− Pj(u)]∥L2(0,1)∥∂x∂tθj∥L2(0,1). Since
2 Z t
0
∥∂t2[u− Pj(u)]∥L2(0,1)∥∂tθj∥L2(0,1)≤ 2
Z t 0
∥∂t2[u− Pj(u)]∥L2(0,1)
2 +1
2
max
h∈[0,t]∥∂tθj(h)∥L2(0,1)
2
and
2 Z t
0
ν∥∂x[u− Pj(u)]∥L2(0,1)∥∂x∂tθj∥L2(0,1)≤ 2
Z t 0
ν∥∂x[u− Pj(u)]∥L2(0,1)
2 +ν
2
max
h∈[0,t]∥∂x∂tθj(h)∥L2(0,1)
2 , we deduce that:
1 2
max
h∈[0,t]∥∂tθj(h)∥2L2(0,1)+ν max
h∈[0,t]∥∂x∂tθj(h)∥2L2(0,1)
≤
∥∂tθj(0)∥2L2(0,1)+ν∥∂xθj(0)∥2L2(0,1)+ 2 Z t
0
∥∂t2[u− Pj(u)]∥L2(0,1)
2 + 2
Z t 0
ν∥∂x[u− Pj(u)]∥L2(0,1)
2 . From (20) we get:
∥∂tθj(t)∥2L2(0,1)+ν∥∂xθj(t)∥2L2(0,1)≤2∥∂tθj(0)∥2L2(0,1)+ 2ν∥∂xθj(0)∥2L2(0,1)+ 4
Z t 0
∥∂t2[u− Pj(u)]∥L2(0,1)
2 + 4
Z t 0
ν∥∂x[u− Pj(u)]∥L2(0,1)
2 . Again using Jackson estimation (7), we get:
∥∂tθj(t)∥2L2(0,1)+ν∥∂xθj(t)∥2L2(0,1)≤ 2∥∂tθj(0)∥2L2(0,1)+2ν∥∂xθj(0)∥2L2(0,1)+C2−2js
"
Z t 0
∥∂t2u∥Hs(0,1)
2 +
Z t 0
∥u∥Hs+1
2# . Poincar´e and the previous estimates give
(21)
∥θj(t)∥2L2(0,1)≤C∥∂xθj(t)∥2L2(0,1)≤2∥∂tθj(0)∥2L2(0,1)+2ν∥∂xθj(0)∥2L2(0,1)+C2−2js. Finally, we have:
(22) ∥ej(t)∥2L2(0,1)≤C
∥∂tθj(0)∥2L2(0,1)+ν∥∂xθj(0)∥2L2(0,1)+ 2−2js . and
(23) ∥∂tej(t)∥2L2(0,1)≤C
∥∂tθj(0)∥2L2(0,1)+ν∥∂xθj(0)∥2L2(0,1)+ 2−2js . To get the desired order of errorsej and ∂tej in O(2−js), then it suffices to take uj(0, x) =Pj(u(0, x)) and∂tuj(0, x) =Pj(∂tu(0, x)). This will be assumed in the
sequel. □
Our goal is now to build a numerical method to reconstruct the initial conditions associated to (14) and with the same accuracy order as on the exact solution given by Proposition 1.
3.2. Wave equation Initial conditions identification. One of the problems encountered in the wave propagation control is to know the origin. We present in this section a method to identify the wavelet coefficients of uj(0, x) =Pj(u(0, x)) and ∂tuj(0, x) = Pj(∂tu(0, x)), where uj is the numerical solution of (14). The method is based on the exact time integration of (15) and supposes that we can mesure the wave amplitude in two different times 0< t1< t2.
Most often, the numerical resolution of (15) required a time discretization schemes [12]. Conversely, we propose here to use a singular value decomposition (SVD) of the stiffness matrix [14], then to get explicitly the solution as in Fourier domain.
To the best of our knowledge, there is no method in the literature that uses this approach well suited to the homogeneous linear model. In the general case of
biorthogonal wavelet basis whereAj is not the identity matrix, the first step of the method is to transform the system (14) in
(24)
d′′j,k(t)
+A−1j Rj[dj,k(t)] = 0.
Then, we apply the SVD to the matrix Mj =A−1j Rj to obtain Dj =C−1j MjCj, where Dj is a diagonal matrix of eigen values of Mj and Cj is a change of basis matrix. Setting :
(25) d¯j,k=Cj−1[dj,k]Cj,
where [dj,k] is the vector of unknown coefficients, it easy to see that:
(26) d¯′′j,k(t)
+Dj
d¯j,k(t)
= 0.
According to the differential equations theory, the solution of (26) is given by:
(27) d¯j,k(t) = cos(tp
Dj) ¯dj,k(0) + 1 pDj
sin(tp
Dj) ¯d′j,k(0), where p
Dj denotes the diagonal matrix constituted by the square root of the diagonal elements ofDj and √1
Dj its inverse: √1
Dj := [p Dj]−1.
Starting from (27), to identify the initial conditions coefficients ¯dj,k(0) and d¯′j,k(0), it suffices to know two different values of ¯dj,k(t). Precisely, suppose that we have two solutions of (27) denoted [ ¯dj,k(t1)]̸= [ ¯dj,k(t2)]:
d¯j,k(t1) = cos(t1p
Dj) ¯dj,k(0) + 1 pDj
sin(t1p
Dj) ¯d′j,k(0) =A1d¯j,k(0) +B1d¯′j,k(0), and
d¯j,k(t2) = cos(t2
pDj) ¯dj,k(0) + 1 pDj
sin(t2
pDj) ¯d′j,k(0) =A2d¯j,k(0) +B2d¯′j,k(0).
Then, a simple calculation gives:
d¯j,k(0) =A−11 d¯j,k(t1)−A−11 B2d¯′j,k(0), (28)
and
d¯′j,k(0) = [A2A−11 B1+B2]−1(A2A−11 d¯j,k(t1)−d¯j,k(t2)).
(29)
Remark 1. The matrices A1, A2, B1 and B2 are diagonal matrices, then their inverses are computed explicitly. Moreover, the times t1 and t2 should be selected in such a way to avoid the zeros line problem or division by zeros.
In terms of error, the reconstructed initial conditions ¯uj(0) and ¯u′j(0) from the wavelet coefficients ¯dj,k(0) and ¯d′j,k(0) satisfy:
Proposition 2. Let u¯j(0) and u¯′j(0) be the reconstructed initial conditions from the wavelet coefficients defined by (28)and (29). Then, we have:
(30) ∥u¯j(0)−u0∥L2(0,1)≤C2−js and ∥¯u′j(0)−u1∥L2(0,1)≤C2−js. Proof. The computation of coefficients ¯dj,k(0) and ¯d′j,k(0) using (28) and (29) is a linear system resolution:
(31)
A1 B1
A2 B2
d¯j,k(0) d¯′j,k(0)
=
d¯j,k(t1) d¯j,k(t2)
.
For a linear systemAx=b, it is known that if we choose slightly different the right hand side vector ˆb then we obtain a different solution vector ˆxsatisfyingAˆx= ˆb and the relative error on the solution is bounded as:
(32) ∥ˆx−x∥
∥x∥ ≤ ∥A∥∥A−1∥∥ˆb−b∥
∥b∥ . Since from Proposition 1, we have:
∥uj(t)− Pj(u(t))∥L2(0,1)≤C2−js,
which is the order of the relative error on the right hand term of system (31),
reporting this in (32) we obtain the proposition. □
4. Heat equation
As done for the wave equation in the previous sections, we are concerned here by the identification of initial temperature in heat conduction problem. The corre- sponding Cauchy’s problem is:
(33)
∂tu(t, x) =ν∂2xu(t, x), u(0, x) =u0(x),
wherex∈[0,1] andt∈[0, T]. Agains, homogeneous Dirichlet boundary conditions are assumed: u(t,0) =u(t,1) = 0. Other boundary conditions can be used without impacting the method as we will see. We also used the wavelet based Galerkin method for the spatial discretization of (33).
4.1. Wavelet Galerkin method for the heat equation. Similar to the wave equation, the wavelet based Galerkin method for the heat equation starts with (Vj)j∈Z a multi-resolution analysis ofL2(0,1). Then, we search a solutionuj ∈Vj
solution of (33) in the following form
(34) uj(t, x) =
Nj
X
k=1
dj,k(t)ψj,k(x), and that satisfies the variational formulation
(35)
Nj
X
k=1
d′j,k(t)
Z 1 0
ψj,k(x)ψj,m(x)dx+νdj,k(t) Z 1
0
ψ′j,k(x)ψ′j,m(x)dx
= 0, form= 1, . . . , Nj.
Thus, the computation ofuj is reduced to computation of its wavelet coefficients (dj,k). According to (35), these coefficients are solution of the following one order ordinary differential equation system:
(36) Aj
d′j,k(t)
+Rj[dj,k(t)] = 0,
whereAj andRj are respectively the mass matrix and the stiffness matrix of the considered wavelet basis defined in (16). The discretization error satisfies:
Proposition 3. Let uj and u be solutions of (33) and (35), respectively. If the initial conditionsu0(x)and the wavelet basis are regular enough, then we have:
(37) ∥uj−u∥L2(0,1)≤C2−js, for allj≥jmin ands >0.
Proof. Letej(t, x) =u(t, x)−uj(t, x) be the discretization error. Then, we have:
∥ej∥L2(0,1)=∥u−uj∥L2(0,1)≤ ∥Pj(u)−uj∥L2(0,1)+∥Pj(u)−u∥L2(0,1). (38)
Using the Jackson inequality (7), we see that the second term in the right hand of (38) is bounded byC2−js for a solutionu∈Hs(0,1).
On the other hand, since:
⟨∂tu−ν∂x2u, ψj,m⟩=⟨∂tuj−ν∂2xuj, ψj,m⟩, ∀ψj,m∈Vj, we have:
⟨∂t[u− Pj(u)]−ν∂x2[u− Pj(u)], ψj,m⟩=⟨∂t[uj− Pj(u)]−ν∂x2[uj− Pj(u)], ψj,m⟩,
∀ψj,m∈Vj. Denotingθj(t, x) =uj(t, x)− Pj(u(t, x)) and replacingψj,m byθj, we get:
1 2
d
dt∥θj(t)∥2L2(0,1)+ν Z 1
0
|∂xθj|2=⟨∂t[u− Pj(u)]−ν∂x2[u− Pj(u)], θj⟩
≤ ∥∂t[u− Pj(u)]∥L2(0,1)∥θj∥L2(0,1)+ν∥∂x[u− Pj(u)]∥L2(0,1)∥∂xθj∥L2(0,1)
≤C∥∂t[u− Pj(u)]∥L2(0,1)∥∂xθj∥L2(0,1)+ν∥∂x[u− Pj(u)]∥2L2(0,1)+ν
4∥∂xθj∥2L2(0,1)
≤ C2
ν ∥∂t[u− Pj(u)]∥2L2(0,1)+ν∥∂x[u− Pj(u)]∥2L2(0,1)+ν
2∥∂xθj∥2L2(0,1), using Cauchy-Schwarz and Poincar´e inequalities. Thus:
1 2
d
dt∥θj(t)∥2L2(0,1)+ν
2∥∂xθj∥2L2(0,1) ≤ C2
ν ∥∂t[u− Pj(u)]∥2L2(0,1)+ν∥∂xu−∂xPj(u)∥2L2(0,1)
≤ C2−2js∥∂tu∥2Hs(0,1)+C2−2js∥u∥2Hs+1. Integration on time gives:
(39)
∥θj(t)∥2L2(0,1)+ν Z t
0
∥∂xθj∥2L2(0,1)≤ ∥θj(0)∥2L2(0,1)+C2−2js Z t
0
∥∂tu∥2Hs(0,1)+∥u∥2Hs+1
. Finally, we have:
(40) ∥ej(t)∥L2(0,1)≤C ∥θj(0)∥L2(0,1)+ 2−js .
Thus to get an errorej inO(2−js) it suffices to takeuj(0, x) =Pj(u(0, x)) and this
will be assumed in the sequel. □
4.2. Initial temperature reconstruction. Heat conduction is a diffusion process and reversing a diffusion is very difficult and often impossible. Like the wave equation, we will give in this section a numerical method that allows to compute the coefficients dj,k(0) of the projection Pj(u(0, .)) from the measurements of the solutionu(t, x) at different positions or in different times.
To get an approximation of the wavelet coefficientsdj,k(0), we consider the or- dinary differential equation system (36) started fromdj,k(0):
(41)
Aj
h d′j,k(t)i
+Rj[dj,k(t)] = 0, dj,k(0)∼ Pj(u(0, .)).
The solution of (41) is given by:
(42) dj,k(t) =e−tMjdj,k(0), with Mj =A−1j Rj, and using (42), it is easy to infer that:
(43) dj,k(0) =etnMjdj,k(tn), ∀tn>0.
In practice, the coefficients dj,k(tn) are computed from some measurements of the temperature u(tn, xn). If the Deslauriers-Dubuc interpolating wavelet basis [7, 8]
is used in the discretization of the solutionu, we have:
(44) u(t, x) =Pj(u(t, x)) +X
ℓ≥j
Qℓ(u(t, x)), with
(45) Pj(u(t, x)) =
Nj
X
k=1
cj,k(t)φj,k(x).
Then, we get:
(46) u(tn, m/2j) =Pj(u(tn, m/2j)) =
Nj
X
k=1
cj,k(tn)φj,k(m/2j) =cj,m(tn), for 1 ≤ m ≤Nj. Therefore, setting dj,k(tn) = cj,k(tn), one can computedj,k(0) according to (43). In the general case, with any regular wavelet basis, the recon- structed initial condition ¯uj(0) satisfies the same order of accuracy as for the wave equation:
Proposition 4. Let ¯uj(0)) be the initial condition reconstructed from the wavelet coefficientsdj,k(0)given by (43). Then, we have:
(47) ∥¯uj(0)−u0∥L2(0,1)≤C2−js.
The proof of Proposition 4 uses the same arguments as for Proposition 2, we leave it to prevent redundancy.
5. Numerical results
This section is devoted to the presentation of some numerical results to eval- uate the performance and efficiency of the proposed identification algorithm. All the simulations are executed with MATLAB and the source codes are available on request. The wavelet basis generator that we used is Daubechies orthogonal generators with 4 vanishing moments [5] and the autocorrelation of this generator will give the Deslauriers-Dubuc generator [7, 8] that interpolates polynomials up to degree 4. Boundary conditions are imposed in the wavelet basis following [13].
As exact solution of the wave propagation problem (12) we used:
(48) u(t, x) = cos(2πt+ 2) sin(2πx).
Then, the initial conditions areu0(x) = cos(2) sin(2πx),u1(x) =−2πsin(2) sin(2πx) andν = 1. Likewise, the exact solution of (33) that we used is:
(49) u(t, x) =e−tsin(2πx),
with the initial conditionu0(x) = sin(2πx) and ν = 1/4π2. These solutions have the advantage to be considered with homogeneous Dirichlet boundary condition or with periodic boundary condition. The simulation final time areT = 1+4(1−π1) for the wave equation andT =πfor the heat equation. For the wave equation initial condition reconstruction, to solve the system defined in (28) and (29), as given data, we used the solution (48) taken at the timest1=T /2 andt2=T. Similarly, the used data in the heat problem corresponds to (49) taken at the simulation final time.
We first study the numerical errors decaying rates for the Galerkin method that provides schemes (26) and (42). Since the theoretical order are given by the Propo- sition 1 and Proposition 3, our objective is to verify if these orders are the same for the numerical solution, especially in the case of the reconstruction of the initial conditions.
Figure 1 shows the plot of the norm of the relative errors on the solutions (48) and (49), according to the number of grid pointsNj ≈2j, where j is the maximal space resolution. The solutions (48) and (49) areC∞, then the slopes of the curves obtained correspond to the maximum regularity of the wavelet basis. This is in good agreement with the theoretical results.
For comparison purposes, we plot the same errors norm on the reconstructed initial conditions. Figure 2 and Figure 3 show these errors in the case of wave equation and Figure 4 and Figure 2 in the case of heat equation. Again, the expected orders given by the Proposition 2 and Proposition 4 are achieved.
To prevent the limited floating point precision, in the reconstruction of heat equation initial condition, we made a thresholding on the coefficients of the matrix Mjbefore to compute its exponential. The threshold level is set to be proportional to the largest eigenvalue ofRj which is about: λmax≈22j [3]. Figure 3 and Figure 5 show the residual errors at grid points k/2j for k = 0 : 2j. Due to the non- symmetry of the Daubechies’s filters, this error are much bigger at boundaryx= 1 than atx= 0.
Remark 2.
As can be seen, adding terms that not depending on the wavelets coefficients cannot affect the steps of the resolution of the systems (28), (29) and (41). Then our approach works for equations with a second member of type:
∂t2u(t, x) =ν∂2xu(t, x) +f(t, x) and ∂tu(t, x) =ν∂x2u(t, x) +g(t, x).
In this case, we will have a time discretization error of the used numerical method to computed the integral of the added new terms.
102 103 10-12
10-10 10-8 10-6 10-4 10-2 100
(a)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(b)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(c)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(d)
Figure 1. Norms of the relative errors on the exact solution (48) (up- per two figures) and on the exact solution (49) (bottom two figures). Daubechies orthogonal generator withr= 4.
6. Conclusion
We have presented a numerical method to reconstruct the initial conditions of systems arising from the Galerkin method applied to the wave equation and the heat equation. Singular value decomposition is applied to diagonalize this system and thus solve the equation without time discretization errors. The approach does not generate more errors than a classical Galerkin method and it is simple and easy to implement. The extension to higher dimensions is immediate. However, the method is limited only to the linear case. In the general non linear case, we do not have the exact expression of the solutions. To the best of our knowledge, we do not know a numerical method that uses this kind of process, except using the Fourier basis with periodic boundary conditions. The method assumes that the data for the reconstruction are given on all the grid points. It would then be interesting to see if we can do the reconstruction with sparse measurements at selected positions or using the non-linear approximation of dense measurements. This will be the subject of future work.
References
[1] G. Beylkin, On the representation of operator in bases of compactly supported wavelets, SIAM J. Numer. Anal.,6(1992), 1716–1740.
102 103 10-12
10-10 10-8 10-6 10-4 10-2 100
(a)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(b)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(c)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(d)
Figure 2. Relative errors norm on the reconstructed initial conditions u0 andu1associated to (48). Daubechies orthogonal gener- ator withr= 4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-3 -2 -1 0 1 2 310-10
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5 -4 -3 -2 -1 0 1 2 3 4 10-9
(b)
Figure 3. Residual errors at grid points for the reconstructed initial conditionsu0andu1associated to (48). Daubechies orthog- onal generator withr= 4.
[2] M. Choulli, Une introduction aux probl`emes inverses elliptiques et paraboliques, Math´ematiques & Applications, 65, Springer, (2009).
[3] A. Cohen, Numerical analysis of wavelet methods, Studies in mathematics and its applica- tions, Elsevier, Amsterdam, (2003).
102 103 10-12
10-10 10-8 10-6 10-4 10-2 100
(a)
102 103
10-12 10-10 10-8 10-6 10-4 10-2 100
(b)
Figure 4. Relative errors norms on the reconstructed initial conditions u0associated to (49). Daubechies orthogonal generator with r= 4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-8 -6 -4 -2 0 2 4 610-9
Figure 5. Residual errors at grid points for the reconstructed initial conditionsu0associated to (49). Daubechies orthogonal gen- erator withr= 4.
[4] A. Cohen, I. Daubechies and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets,Comm. Pure . Appli. Maths.,45(1992), 485–560.
[5] I. Daubechies,Ten lectures on Wavelets, Book, SIAM, Philadelphia, Pennsylvania, (1992).
[6] R. DeVore and E. Zuazua, Recovery of an Initial Temperature from Discrete Sampling,Math.
Models. Meth. Appl. Sci.,12(2014), 2487–2501.
[7] G. Deslauriers and S. Dubuc, Symmetric iterative interpolation processes,Constructive Ap- proximation,5(1989), 49–68.
[8] S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl., 114 (1986), 185–204.
[9] S. Imperiale, P. Moireau, A. Tonnoir, Analysis of an observer strategy for initial state recon- struction of wave-like systems in unbounded domains,ESAIM: COCV,26(2020), 45.
[10] S. Kadri Harouna and V. Perrier, Homogeneous Dirichlet wavelets on the interval diagonal- izing the derivative operator, and related applications,hal-01568431, version 3, (2021).
[11] V. Komornik and P. Loreti, Fourier series in control theory, Springer Science & Business Media, (2005).
[12] S. Larsson and V. Thomee,Partial Differential Equations with Numerical Methods, Springer Science & Business Media, (2003).
[13] P. Monasse and V. Perrier, Orthogonal Wavelet Bases Adapted For Partial Differential Equa- tions With Boundary Conditions,SIAM J. Math. Anal.,29(1998), 1040–1065.
[14] G. Strang,Introduction to linear algebra, Wellesley, Wellesley-Cambridge Press, (1993).
UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics Faculty of Sciences of Monastir, University of Monastir 5019 Monastir, Tunisia
Email address:kais.ammari@fsm.rnu.tn
Laboratoire de Math´ematiques, Image et Applications, Avenue Michel Cr´epeau, 17042 La Rochelle cedex1France
Email address:souleymane.kadri harouna@univ-lr.fr
