Inverse problems for linear hyperbolic equations via a mixed formulation

Transcription

1 Inverse problems for linear hyperbolic equations via a mixed formulation Nicolae Cindea, Arnaud Münch To cite this version: Nicolae Cindea, Arnaud Münch. Inverse problems for linear hyperbolic equations via a mixed formulation. 5. <hal-47v3> HAL Id: hal-47 Submitted on 9 Jan 5 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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, est destinée 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.

2 Inverse problems for linear hyperbolic equations via a mixed formulation Nicolae Cîndea Arnaud Münch January 9, 5 Abstract We introduce in this document a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in Ω (, T ) - Ω a bounded subset of R N - from a partial distributed observation. We employ a least-squares technique and minimize the L -norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discussed several examples for N = and N =. The problem of the reconstruction of both the state and the source term is also addressed. Keywords Linear wave equation Inverse problem Finite elements methods Mixed formulation Mathematics Subject Classification () 35L 65M 93B4 Nicolae Cîndea Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand ), UMR CNRS 66, Campus de Cézeaux, 6377, Aubière, France. Nicolae.Cindea@math.univ-bpclermont.fr Arnaud Münch Laboratoire de Mathématiques, Université Blaise Pascal (Clermont-Ferrand ), UMR CNRS 66, Campus de Cézeaux, 6377, Aubière, France. Arnaud.Munch@math.univ-bpclermont.fr

3 Nicolae Cîndea, Arnaud Münch Introduction - Inverse problems for the wave equation Let Ω be a bounded domain of R N (N ) whose boundary Ω is Lipschitz and let T >. We note Q T := Ω (, T ) and Σ T := Ω (, T ). We are concerned in this work with inverse type problems for linear hyperbolic equation of the following type 8 >< y tt (c(x) y) + d(x, t)y = f, (x, t) Q T y =, (x, t) Γ T () >: (y(, ), y t (, )) = (y, y ), x Ω. We assume that c C (Ω, R) with c(x) c > in Ω, d L (Q T ), (y, y ) H := L (Ω) H (Ω) and f X := L (, T ; H (Ω)). For any (y, y ) H and any f X, there exists exactly one solution y to (), with y C ([, T ]; L (Ω)) C ([, T ]; H (Ω)) (see []). In the sequel, for simplicity, we shall use the following notation: L y := y tt (c(x) y) + d(x, t)y. () and X := L (, T ; H (Ω)). Let now ω be any non empty open subset of Ω and let q T := ω (, T ) Q T. A typical inverse problem for () is the following one : from an observation or measurement y obs in L (q T ) on the sub-domain q T, we want to recover a solution y of the boundary value problem () which coincides with the observation on q T. Introducing the operator P : L (Q T ) X L (q T ) defined by P y := (Ly, y qt ), the problem is reformulated as : find y L (Q T ) solution of P y = (f, y obs ). (IP ) From the unique continuation property for (), if the set q T satisfies some geometric conditions and if y obs is a restriction to q T of a solution of (), then the problem is well-posed in the sense that the state y corresponding to the pair (y obs, f) is unique. In view of the unavoidable uncertainties on the data y obs (coming from measurements, numerical approximations, etc), the problem needs to be relaxed. In this respect, the most natural (and widely used in practice) approach consists to introduce the following extremal problem (of least-squares type) 8 < minimize over H J(y, y ) := y y obs L (q T ) (LS) : where y solves (), since y is uniquely and fully determined from f and the data (y, y ). Here the constraint y y obs = in L (q T ) is relaxed; however, if y obs is a restriction to q T of a solution of (), then problems (LS) and (IP ) obviously coincide. A minimizing sequence for J in H is easily defined in term of the solution of an auxiliary adjoint problem. Apart from a possible low decrease of the sequence near extrema, the main drawback, when one wants to prove the convergence of a discrete approximation is that, it is in general not possible to minimize over a discrete

4 Inverse problems for linear hyperbolic equations via a mixed formulation 3 subspace of {y; Ly f = } subject to the equality (in X) Ly f =. Therefore, the minimization procedure first requires the discretization of the functional J and of the system (); this raised the issue of uniform coercivity property (typically here some uniform discrete observability inequality for the adjoint solution) of the discrete functional with respect to the approximation parameter. As far as we know, this delicate issue has received answers only for specific and somehow academic situations (uniform Cartesian approximation of Ω, constant coefficients in ()). We refer to [,7,9,] and the references therein. More recently, a different method to solve inverse type problems like (IP ) has emerged and use so called Luenberger type observers: this consists in defining, from the observation on q T, an auxiliary boundary value problem whose solution possesses the same asymptotic behavior in time than the solution of (): the use of the reversibility of the hyperbolic equation then allows to reconstruct the initial data (y, y ). We refer to [8,4] and the references therein. But, for the same reasons, on a numerically point of view, these method require to prove uniform discrete observability properties. In a series of works, Klibanov and co-workers use different approaches to solve inverse problems (we refer to [8] and the references therein): they advocate in particular the quasi-reversibility method which reads as follows : for any ε >, find y ε A the solution of P y ε, P y X L (q T ) + ε y ε, y A = (f, y obs ), P y X L (q T ),X L (q T ), (QR) for all y A, where A denotes a Hilbert space subset of L (Q T ) so that P y X L (q T ) for all y A and ε > a Tikhonov like parameter which ensures the well-posedness. We refer for instance to [3] where the lateral Cauchy problem for the wave equation with non constant diffusion is addressed within this method. Remark that (QR) can be viewed as a least-squares problem since the solution y ε minimizes over A the functional y P y (f, y obs ) X L (q T ) +ε y A. Eventually, if y obs is a restriction to q T of a solution of (), the corresponding y ε converges in L (Q T ) toward to the solution of (IP ) as ε. There, unlike in Problem (LS), the unknown is the state variable y itself (as it is natural for elliptic equations) so that any standard numerical methods based on a conformal approximation of the space A together with appropriate observability inequalities allow to obtain a convergent approximation of the solution. In particular, there is no need to prove discrete observability inequalities. We refer to the book []. We also mention [6, 5] where a similar technique has been used recently to solve the inverse obstacle problem associated to the Laplace equation, which consists in finding an interior obstacle from boundary Cauchy data. In the spirit of the works [8,6,3], we explore the direct resolution of the optimality conditions associated to the extremal problem (LS), without Tikhonov parameter while keeping y as the unknown of the problem. This strategy, which avoids any iterative process, has been successfully applied in the closed context of the exact controllability of () in [] and [7,]. The idea is to take into account the state constraint Ly f = with a Lagrange multiplier. This allows to derive

5 4 Nicolae Cîndea, Arnaud Münch explicitly the optimality systems associated to (LS) in term of an elliptic mixed formulation and therefore reformulate the original problem. Well-posedness of such new formulation is related to an observability inequality for the homogeneous solution of the hyperbolic equation. The outline of this paper is as follow. In Section, we consider the leastsquares problem (P) and reconstruct the solution of the wave equation from a partial observation localized on a subset q T of Q T. For that, in Section., we associate to (P) the equivalent mixed formulation (7) which relies on the optimality conditions of the problem. Assuming that q T satisfies the classical geometric optic condition (Hypothesis, see (H)), we then show the well-posedness of this mixed formulation, in particular, we check the Babuska-Brezzi inf-sup condition (see Theorem ). Interestingly, in Section., we also derive a equivalent dual extremal problem, which reduces the determination of the state y to the minimization of an elliptic functional with respect to the Lagrange multiplier. In Section 3, we apply the same procedure to recover from a partial observation both the state and the source term. Section 4 is devoted to the numerical approximation, through a conformal space-time finite element discretization. The strong convergence of the approximation (y h, f h ) is shown as the discretization parameter h tends to zero. In particular, we discuss the discrete inf-sup property of the mixed formulation. We present numerical experiments in Section 5 for Ω = (, ) and Ω R, in agreement with the theoretical part. We consider in particular time dependent observation zones. Section 6 concludes with some perspectives. Recovering the solution from a partial observation: a mixed re-formulation of the problem In this section, assuming that the initial (y, y ) H are unknown, we address the inverse problem (IP ). Without loss of generality, in view of the linearity of the system (), we assume that the source term f. We consider the non empty vectorial space Z defined by Z := {y : y C([, T ], L (Ω)) C ([, T ], H (Ω)), Ly X}. (3) and then introduce the following hypothesis : Hypothesis There exists a constant C obs = C(ω, T, c C (Ω), d L (Ω)) such that the following estimate holds : «y(, ), y t (, ) H C obs y L (q T ) + Ly X, y Z. (H) Condition (H) is a generalized observability inequality for the solution of the hyperbolic equation: for constant coefficients, this estimate is known to hold if the triplet (ω, T, Ω) satisfies a geometric optic condition. We refer to []. In particular, T should be large enough. Upon the same condition, (H) also holds in the noncylindrical situation where the domain ω varies with respect to the time variable:

6 Inverse problems for linear hyperbolic equations via a mixed formulation 5 we refer to [7] for the one dimensional case. For non constant velocity c and potential d, we refer to [] and the references therein. Then, within this hypothesis, for any η >, we define on Z the bilinear form ZZ y, y Z := y y dxdt + η q T Z T Ly, Ly H (Ω) dt y, y Z. (4) In view of (H), this bilinear form defines a scalar product over Z. Moreover, endowed to this scalar product, we easily obtain that Z is a Hilbert space (see [7], Corollary.4). We note the corresponding norm by y Z := p y, y Z. Then, we consider the following extremal problem : 8 < inf J(y) := y y obs L (q T ), : subject to y W where W is the closed subspace of Z defined by W := {y Z; Ly = in X} and endowed with the norm of Z. The extremal problem (P) is well posed : the functional J is continuous over W, is strictly convex and is such that J(y) + as y W. Note also that the solution of (P) in W does not depend on η. Remind that from the definition of Z, Ly belongs to X. Similarly, the uniqueness of the solution is lost if the hypothesis (H) is not fulfilled, for instance if T is not large enough. Eventually, from (H), the solution y in Z of (P) satisfies (y(, ), y t (, )) H, so that problem (P) is equivalent to the minimization of J with respect to (y, y ) H as in problem (IP ), Section. We also recall that for any z Z there exists a positive constant C Ω,T such that z L (Q T ) C Ω,T z(, ), z t (, ) H + Lz X This equality and (H) imply that z L (Q T ) C Ω,T C obs z L (q T ) + ( + C obs ) Lz X (P) «. (5) «, z Z. (6). Direct approach In order to solve (P), we have to deal with the constraint equality which appears in the space W. Proceeding as in [], we introduce a Lagrangian multiplier λ X and the following mixed formulation: find (y, λ) Z X solution of 8 < a(y, y) + b(y, λ) = l(y), y Z : b(y, λ) =, λ X, (7)

7 6 Nicolae Cîndea, Arnaud Münch where ZZ a : Z Z R, a(y, y) := y y dxdt, (8) q T Z T b : Z X R, b(y, λ) := λ, Ly H (Ω),H (Ω)dt, (9) ZZ l : Z R, l(y) := y obs y dxdt. () q T System (7) is nothing else than the optimality system corresponding to the extremal problem (P). Precisely, the following result holds : Theorem Under the hypothesis (H),. The mixed formulation (7) is well-posed.. The unique solution (y, λ) Z X to (7) is the unique saddle-point of the Lagrangian L : Z X R defined by L(y, λ) := a(y, y) + b(y, λ) l(y). 3. We have the estimate q y Z = y L (q T ) y obs L (q T ), λ X C Ω,T + η y obs L (q T ). () Proof- We use classical results for saddle point problems (see [4], chapter 4). We easily obtain the continuity of the bilinear form a over Z Z, the continuity of bilinear b over Z X and the continuity of the linear form l over Z. In particular, we get l Z = y obs L (q T ), a (Z Z) =, b (Z X ) = η /. () Moreover, the kernel N (b) = {y Z; b(y, λ) = λ X } coincides with W : we easily get a(y, y) = y Z, y N (b) = W. Therefore, in view of [4, Theorem 4..], it remains to check the inf-sup constant property : δ > such that inf λ X sup y Z b(y, λ) y Z λ X δ. (3) We proceed as follows. For any fixed λ X, we define y as the unique solution of Ly = λ in Q T, (y(, ), y t (, )) = (, ) on Ω, y = on Σ T. (4) We get b(y, λ) = λ X and y Z = y L (q T ) + η λ X.

8 Inverse problems for linear hyperbolic equations via a mixed formulation 7 Using (5), the estimate y L (q T ) p C Ω,T λ X implies that y Z and that sup y Z b(y, λ) y Z λ X p CΩ,T + η > leading to the result with δ = (C Ω,T + η) /. The third point is the consequence of classical estimates (see [4], Theorem 4..3.) : where y Z α l Z, λ X δ α := inf y N (b) + a α «l Z a(y, y) y. (5) Z Estimates () and the equality α = lead to the results. Eventually, from (), we obtain that λ X δ y obs L (q T ) and that δ (C Ω,T + η) / to get (). In practice, it is very convenient to augment the Lagrangian (see [6]) and consider instead the Lagrangian L r defined for any r > by L r (y, λ) := a r(y, y) + b(y, λ) l(y), a r (y, y) := a(y, y) + r Ly X. Since a r (y, y) = a(y, y) on W, the Lagrangian L and L r share the same saddlepoint. The positive number r is an augmentation parameter. Remark Assuming additional hypotheses on the regularity of the solution λ, precisely Lλ L (Q T ) and (λ, λ t ) t=,t H (Ω) L (Ω), we easily prove, writing the optimality condition for L, that the multiplier λ satisfies the following relations : ( Lλ = (y yobs ) ω in Q T, λ = in Σ T, λ = λ t = on Ω {, T }. (6) Therefore, λ (defined in the weak sense) is an exact controlled solution of the wave equation through the control (y y obs ) ω L (q T ). If y obs is the restriction to q T of a solution of (), then the unique multiplier λ must vanish almost everywhere. In that case, we have sup λ Λ inf y Y L r (y, λ) = inf y Y L r (y, ) = inf y Y J r (y) with J r (y) := y y obs L (Q T ) + r Ly X. (7) The corresponding variational formulation is then : find y Z such that ZZ a r (y, y) = y y dxdt + r q T Z T Ly, Ly H (Ω) dt = l(y), y Z.

9 8 Nicolae Cîndea, Arnaud Münch In the general case, the mixed formulation can be rewritten as follows: find (z, λ) Z X solution of ( Pr y, P r y X L (q T ) + Ly, λ X,X = (, y obs ), P r y X L (q T ), y Z, Ly, λ X,X =, λ X (8) with P r y := ( rl y, y qt ). This approach may be seen as generalization of the (QR) problem (see (QR)), where the variable λ is adjusted automatically (while the choice of the parameter ε in (QR) is in general a delicate issue). System (6) can be used to define a equivalent saddle-point formulation, very suitable at the numerical level. Precisely, we introduce - in view of (6) - the space Λ by Λ := {λ : λ C([, T ];H (Ω)) C ([, T ]; L (Ω)), Lλ L (Q T ), λ(, ) = λ t (, ) = }. ZZ Endowed with the scalar product λ, λ Λ := (λ λ+lλlλ) dxdt, we check that Q T Λ is a Hilbert space. Then, for any parameter α (, ), we consider the following mixed formulation : find (y, λ) Z Λ such that where 8 < a r,α (y, y) + b α (y, λ) = l,α (y), y Z : b α (y, λ) c α (λ, λ) = l,α (λ), λ Λ, ZZ Z T a r,α : Z Z R, a r,α (y, y) := ( α) yy dxdt + r (Ly, Ly) H (Ω)dt, q T Z T ZZ b α : Z Λ R, b α (y, λ) := λ, Ly H (Ω),H (Ω)dt α y Lλ dxdt, q ZZ T c α : Λ Λ R, c α (λ, λ) := α Lλ Lλ, dxdt Q ZZ T l,α : Z R, l,α (y) := ( α) y obs y dxdt, q ZZ T l,α : Λ R, l,α (λ) := α y obs Lλ dxdt. q T From the symmetry of a r,α and c α, we easily check that this formulation corresponds to the saddle point problem : 8 >< sup inf L r,α(y, λ), y Z λ Λ >: L r,α (y, λ) := L r (y, λ) α Lλ + (y y obs) ω L (Q T ). (9)

10 Inverse problems for linear hyperbolic equations via a mixed formulation 9 Proposition Under the hypothesis (H), for any α (, ), the formulation (9) is well-posed. Moreover, the unique pair (y, λ) in Z Λ satisfies with θ y Z + θ λ ( α) Λ θ θ := min α, r «, θ := η min α, «+ α y obs L θ (q T ). () C Ω,T Proof- We easily get the continuity of the bilinear forms a r,α, b α and c α : «. a r,α (y, y) max( α, r η ) y Z y Z, y, y Z, b α (y, λ) max(α, c α (λ, λ) α λ Λ λ Λ, λ, λ Λ η ) y Z λ Λ, y Z, λ Λ, and of the linear form l and l : l Z = ( α) y obs L (q T ) and l Λ = α y obs L (q T ). Moreover, since α (, ), we also obtain the coercivity of a r,α and of c α : precisely, a r,α (y, y) min α, r «y Z, y Z, η c α (λ, λ) min αm, m C Ω,T «λ Λ λ Λ, m (, ). The result [4, Prop 4.3.] implies the well-posedness and the estimate () taking m = /. The α-term in L r,α is a stabilization term: it ensures a coercivity property of L r,α with respect to the variable λ and automatically the well-posedness. In particular, there is no need to prove any inf-sup property for the application b α. Proposition If the solution (y, λ) Z X of (7) enjoys the property λ Λ, then the solutions of (7) and (9) coincide. Proof- The hypothesis of regularity and the relation (6) imply that the solution (y, λ) Z X of (7) is also a solution of (9). The result then follows from the uniqueness of the two formulations.. Dual formulation of the extremal problem (7) As discussed at length in [], we may also associate to the extremal problem (P) an equivalent problem involving only the variable λ. Again, this is particularly interesting at the numerical level. This requires a strictly positive augmentation parameter r.

11 Nicolae Cîndea, Arnaud Münch For any r >, let us define the linear operator P r from X into X by where y Z is the unique solution to P r λ := (Ly), λ X a r (y, y) = b(y, λ), y Z. () The assumption r > is necessary here in order to guarantee the well-posedness of (). Precisely, for any r >, the form a r defines a norm equivalent to the norm on Z. The following important lemma holds: Lemma For any r >, the operator P r is a strongly elliptic, symmetric isomorphism from X into X. Proof- From the definition of a r, we easily get that P r λ X r λ X and the continuity of P r. Next, consider any λ X and denote by y the corresponding unique solution of () so that P r λ := (Ly ). Relation () with y = y then implies that Z T P r λ, λ H (Ω) dt = a r (y, y ) () and therefore the symmetry and positivity of P r. The last relation with λ = λ and the observability estimate (H) imply that P r is also positive definite. Finally, let us check the strong ellipticity of P r, equivalently that the bilinear functional (λ, λ ) R T P rλ, λ H (Ω),H (Ω) dt is X -elliptic. Thus we want to show that Z T P r λ, λ H (Ω) dt C λ X, λ X (3) for some positive constant C. Suppose that (3) does not hold; there exists then a sequence {λ n } n of X such that λ n X =, n, lim n Z T P r λ n, λ n H (Ω) dt =. Let us denote by y n the solution of () corresponding to λ n. From (), we then obtain that lim r Ly n X + y n L n (q T ) =. (4) From () with y = y n and λ = λ n, we have Z T D ZZ r( )Ly n λ n, ( )Ly EH dt + y n ydx dt =, y Z. (Ω) q T We define the sequence {y n } n as follows : 8 Ly >< n = r Ly n + λ n, in Q T, y n =, in Σ T, >: y n (, ) = y n,t (, ) =, in Ω, (5)

12 Inverse problems for linear hyperbolic equations via a mixed formulation so that, for all n, y n is the solution of the wave equation with zero initial data and source term rly n + λ n in X. Using again (5), we get y n L (q T ) p C Ω,T rly n + λ n X, so that y n Z. Then, using (5) with y = y n we get q r( )Ly n λ n X C Ω,T y n L (q T ). Then, from (4), we conclude that lim n + λ n X = leading to a contradiction and to the strong ellipticity of the operator P r. The introduction of the operator P r is motivated by the following proposition : Proposition 3 For any r >, let y Z be the unique solution of a r (y, y) = l(y), y Z and let J r : X X be the functional defined by J r (λ) = The following equality holds : Z T sup inf L r(y, λ) = inf y Z λ X P r λ, λ H (Ω)dt b(y, λ). λ X J r (λ) + L r (y, ). The proof is classical and we refer for instance to [] in a similar context. This proposition reduces the search of y, solution of problem (P), to the minimization of J r. The well-posedness is a consequence of the ellipticity of the operator P r. Remark The results of this section apply if the distributed observation on q T is replaced by a Neumann boundary observation on a sufficiently large subset Σ T of Ω (, T ) (i.e. assuming y ν = y obs L (Σ T ) is known on Σ T ). This is due to the following generalized observability inequality: there exists a positive constant C obs = C(ω, T, c C (Ω), d L (Ω)) such that the following estimate holds : y(, ), y t (, ) y H (Ω) L (Ω) C obs ν L (Σ T ) «+ Ly L (Q T ), y Z (6) which holds if the triplet (Q T, Σ T, T ) satisfies the geometric condition as before (we refer to [] and the references therein). Actually, it suffices to re-define the form a in (8) by a(y, y) := RR y y Σ T ν ν dσdx and the form l by l(y) := RR y Σ T ν y obs dσdx for all y, y Z. Remark 3 We emphasize that the mixed formulation (7) has a structure very closed to the one we get when we address - using the same approach - the null controllability of (): more precisely, the control of minimal L (q T )-norm which drives to rest the initial data (y, y ) H (Ω) L (Ω) is given by v = ϕ qt where (ϕ, λ) Φ L (, T ; H (Ω)) solves the mixed formulation 8 < a(ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ (7) : b(ϕ, λ) =, λ L (, T ; H (Ω)),

13 Nicolae Cîndea, Arnaud Münch where ZZ a : Φ Φ R, a(ϕ, ϕ) = ϕ(x, t)ϕ(x, t) dx dt q T b : Φ L (, T ; H (Ω)) R, b(ϕ, λ) = Z T l : Φ R, l(ϕ) = ϕ t (, ), y H (Ω),H (Ω) + Lϕ, λ H,H dt Z ϕ(, ) y dx. with Φ = ϕ L (Q T ), ϕ = on Σ T such that Lϕ L (, T ; H (Ω)). We refer to []. Remark 4 Reversing the order of priority between the constraint y y obs = in L (q T ) and Ly f = in X, a possibility could be to minimize the functional y Ly f X over y Z subject to the constraint y y obs = in L (q T ) via the introduction of a Lagrange multiplier in L (q T ). The proof of the following inf-sup property : there exists δ > such that RR inf sup q T λy dxdt λ L (q T ) y Z λ L (q T ) y Y δ associated to the corresponding mixed-formulation is however unclear. If a ε-term is added as in (QR), this difficulty disappears (we refer again to the book [8]). 3 Recovering the source and the solution from a partial observation: a mixed re-formulation of the problem Given a partial observation y obs of the solution on the subset q T Q T, we now consider the reconstruction of the full solution as well as the source term f assumed in X. We assume that the initial data (y, y ) H are unknown. The situation is different with respect to the previous section, since without additional assumption on f, the couple (y, f) is not unique. Consider the case of a source f supported in a set which is near Ω (, T ) and disjoint from q T : from the finite propagation of the solution, the source f will not affect the solution y in q T. On the other hand, the determination of a couple (y, f) which solves () such that y coincides with y obs is straightforward : it suffices to extend y on Q T \ q T appropriately to preserve the boundary conditions, then compute Ly and recover a source term. However, we emphasize that, on a practical viewpoint, the extension of y obs out of q T is not obvious. Moreover, this strategy does not offer any control on the object f. We briefly show that we can apply the method developed in Section which allows a robust reconstruction and then consider the case of uniqueness via additional condition on f.

14 Inverse problems for linear hyperbolic equations via a mixed formulation 3 We assume again that (H) holds. We note Y := Z X and define on Y the bilinear form, for any ε, η > ZZ (y, f), (y, f) Y := y y dxdt + η q T + ε Z T Z T Ly f, Ly f H (Ω)dt f, f H (Ω)dt, (y, f), (y, f) Y. In view of (H), this bilinear form defines a scalar product over Y. Moreover, endowed to this scalar product, we easily obtain that Y is a Hilbert space (we refer to [7]). We note the corresponding norm by (y, f) Y := p ((y, f), (y, f)) Y. Then, for any ε >, we consider the following extremal problem : (P ε ) 8 < inf J ε (y, f) := y y obs L (q T ) + ε f X, : subject to (y, f) W where W is the closed subspace of Y defined by W := {(y, f) Y ; Ly f = in X} and endowed with the norm of Y : precisely, it follows that (y, f) W := q y L (q T ) + ε f X, (y, f) W. The extremal problem (P ε ) is well posed : the functional J ε is continuous over W, is strictly convex and is such that J ε (y, f) + as (y, f) W. Note also that the solution of (P ε ) in W, depends on ε but not on η. Remark also that if ε =, then J ε is a priori only convex leading possibly to distinct minima. This justifies the introduction of the ε-term in the functional J ε. We emphasize however that the ε-term is not a regularization term as it does not improve the regularity of the state y. Eventually, from (H), the solution (y ε, f ε ) in W of (P ε ) satisfies (y ε (, ), y ε,t (, )) H, so that problem (P ε ) is again equivalent to the minimisation of J ε with respect to (y, y, f) H X. Proceeding as in Section, we introduce a Lagrangian multiplier λ ε X and the following mixed formulation: find ((y ε, f ε ), λ ε ) Y X solution of 8 < a ε ((y ε, f ε ), (y, f)) + b((y, f), λ ε ) = l(y, f), (y, f) Y (9) : b((y ε, f ε ), λ) =, λ X, (8) where ZZ a ε : Y Y R, a ε ((y, f), (y, f)) := yy dxdt + ε(f, f) X, (3) q T Z T b : Y X R, b((y, f), λ) := λ, Ly f H (Ω),H (Ω) dt, (3) ZZ l : Y R, l(y, f) := y obs y dxdt. (3) q T

15 4 Nicolae Cîndea, Arnaud Münch Theorem Under the hypothesis (H), the following hold :. The mixed formulation (9) is well-posed.. The unique solution ((y ε, f ε ), λ ε ) Y X is the saddle-point of the Lagrangian L ε : Y X R defined by L ε ((y, f), λ) := a ε((y, f), (y, f)) + b((y, f), λ) l(y, f). Moreover, the pair (y ε, f ε ) solves the extremal problem (P ε ). 3. The following estimates hold : (y ε, f ε ) Y = y ε L (q T ) + ε f ε / yobs L (q T ) (33) and q λ ε L (Q T ) C Ω,T + η y obs L (q T ) (34) for some constant C Ω,T >. The proof is very closed to the proof of Theorem. In particular, the obtention of the inf-sup property is obtained by taking, for any λ X, f = and y as in (4) so that the inf-sup constant δ ε := inf sup λ X (y,f) Y b((y, f), λ) (y, f) Y λ X (35) is bounded by above by (C Ω,T + η) / uniformly with respect to ε. Remark in particular that the inequality (33) implies that, at the optimality, since ε >, the equality y y obs L (q T ) = can not hold if f ε. Remark 5 We may also prove the inf-sup property using the variable f: for any λ X, we set y = and f = λ X. We get sup (y,f) Y b((y, f), λ) (y, f) Y λ X b((, λ), λ) (, λ) Y λ X = ε + η so that δ ε (ε + η) /. Therefore, the estimate λ ε X δ ε y obs L (q T ) implies that λ ε X ε + η y obs L (q T ). (36) This argument is valid if and only if f is distributed everywhere in Q T. Remark 6 The estimate (36) implies that the multiplier λ ε vanishes in X as ε + η + (recall that ε and η can be chosen arbitrarily small in (4)). Remark 7

16 Inverse problems for linear hyperbolic equations via a mixed formulation 5 (a) Assuming enough regularity on the solution λ ε, precisely that Lλ ε L (Q T ) and (λ, λ t ) t=,t H (Ω) L (Ω), we easily check that the multiplier λ ε satisfies the following relations : 8 Lλ >< ε = (y ε y obs ) ω, Ly ε f ε =, εf ε + λ ε = in Q T, λ ε = in Σ T, >: λ ε = λ ε,t = on Ω {, T }. Therefore, λ ε is an exact controlled solution of the wave equation through the control (y ε y obs ) ω and from (36) implies that y ε y obs L (q T ) as ε +. (37) Remark however that f ε may not be bounded in X uniformly w.r.t. ε (contrarily to the sequence ( εf ε ) ε> ). (b) The equality Ly ε = f ε becomes εly ε = λ ε and leads to L(ε Ly ε ) = Lλ ε = (y ε y obs ) ω. Finally, y ε solves, at least in D, the boundary value problem 8 L(ε( >< )Ly ε ) + y ε ω = y obs ω, in Q T, (εly ε ) = (εly ε ) t =, in Ω {, T } >: y ε =, on Σ T or equivalently the variational formulation: find y ε Z (see (3)) solution of ε Z T ZZ ZZ Ly ε, Ly H (Ω)dt + y ε y dxdt = y obs y dxdt, y Z (38) q T q T which actually can be obtained directly from the cost J ε, replacing from the beginning f by the term Ly. From the Lax-Milgram lemma, (38) is well-posed and the following estimates hold : y ε L (q T ) y obs L (q T ), ε Lyε X y obs L (q T ). This kind of variational formulation involving the fourth order term Ly ε Ly has been derived and used in [] in a controllability context. For any ε > and any y obs L (q T ), the method allows to recover a couple (y ε, f ε ) such that Ly ε = f ε in Q T and y ε is closed to y obs (see (37)). In view of the loss of uniqueness, we have no information on the limit of the sequence as ε : the sequence may be unbounded at the limit in L (Q T ) L (Q T ) even if y obs is the restriction to q T of a solution of (). Remark 8 Contrarily to the inf-sup property, the coercivity of a ε over N (b) does not hold uniformly with respect to ε. Recall that the ε-term has been introduced to get a norm for Y. This enforces us to add this term in the mixed formulation.

17 6 Nicolae Cîndea, Arnaud Münch Remark 9 A fortiori, if the initial condition (y, y ) H is known, one may recover the pair (y, f) Y from y obs and (y, y ). The procedure is similar; it suffices to define two additional Lagrange multipliers (λ, λ ) L (Ω) H (Ω) to deal with the constraint y(, ) = y and y t (, ) = y respectively. The extremal problem is now : inf (y,f) W J ε(y, f) := y y obs L (q T ) + ε f X where W is the closed subspace of Y defined by W := {(y, f) Y ; Ly f = in X, (y(, ), y t (, )) = (y, y ) in H}. The corresponding mixed formulation is : find ((y ε, f ε ), (λ ε, λ ε,, λ ε, )) Y Λ solution of 8 < a ε ((y ε, f ε ), (y, f)) + b((y, f), (λ ε, λ ε,, λ ε, )) = l (y, f), (y, f) Y : b((y ε, f ε ), (λ, λ, λ )) = l (λ, λ, λ ), (λ, λ, λ ) Λ, (39) where a ε is given by (3) and ZZ b : Y Λ R, b((y, f), (λ, λ, λ )) := λ(ly f) dxdt Q T + y(, ), λ L (Ω) + y t (, ), λ H (Ω),H ZZ (Ω) l : Y R, l (y, f) := y obs y dxdt q T l : Λ R, l (λ, λ, λ ) := y, λ L (Ω) + y, λ H (Ω),H (Ω) with Λ := X L (Ω) H (Ω). Using the estimate (H), we easily show that this formulation is well-posed. In view of Remark 7 (a), we may also associate to the mixed formulation (9) a stabilized version, similarly to (9). Again, it is very convenient to augment the Lagrangian (see [6]) and consider instead the Lagrangian L ε,r defined for any r > by L ε,r ((y, f), λ) := a ε,r((y, f), (y, f)) + b(y, λ) l(y, f), a ε,r ((y, f), (y, f)) := a ε ((y, f), (y, f)) + r Ly f X. Since a ε (y, y) = a ε,r (y, y) on W, the Lagrangian L ε and L ε,r share the same saddle-point. The positive number r is an augmentation parameter. Similarly, proceeding as in Section., we may also associate to the saddle-point problem sup λ X inf (y,f) Y L r,ε ((y, f), λ) a dual problem, which again reduces the search of the couple (y ε, f ε ), solution of problem (P ε ), to the minimization of a elliptic functional in λ ε.

18 Inverse problems for linear hyperbolic equations via a mixed formulation 7 Proposition 4 For any r >, let (y, f ) Y be the unique solution of a ε,r ((y, f ), (y, f)) = l(y, f), (y, f) Y and let P ε,r be the strongly elliptic and symmetric operator from X into X defined by P ε,r λ := (Ly f) where (y, f) Y is the unique solution to Then, the following equality holds a ε,r ((y, f), (y, f)) = b((y, f), λ), (y, f) Y. (4) sup inf L ε,r((y, f), λ) = inf (y,f) Y λ X λ X J where J ε,r : X X is the functional defined by J ε,r(λ) = Z T ε,r(λ) + L ε,r ((y, f ), ). (P ε,r λ, λ) H (Ω) dt b((y, f ), λ). Compared to the previous section, the additional unknown f ɛ on the problem guarantees that the term y ε y obs L (q T ) vanishes at the limit in ε, for any y obs L (q T ), be a restriction of a solution of () or not. The situation is different if additional assumption on f enforces the uniqueness of the pair (y, f) (we refer to [5] and the references therein). 4 Numerical Analysis of the mixed formulations 4. Numerical approximation of the mixed formulation (7) We consider the numerical analysis of the mixed formulation (7), assuming r >. We follow [], to which we refer for the details. Let Z h and Λ h be two finite dimensional spaces parametrized by the variable h such that Z h Z, Λ h X for every h >. Then, we can introduce the following approximated problems: find the (y h, λ h ) Z h Λ h solution of 8 < a r (y h, y h ) + b(y h, λ h ) = l(y h ), y h Z h (4) : b(y h, λ h ) =, λ h Λ h. The well-posedness of this mixed formulation is again a consequence of two properties: the coercivity of the bilinear form a r on the subset N h (b) = {y h Z h ; b(y h, λ h ) = λ h Λ h }. Actually, from the relation a r (y, y) (r/η) y Z for all y Z, the form a r is coercive on the full space Z, and so a fortiori on N h (b) Z h Z. The second property is a discrete inf-sup condition. We note δ h > by δ h := inf λ h Λ h b(y h, λ h ) sup. (4) y h Z h λ h X y h Z

19 8 Nicolae Cîndea, Arnaud Münch For any fixed h, the spaces Z h and Λ h are of finite dimension so that the infimum and supremum in (4) are reached: moreover, from the property of the bilinear form a r, it is standard to check that δ h is strictly positive. Consequently, for any fixed h >, there exists a unique couple (y h, λ h ) solution of (4). We then have the following estimate. Proposition 5 Let h >. Let (y, λ) and (y h, λ h ) be the solution of (7) and of (4) respectively. Let δ h the discrete inf-sup constant defined by (4). Then, y y h Z + «d(y, Z h ) + d(λ, Λ h ), (43) ηδh η λ λ h X + ηδh «δ h d(y, Z h ) + 3 ηδh d(λ, Λ h ) (44) where d(λ, Λ h ) := inf λh Λ h λ λ h X and d(y, Z h ) := inf y y h Z y h Z h / = inf y y h L y h Z (q T ) + η L(y y h ) X«. h Proof- From the classical theory of approximation of saddle point problems (see [4, Theorem 5..]) we have that B y y h a r (Z Z) α + a r (Z Z) b (Z X ) C A d(y, Z α h ) δ h + b (Z X ) α d(λ, Λ h ) (45) and B a r 3 (Z Z) λ λ h α δ h + a r (Z Z) b (Z X ) δ h C A d(y, Z h ) + 3 a r b (Z X ) α δ h d(λ, Λ h ). (46) Since, a r (Z Z) = α = ; b (Z Λ) = η, the result follows. Remark For r =, the discrete mixed formulation (4) is not well-posed over Z h Λ h because the form a r= is not coercive over the discrete kernel of b: the equality b(y h, λ h ) = for all λ h Λ h does not imply in general that Ly h vanishes. Therefore, the term r Ly h X, which appears in the Lagrangian L r, may be understood as a stabilization term: for any h >, it ensures the uniform coercivity of the form a r and vanishes at the limit in h. We also emphasize that this term is not a regularization term as it does not add any regularity on the solution y h.

20 Inverse problems for linear hyperbolic equations via a mixed formulation 9 Let n h = dim Z h, m h = dim Λ h and let the real matrices A r,h R n h,n h, B h R m h,n h, J h R m h,m h and L h R n h be defined by 8 a r (y h, y h ) = A r,h {y h }, {y h } R n h,r n h y h, y h Z h, >< b(y h, λ h ) = B h {y h }, {λ h } R m h,r m h y h Z h, λ h Λ h, ZZ λ h λ h dx dt = J h {λ h }, {λ h } R m h,r m h λ h, λ h Λ h, Q T >: l(y h ) = L h, {y h } R n h y h Z h, where {y h } R n h denotes the vector associated to y h and, R n h,r n h the usual scalar product over R n h. With these notations, the problem (4) reads as follows: find {y h } R n h and {λ h } R m h such that A r,h B T h B h! R n h +m h,n h +m h (47)! {y h } = L! h. (48) {λ h } R n h +m h R n h +m h The matrix A r,h as well as the mass matrix J h are symmetric and positive definite for any h > and any r >. On the other hand, the matrix of order m h + n h in (48) is symmetric but not positive definite. We use exact integration methods developed in [5] for the evaluation of the coefficients of the matrices. The system (48) is solved using the direct LU decomposition method. 4.. C -finite elements and order of convergence for N = The finite dimensional and conformal space Z h must be chosen such that Ly h belongs to X = L (, T ; H (Ω)) for any y h Z h. This is guaranteed, for instance, as soon as ϕ h possesses second-order derivatives in L loc(q T ). As in [], we consider a conformal approximation based on functions continuously differentiable with respect to both variables x and t. We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. We note h := max{diam(k), K T h }, where diam(k) denotes the diameter of K. Then, we introduce the space Z h as follows : Z h = {y h Z C (Q T ) : z h K P(K) K T h, z h = on Σ T }, (49) where P(K) denotes an appropriate space of functions in x and t. In this work, we consider two choices, in the one-dimensional setting (for which Ω R, Q T R ):. The Bogner-Fox-Schmit (BFS for short) C -element defined for rectangles. It involves 6 degrees of freedom, namely the values of y h, y h,x, y h,t, y h,xt on the four vertices of each rectangle K. Therefore P(K) = P 3,x P 3,t where P r,ξ is by definition the space of polynomial functions of order r in the variable ξ. We refer to [9, ch. II, sec. 9, p. 94].

21 Nicolae Cîndea, Arnaud Münch. The reduced Hsieh-Clough-Tocher (HCT for short) C -element defined for triangles. This is a so-called composite finite element and involves 9 degrees of freedom, namely, the values of y h, y h,x, y h,t on the three vertices of each triangle K. We refer to [9, ch. VII, sec. 46, p. 85] and to [3, ] where the implementation is discussed. We also define the finite dimensional space Λ h = {λ h C (Q T ), λ h K Q(K) K T h }. where Q(K) denotes the space of affine functions both in x and t on the element K. For any h >, we have Z h Z and Λ h X. We then have the following result: Proposition 6 (BFS element for N = - Rate of convergence for the norm Z) Let h >, let k {, } be a positive integer. Let (y, λ) and (y h, λ h ) be the solution of (7) and (4) respectively. If the solution (y, λ) belongs to H k+ (Q T ) H k (Q T ), then there exists two positives constants K i = K i ( y H k+ (Q T ), c C (Q T ), d L (Q T )), i {, }, independent of h, such that h k y y h Z K ( η + )(h 3 + «ηh) +, η δ h (5) h k λ λ h X K ( η + )(h 3 + «nh) +. ηδh δ h (5) Proof - From [9, ch. III, sec. 7], for any λ H k (Q T ), k, there exists C = C ( λ H k (Q T )) such that λ Π Λh,T h (λ) X C h k, h > (5) where Π Λh,T h designates the interpolant operator from X to Λ h associated to the regular mesh T h. Similarly, for any y H k+ (Q T ), there exist C = C ( y H k+ (Q T )) such that for every h > we have y Π Zh,T h (y) L (Q T ) C h k+, y Π Zh,T h (y) H (Q T ) C h k. (53) Then, observing that Ly Ly h X K( c C (Q T ), d L (Q T )) y y h H (Q T ), (54) for some positive constant K, we get that d(y, Z h ) = inf y y h L y h Z (q T ) + η Ly Ly h X h C (h k+ ) + ηk (h k ) «/ (55) C (h k+ + nk h k )

22 Inverse problems for linear hyperbolic equations via a mixed formulation and then from Proposition 5, we get that y y h Z + «C (h k+ + nk h k ) + C h k. (56) ηδh η Similarly, λ λ h X + ηδh «δ h C (h k+ + nk h k ) + 3 ηδh C h k. From the last two estimates, we obtain the conclusion of the proposition. It remains now to deduce the convergence of the approximated solution y h for the L (Q T ) norm: this is done using the observability estimate (H). Precisely, we write that (y y h ) solves 8 L(y y >< h ) = Ly h in Q T ((y y h ), (y y h ) t )() H >: y y h = on Σ T. Therefore using (6), there exists a constant C(C Ω,T, C obs ) such that y y h L (Q T ) C(C Ω,T, C obs )( y y h L (q T ) + Ly h X) from which we deduce, in view of the definition of the norm Y, that y y h L (Q T ) C(C Ω,T, C obs ) max(, η ) y y h Z. (57) Eventually, by coupling (57) and Proposition 6, we obtain the following result : Theorem 3 (BFS element for N = - Rate of convergence for the norm L (Q T )) Assume that the hypothesis (H) holds. Let h >, let k {, } be a positive integer and let η <. Let (y, λ) and (y h, λ h ) be the solution of (7) and (4) respectively. If the solution (y, λ) belongs to H k+ (Q T ) H k (Q T ), then there exists two positives constant K = K( y H k+ (Q T ), c C (Q T ), d L (Q T ), C Ω,T, C obs ), independent of h, such that y y h L (Q T ) K max(, ) hk ( η + )(h 3 + «ηh) +. (58) η η δ h Remark Estimate (58) is not fully satisfactory as it depends on the constant δ h. In view of the complexity of both the constraint Ly = and of the structure of the space Z h, the theoretical estimation of the constant δ h with respect to h is a difficult problem. However, as discussed at length in [, Section.], δ h can be evaluated numerically for any h, as the solution of the following generalized eigenvalue problem (taking η = r, so that a r (y, y) is exactly y Z): j δ ff δ h = inf : Bh A r,h BT h {λ h } = δ J h {λ h }, {λ h } R m h \ {} (59) where the matrix A r,h, B h and J h are defined in (47).

23 Nicolae Cîndea, Arnaud Münch Table reports the values of δ h for r = and r = h for several values of h, T =, ω = (.,.3) and the BFS element. As in [] where the boundary situation is considered with more details, these values suggests that, asymptotically in h, the constant δ r,h behaves like : with C r >, a uniformly bounded constant w.r.t. h. δ r,h C r r as h + (6) h r = r = h Table ε = : T = - δ r,h for r = and r = h with respect to h. Consequently, in view of 6, the right hand side of the estimate (58) of y y h L (Q T ) behaves, taking η = r and r > so that max(, r ) =, like y y h L (Q T ) Kh k rh + r «and reaches its minimum for r = /h, leading to y y h L (Q T ) Kh k /. Eventually, when the space Z h is based on the HCT element, Theorem 3 and Remark still hold for k =. From [9, ch. VII, sec. 48, p. 95], we use that, for k {, }, there exists a constant C > such y Π Zh,T h (y) L (Q T ) C h k+, y Π Zh,T h (y) H (Q T ) C h k. (6) Then, we use that the error y y h L (Q T ) is again controlled by the error on the Lagrange multiplier λ through the term d(λ, Λ h ) in (43) to conclude. 4. Numerical approximation of the mixed formulation (9) We address the numerical approximation of the stabilized mixed formulation (9) with α (, ) and r >. Let h be a real parameter. Let Z h and e Λ h be two finite dimensional spaces such that Z h Z, e Λ h Λ, h >. The problem (9) becomes : find (y h, λ h ) Z h Λ e h solution of 8 < a r,α (y h, y h ) + b α (λ h, y h ) = l,α (y h ), y h Z h : b α (λ h, y h ) c α (λ h, λ h ) = l,α (λ h ), λ h Λ e h, (6) Proceeding as in the proof of [4, Theorem 5.5.], we first easily show that the following estimate holds.

24 Inverse problems for linear hyperbolic equations via a mixed formulation 3 Lemma Let (y, λ) Y Λ be the solution of (9) and (y h, λ h ) Z h f Λ h be the solution of (6). Then we have, 4 θ y y h Z + 4 θ λ λ h ar,α Λ e + b α + θ α a α c bα «+ θ + α + θ θ «inf y h Z h y h y Z inf λ h λ λ h Λ f Λ (63) h with a r,α max( α, η r), b α max(η /, α). Parameters θ and θ are defined in (). Concerning the space e Λ h, since Lλ h should belong to L (Q T ), a natural choice is eλ h = {λ Z h ; λ(, ) = λ t (, ) = }. (64) where Z h Z is defined by (49). Then, using Lemma and the estimate (55), we obtain the following result. Proposition 7 (BFS element for N = - Rates of convergence - Stabilized mixed formulation) Let h >, let k be a positive integer and let α (, ). Let (y, λ) and (y h, λ h ) be the solution of (9) and (6) respectively. If (y, λ) belongs to H k+ (Q T ) H k+ (Q T ), then there exists a positive constant K = K( y H k+ (Q T ), c C (Q T ), d L (Q T ), α, r, η) independent of h, such that y y h Z + λ λ h Λ Kh k. (65) In particular, arguing as in the previous section, we get Theorem 4 (BFS element for the N = - Rates of convergence for the norm L (Q T ) - Stabilized version) Assume that the hypothesis (H) holds. Let h >, let an integer k. Let (y, λ) and (y h, λ h ) be the solution of (9) and (6) respectively. If the solution (y, λ) belongs to H k+ (Q T ) H k+ (Q T ), then there exist a positive constant K = K( y H k+ (Q T ), λ H k+ (Q T ), c C (Q T ), d L (Q T ), α, r, η) independent of h such that y y h L (Q T ) K hk η. (66) 5 Numerical experiments We now report and discuss some numerical experiments corresponding to mixed formulation (4) and (6) for N = and N =.

25 4 Nicolae Cîndea, Arnaud Münch 5. One dimensional case (N = ) We take Ω = (, ). In order to check the convergence of the method, we consider explicit solutions of (). We define the smooth initial condition (see [8]): (EX) ( y (x) = 6x ( x), y (x) = (3x 4x 3 ) (,.5) (x) + (4x 3 x x (, ) + 9x ) (.5,) (x), and f =. The corresponding solution of () with c, d is given by y(x, t) = X a k cos(kπt) + b «k kπ sin(kπt) sin(kπx) k> with a k = 3 (π k ) π 5 k 5 (( ) k ), b k = 48 sin(πk/) π 4 k 4, k >. We also define the initial data in H (Ω) L (Ω) (EX) y (x) = x, y (x) = (/3,/3) (x), x (, ) for which the Fourier coefficients are a k = 4 π k sin(πk/), b k = (cos(πk/3) cos(πk/3)), k >. πk 5.. The cylindrical case: q T = ω (, T ) We consider the case ε = described in Section. We take ω = (.,.3) and T = for which the inequality (H) holds true. We consider the BFS finite element with uniform triangulation (each element K of the triangulation T h is a rectangle of lengths x and t so that h = p ( x) + ( t) ). We recall that the direct method amounts to solve, for any h, the linear system (48). We use the LU decomposition method. Table collects some norms with respect to h for the initial data (EX) for r = and for x = t. We observe a linear convergence for the variables y h, λ h for the L -norm: y y h L (Q T ) y L (Q T ) = O(h.3 ), y y h L (q T ) y L (q T ) = O(h.98 ), λ h L (Q T ) = O(h.98 ). In agreement with Remark, since y obs is by construction the restriction to q T of a solution of (), the sequence λ h, approximation of λ, vanishes as h. The L -norm of Ly h do also converges to with h, with a lower rate: (67) Ly h L (Q T ) = O(h.7 ). (68) We also check that the minimization of the functional Jr introduced in Proposition 3 leads exactly to the same result: we recall that the minimization of the

26 Inverse problems for linear hyperbolic equations via a mixed formulation 5 h y y h L (QT ) y L (QT ) y y h L (qt ) y L (qt ) Ly h L (Q T ) λ h L (Q T ) κ card({λ h }) CG iterates Table Example EX - r = - T = - y L (q T ) = y L (Q T ) =.59. functional Jr corresponds to the resolution of the associate mixed formulation by an iterative Uzawa type method. The minimization is done using a conjugate gradient algorithm ( we refer to [, Section.] for the algorithm). Each iteration amounts to solve a linear system involving the matrix A r,h which is sparse, symmetric and positive definite. The Cholesky method is used. The performance of the algorithm depends on the conditioning number of the operator P r : precisely, it is known that (see for instance [4]), q p «λ n ν(pr ) n λ L (Q T ) ν(p r ) p λ λ L ν(pr ) + (Q T ), n. ν(p r ) = P r P denotes the condition number of where λ minimizes Jr r the operator P r. As discussed in [, Section 4.4], the conditioning number of P r restricted to Λ h L (Q T ) behaves asymptotically as Cr h. Table reports the number of iterations of the algorithm, initiated with λ = in Q T. We take ɛ = as a stopping threshold for the algorithm (the algorithm is stopped as soon as the norm of the residue g n given here by Ly n satisfies g n L (Q T ) g L (Q T )). Table reports the number of iterates to reach convergence, with respect to h. We observe that this number is sub-linear with respect to h, precisely, with respect to the dimension m h = card({λ h }) of the approximated problems. This renders this method very attractive from a numerical point of view. From Remark 6, we also check the convergence w.r.t. h when we assume from the beginning that the multiplier λ vanishes (see Table 3). This amounts to minimize the functional J r given by (7) or, equivalently, to perform exactly one iteration of the conjugate gradient algorithm we have just discussed. With r =, we observe a weaker convergence : y y h L (Q T ) y L (Q T ) = O(h.574 ), y y h L (q T ) y L (q T ) = O(h.94 ). (69) This example illustrates that the convergence of Ly h to in the norm L (, T, H (, )) is enough here to guarantee the convergence of the approximation y h : we get that h Ly h L (Q T ) Ly h L (,T ;H (,) = O(h.3 ) while Ly h L (Q T ) slightly increases. Obviously, in this specific situation, a larger r (acting as a penalty term) independent of h yields a lower Ly h L (Q T ) norm.