FLUX IDENTIFICATION IN CONSERVATION LAWS 2 It is quite natural to formulate this problem more or less like an optimal control problem: for any functio

Transcription

1 CONVERGENCE RESULTS FOR THE FLUX IDENTIFICATION IN A SCALAR CONSERVATION LAW FRANCOIS JAMES y AND MAURICIO SEP ULVEDA z Abstract. Here we study an inverse problem for a quasilinear hyperbolic equation. We start by proving the existence of solutions to the problem which is posed as the minimization of a suitable cost function. Then we use a Lagrangian formulation in order to formally compute the gradient of the cost function introducing an adjoint equation. Despite the fact that the Lagrangian formulation is formal and that the cost function is not necessarily dierentiable, a viscous perturbation and a numerical approximation of the problem allow us to justify this computation. When the adjoint problem for the quasilinear equation admits a smooth solution, then the perturbed adjoint states can be proved to converge to that very solution. The sequences of gradients for both perturbed problems are also proved to converge to the same element of the subdierential of the cost function. We evidence these results for a large class of numerical schemes and particular cost functions which can be applied to the identication of isotherms for conservation laws modelling distillation or chromatography. Key words. method Inverse problem { Scalar conservation laws { Adjoint state { Gradient AMS subject classications. 35R30, 35L65, 65K10, 49M07 1. Introduction In this paper, we are interested in the following inverse problem: consider the scalar hyperbolic conservation law t w x f(w) = 0; x 2 IR; t > 0; together with the Cauchy data (2) w(x; 0) = w 0 (x) 2 BV (IR) \ L 1 (IR): It is well-known that there exists one and only one entropy solution in L 1 (IR + ; BV (IR)) \ L 1 (IR (?1; +1)) of (1)-(2) (see [6], [18]), and we emphasize the fact that the unique entropy solution to (1) depends continuously (in a sense which we shall precise) on the smooth function f by denoting it w f. The question we address is whether, an observation w obs at time T > 0 being given, one can identify the non linearity f such as w f at time T is as close as possible to w obs. This work has been partially supported by the Program A on Numerical Analysis of FONDAP in Applied Mathematics and the Universidad de Concepcion (P.I and In). y Mathematiques, Applications et Physique Mathematique d'orleans, UMR CNRS 6628, Universite d'orleans, B.P. 6759, F Orleans Cedex 2, FRANCE (james@cmapx.polytechnique.fr). z Departamento de Ingeniera Matematica, Facultad de Ciencias Fsicas y Matematicas, Universidad de Concepcion, Casilla 4009, Concepcion, CHILE (mauricio@ing-mat.udec.cl). 1

2 FLUX IDENTIFICATION IN CONSERVATION LAWS 2 It is quite natural to formulate this problem more or less like an optimal control problem: for any function v : IR! IR we dene a cost function J(v), and we look for an f solving (3) min f J(w f (:; T )); thus giving a precise meaning to the sentence \as close as possible". Therefore we are led to the constrained optimization problem of minimizing J(w(:; T )) under the constraint for w to satisfy the partial dierential equation (1)-(2). This problem can be viewed as well as an unconstrained minimization problem: if we set J(f) ~ = J(wf ), then problem (3) boils down to minimizing J ~ on a suitable set of functions. In theory, this inverse problem is in general ill-posed in uniqueness when there are discontinuities in the solution. For instance, a well-known undesirable case appears when we try to identify f over a shock wave with a propagation speed equal to : there are innitely many functions f giving the same entropic solution w f of (1)-(2) equal to the shock wave (see [4] for more details). Yet, as far as applications are concerned some interesting practical problems can be found: it is possible to resolve the identication of f (or \a part of f") via a gradient technique in order to compute numerically the minimum of J. ~ This was achieved in a preceding paper [9] in which we considered the identication problem arising from a model of diphasic propagation in chromatography. Therefore we dealt with a system of conservation laws, and we obtained successful numerical results because the function f was given a precise analytic form, so the minimization occurred on IR n, and we chose adequate criteria for the cost function linked to the physical parameter of the problem. The classical gradient technique used in order to obtain the gradient of J, ~ consists in writing a Lagrangian formulation for the constrained problem and in introducing the adjoint state. This has to be done at two levels. We rst consider a formal level, that is we take a solution of the continuous equation (1), and perform the computations. We obtain a backward linear hyperbolic equation for the adjoint state. The trouble is that this equation is ill-posed as soon as the solution of (1) is not smooth { which is of course the case in most of the applications. This is related to the fact that the inverse problem is ill-posed in uniqueness when there are discontinuities in the solution. Thus, in general, the computation of the gradient of J ~ remains formal. Furthermore, it is easy to nd some counterexamples where the gradient does not exist. On the other hand, we can perform the same computations at a discrete level, that is when both the equation (1) and the cost function J are discretized. This introduces a \discrete adjoint state" which we call adjoint scheme, and we obtain the gradient for the discretization of J, ~ which is well-dened. Thus we are able to perform numerical computations, using standard conjugate gradient techniques, and the numerical evidence is that the method seems to converge (see [9], Section 5 and [11] for application on real data). The aim of this paper is to interprete and justify the convergence of the method in the scalar case, and in a particular case, namely when the solution

3 FLUX IDENTIFICATION IN CONSERVATION LAWS 3 of the adjoint state is Lipschitz continuous. We shall consider two modied problems : rst we add a viscous term to (1), then we turn to the discretized problem. In both cases, we prove that the perturbed adjoint states converge to the solution of the original problem. That enables us to pass to the limit in the approximation of the gradient, and we prove that both approximations tend to the same limit. This limit is not necessarily the gradient of the cost function because the gradient does not exist a priori. In fact, we also prove by means of convexity hypotheses that it is an element of the sub-dierential of J. ~ This result gives an interpretation of the formal computation of the gradient for continuous cost functions including some cases when the gradient does not exist. Therefore the paper is organized as follows. First we precisely state the problem, in particular concerning the cost function we consider, which is not the standard least square function. Then we consider the identication problem for a parabolic regularization of the conservation equation, and in particular, we prove the dierentiability of the cost function. We also prove the convergence of the sequence of the perturbed gradients to an element of the sub-dierential of ~J. Finally we prove that we can obtain the same element of the sub-dierential via a discretized problem and for a large class of numerical schemes, and we illustrate these results by a numerical application on experimental data. 2. The identication problem 2.1. The cost function A classical example of cost function J arises in the well-known output least square method (see [4] for instance) : (4) J 0 (w) = 1 2 x2ir jw(x; T )? w obs (x)j 2 dx: For practical reasons, in [9] the following modied cost function J was used : (5) J (w) = J 0 (w) + 2 j 1? w(; T )? 1 (w obs )j 2 ; where is a constant parameter to be adjusted, and where 1 (X) is the rst moment of the function X : IR! IR: (6) 1 (X) = IR xx(x)dx: Roughly speaking, the advantage of J over J 0 is that it is more sensitive to the localization of the observed signal on the x-axis, whereas J 0 essentially takes into account the shape of the signal, independently of its localization. Notice that we shall always consider initial data with compact support so that, by the nite velocity of propagation, the solution at any time t > 0 will

4 FLUX IDENTIFICATION IN CONSERVATION LAWS 4 also have a compact support. Thus all the integrals in the denitions of J 0 and of J have a meaning as soon as w is in L 1 (IR). For practical reasons, what follows is essentially focused on the study of criteria (4) or (5)-(6) Existence and Lipschitz continuous dependence We can assure the existence of at least one solution f of our identication problem (3) when we search it in a compact of the Lipschitz continuous functions. In fact, this is a consequence of the following theorem, the proof of which is contained in a paper by B. Lucier ([13]): Theorem 2.1. The application f 7! w f is Lipschitz continuous from the space of Lipschitz functions to L 1, that is (7) kw f (:; t)? w g (:; t)k L 1 tkf? gk Lip kw 0 k BV : following result of existence : Corollary 2.2. The function J is Lipschitz continuous from Lip(IR) to IR +. Proof. Concerning the function J 0, the estimate follows immediately from the L 1 estimate which holds uniformly in g: (8) 8t > 0; kw g (:; t)k L 1 kw 0 k L 1: Indeed we have jj 0 (g)? J 0 (f)j 1 2 IR j(w g? w f )(x; T )j j(w g + w f )(x; T )? 2w obs (x)j dx max(kw 0 k L 1; kw obs k L 1) kw g (; :T )? w f (:; T )k L 1 by Holder's inequality. The result follows by (7). Now for the momentum criterion, if we assume all the supports included in [?L; L] for some L large enough, we get, using again (8): jj 1 (g)? J 1 (f)j 1 2 j 1(w g (:; T ))? 1 (w f (:; T ))j j 1 (w g + w f (:; T ))? 2 1 (w obs )j L 3 max(kw 0 k L 1; kw obs k L 1) kw g (; :T )? w f (:; T )k L 1: By considering a minimizing sequence for J, we easily deduce form Corollary 2.2 the following existence result: Corollary 2.3. If f 2 F a compact set of the Lipschitz continuous functions, then there exists at least one solution of the identication problem (3), with the cost function dened by (4) or (5)-(6). Remarks. The compactness of the set F is a necessary hypothesis, but it is not a restrictive condition for a lot of practical identication problems : the

5 FLUX IDENTIFICATION IN CONSERVATION LAWS 5 function f can have a precise analytic form so that the minimization occurs on a bounded subset of IR n (for instance, see [9]). Another way to obtain Lipschitz compactness is to seek f for instance in W 2;1 (IR). We cannot ensure the uniqueness of the solution in the corollary 2.3 for the reasons exposed in the introduction. Obviously, we can try to modify the cost functions to obtain a strictly convex functional and, for instance, search the ux f with minimal W 2;1 (IR) norm. Yet in general, this is an arbitrary mathematical condition. Thus we prefer to deal with the cost functions dened by (4), (5) and (6), which have a realistic physical sense, and to leave the uniqueness problem as an open problem Remarks on dierentiability Since we are concerned with the problem of minimizing ~ J(f) = J (w f ) with respect to f, dierentiability is of course of crucial importance: optimality conditions, gradient algorithms rely on it. For the function ~ J, the problem is open in general, and we are going to point out precisely the diculties. These will come of course from the operator f 7! w f, since the functions J 0 and J 1 are smooth convex functions. Let us rst consider this operator in the nice case where all the involved functions are smooth. Indeed, if w 0 2 C 1 (IR) and f 2 C 1 (IR), then there exists T > 0 such that w f 2 C 1 (IR [0; T ]). Let us use the notations of the calculus of variations, denoting by w the variation of w f corresponding to some variation f of f. Then w has to solve (9) 8 < t w x [f 0 (w f )w] x [f(w f )] = 0 w(:; 0) = 0: Under the above assumtions, we are faced with a standard linear conservation equation with smooth coecients. Actually, applying the inverse functions theorem to F : C 1 (IR) Lip(IR)! C 0 (IR) C 0 (IR) (w; f) 7! (@ t w x f(w); w(:; 0)? w obs ) we can prove that f 7! w f is dierentiable in a strong sense from Lip(IR) to C 1 (IR). Thus ~ J is also dierentiable, with (10) ~J 0 (f)f = IR (w f (x; T )? w obs (x))w(x; T ) dt + + ( 1 (w f (:; T ))? 1 (w obs )) 1 (w(:; T )): This by the way justies in this case the integrations by parts we perform in the next section. We would like to point out now that, if w f happens to be discontinuous, then the resolution of (9) is much more dicult. Indeed it becomes a conservation law with discontinuous coecient (f 0 (w f )) and a measure-valued source term

6 FLUX IDENTIFICATION IN CONSERVATION LAWS 6 (@ x f(w f )). The solution has therefore to be seeked in the class of measures on IR, generalizing the results obtained by Bouchut and James [2, 3] in the absence of source term. Up to now, this is possible only if f is convex. In this context, it is reasonable to hope for some very weak dierentiability result, the involved topology being the usual weak convergence of measures. The general case of a non convex f is still completely open. However, even the dierentiability of f 7! w f does not settle the problem for J(f). ~ Indeed since w is a measure, (10) is meaningful if wf (:; T )? w obs 2 L 1 (w(:; T )), which is a priori not obvious. In the same way, 1 (w(:; T )) has to be dened. We refer to the next section for an explicit example and further comments. For all these reasons, we leave the problem of a rigorous dierentiation of ~J (and possibly the choice of theoretically more convenient cost functions) to a future work. We focus in the sequel on the study of two approximated problems where all the involved quantities are well-dened. We prove when it is possible the convergence of these problems to the original one. We begin by a formal computation of the gradient of J, ~ as it was done in [9]. The basic tool for that is to consider the constrained minimization problem and its Lagrangian formulation Lagrangian formulation and adjoint problem In [9] and [17], we formally obtain the gradient through the following Lagrangian formulation for the constrained minimization problem: (11) L (w; p; f) def = J(w)? E (w; p; f) ; where E (w; p; f) is a weak form of (1), dened by (12) E (w; p; f) =? T L 0 0 (w@ t p + f(w)@ z p) + t=t wp? t=0 w 0 p: We are interested For that purpose, we take p solution to the following backward adjoint problem (13) 8 < t p + f 0 (w f )@ x p = 0; x 2 IR; t T; p(x; T ) = p T (x); where f 0 (w f ) represents the derivative of f with respect to w evaluated on the solution w f of (1). The function p T depends on J, w, and w obs. More precisely, we have (14) p T w = D w J(w f )w; 8w 2 D(IR); IR where D w Jw represents the derivative of J in the direction w. The problem (13) is called the adjoint problem associated to the direct problem (1).

7 FLUX IDENTIFICATION IN CONSERVATION LAWS 7 Thus we can compute the gradient of ~ J by the formula (15) D ~ J(f)f = T +1 x p f(w f )dxdt where p is solution of equation (13). Remark. Formula (15) is a formal result, since the derivative of function J e does not necessarily exist, as a rule. For instance, consider the entropic solution of the Riemann problem 1 w k (x; t) = if x kt; 0 otherwise; for the Burger's equation { f(w) = kw 2 { and suppose w obs (x) = w k0 (x; T ) for some given k 0. Then we have that the cost function given by the criterion of the norm L 2 (4) is equal to (16) ~J 0 (k) = J 0 (w k ) = T jk? k 0 j; which is not dierentiable in k 0. Moreover the backward equation dening p is ill-posed as soon as discontinuities occur in the solution of the direct problem: actually the solution is not uniquely dened by the characteristics. If we assume that we are in the neighbourhood of a minimum, the function ~ J is locally convex, so the ~ J is a non-empty set. We may hope that D ~ J(f) dened by (15) is an element ~ J when the adjoint equation is ill-posed. We are not going to answer this question here. We will restrict ourselves to the particular case where there exists a smooth solution to the adjoint equation. In this case we shall study whether (15) is well-dened and whether it is an element of the subdierential. First, we give an existence result for Lipschitz solutions to the adjoint problem (13), then, conditions for the uniqueness. The function a(x; t) veries the One-Side-Lipschitz-Continuous condition (OSLC) when there exists a function m 2 L 1 (0; T ) such as (17) + L + (a(x; t)) def a(x; t)? a(y; t) = ess sup m(t) x? y x6=y In other words, the condition (OSLC) means that the function a(; t) must be Lipschitz continuous for all t, when a(; t) is increasing, and it allows decreasing jumps a(x?; t) > a(x+; t). This condition has been used by several authors, e.g. Oleinik [14], Conway [5], Ho [8], Tadmor [19] to prove the existence of at least one Lipschitz continuous solution to the adjoint problem (13), when p T 2 W 1;1 (IR). A rened version of Oleinik's entropy condition (Ho [8]) states that f 0 (w) veries (OSLC) condition when w accepts only entropic shocks as discontinuties and when f is convex. We need this result to make sense out of solving p by the characteristics method. So far, we do not know a general existence result for any f. The problem with this result is that its solution is

8 FLUX IDENTIFICATION IN CONSERVATION LAWS 8 ill-posed in uniqueness : for instance, if we assume that f 0 (w f )(x; t) =?sign(x), and kp T k W 1;1 (IR) > 0, then it is easy to verify that p p(x; t) = T (x? sign(x)(t? t)) when t + jxj T; '(t + jxj) otherwise ; are Lipschitz continuous solutions of adjoint problem (13), for any '(t) Lipschitz continuous function such that '(T ) = p T (0). We recall briey here the denition of the so-called \reversible solutions" introduced in [2, 3], for which uniqueness holds. First dene the set E of \exceptional solutions" as the space of all the Lipschitz continuous solutions of (13) with p T = 0. Next, we introduce the open set called \support of exceptional solutions": (18) V def = f(t; x) 2]0; T [IR j 9p e 2 E; p e (t; x) 6= 0g; The following result is obtained in [3]: Theorem 2.4. Let p a Lipschitz continuous solutions of (13). Then, both properties are equivalent (i) p is locally constant in V ; (ii) there exists p 1 and p 2 in Lip loc ([0; T ]IR), t p i +f 0 (w)@ x p i = 0 x p i 0, such as p = p 1? p 2. Furthermore, for all p T 2 W 1;1 (IR) there exist one and only one Lipschitz continuous solution p of (13) verifying one of these properties and the following estimate (19) kp(; t)k W 1;1 (IR) kp T k W 1;1 (IR) exp ( T t m()d ) 8t T: This function p is called the reversible solution of (13). Remarks. According to property (i) of theorem 2.4, we can choose a constant for p in each fan-wise set dened by all the characteristic straight lines which converge to a discontinuity of w. This constant is equal to the value of p T (x), where (x; T ) is a point in the discontinuity of w. Property (ii) is a monotonicity property of the reversible solution. Choosing the reversible solution in counterexample (16) is equivalent to choosing the characteristic element of the subdierential of J0 ~ (k 0 ) equal to zero. This is not an arbitrary choice. At the limit, we will see that we obtain an element of the subdierential of J ~ characterized by the reversible solution, when we consider viscosity approximation and some classical numerical schemes. When we consider the cost function (4) or (6), the hypothesis p T 2 W 1;1 (IR) is equivalent to (w(; T )?w obs ) 2 W 1;1 (IR). In practice, this is a very restrictive hypothesis in the sense that it means a regular solution w, or at least a solution in which the shocks at t = T are cancelled with the shocks of the observation w obs. When w is regular (without shocks), it is clear that we do not need the notion of reversible solution: in this case we have seen that the cost function is dierentiable. The resolution of (13) when p T =2 W 1;1 (IR) remains an open problem.

9 FLUX IDENTIFICATION IN CONSERVATION LAWS 9 3. Articial viscosity We introduce the classical viscous regularization of the equation (1): (20) 8 < t w " x f(w " ) = "@ 2 xxw " ; x 2 IR; t > 0 w " (x; 0) = w 0 (x); It is well-known that (20) admits a unique smooth solution which approaches the entropy solution of (1) in the following sense (see e.g. Smoller [18]): Theorem 3.1. We suppose that w 0 2 BV (IR). Then, (i) Problem (20) has a solution w " (; t) 2 BV (IR) for all t > 0. This solution is C 1 on IR (0; 1). Furthermore, for all t > 0, kw " k L 1 (IR) constant; T V (w " (; t)) T V (w 0 ) (ii) The only accumulation point in L 1 (IR (0; +1))-weakly, and L 1 loc (IR (0; +1))-strong, of the sequence w " is the entropic solution w of (1). Now we shall limit ourselves to the (restrictive) case where the nal data for the adjoint state is in W 1;1 (IR), and consider the following minimization problem (21) min J e " (w f " ); where J " (w f " ) = J (w f " ) f and w " f is the solution of the parabolic problem (20), and the cost function J is dened by (5). We are going to proceed in three steps. First we compute the exact derivative of the function e J ". Then we shall prove that the adjoint equation is wellposed, which will give a characterization of the derivative. Finally, we are going to prove the convergence of the adjoint state and of the derivative of the viscous problem to the corresponding quantities for the hyperbolic problem, when " tends to Derivative of the viscous cost function We want to determine the Gateaux-derivative of e J ". We shall prove that the directional derivative (22) D J e " (f)f = lim h e J e J " " (f + hf)? J e " (f) (f) = lim h!0 h!0 exists for all Lipschitz continuous function f (f is called a Lipschitz direction). We have Proposition 3.2. Let w f " be the solution of the viscous problem (20). We suppose that the ux f = f(w " ) is of class C 1 and Lipschitz continuous with respect to w " with a Lipschitz constant C Lip. Then the limit (22) exists for all Lipschitz direction f (which we can suppose with the same Lipschitz constant C Lip ). It is characterized by h ; (23) D e J " (f)f = D w J (w " f )w " 1 ;

10 FLUX IDENTIFICATION IN CONSERVATION LAWS 10 where w " 1 is the solution in L2 (0; T; L 2 (IR)) \ C 1 (IR (0; +1)) of the following linear parabolic problem (24) (25) 8 < t (w " 1) x? f 0 (w " f )w " 1 + f(w " f ) = "@ 2 xx w" 1 ; w " 1(x; 0) = 0; Proof. Recall that J = J 0 + J 1. Then we can write 8 >< >: D w J 0 (w " f )w " 1 = IR? w " f (x; T )? w obs (x) w " 1(x; T )dx; D w J 1 (w " f )w " 1 =? 1? w " f (; T )? 1 (w obs ) 1? w " 1 (; T ) ; and we remark that 8 w " h J 0 (w f >< " f + w f " ) = h (x; T )? w obs (x) h w " (x; T )dx IR 2 (26)! >: h J 1 (w f " ) = 1(w f " (; T )) + 1(w f " (; T )) h? 1 (w obs ) 1 ( h w " (; T )); 2? w" f where f h = f + hf and h w " = w" f h. h We rst prove the convergence of the derivative of the L 2 -criterion: (27) hj0 e (w f " )! D w J 0 (w f " )w " 1 ; when h! 0: We have that w f " and w" f are solutions of the parabolic problem (20) with the h respective non-linear ux f and f h. Thus, if we take the dierence between the equation in w f " and the equation in w" f, we deduce that h h w " is solution of t h w " x f h (w " f h )? f h (w " f ) h + f(w " f )! = "@ 2 xx h w " : We multiply this equation by h w " and we integrate by parts. Using classical estimate arguments and the Gronwall lemma, we prove (29) k h w " (; t)k H 1 C(") 8t 2 (0; T ); where C(") is a constant which does not depend on h. Thus, up to a subsequence, there exists w " 1(; t), such that h w " (; T ) * w " 1(; T ) in H 1 (IR)? weak when h! 0: We can easily verify that w " f h (; T )! w " f (; T ) in L2 (IR)?strong when h! 0. Hence we deduce (27).

11 FLUX IDENTIFICATION IN CONSERVATION LAWS 11 Next we prove the convergence of the derivative of the rst moment criterion: (30) hj1 e (w f " )! D w J 1 (w f " )w " 1 ; when h! 0: We deduce from the compact support of w 0, and from the maximum principle applied to the equations (20), (24) and (28) that the functions jw f " (; T )j, jw f " (; T )j, j h h w " (; T )j and jw 1(; " T )j are bounded by a function g(x) = Ce?rjxj where C; r are constants which depend only on T and kw 0 k L 1, and thus are independent of h. We obtain that j 1 ( h w " (; T ))? 1 (w " 1(; T ))j R?R jx ( h w "? w " 1) (x; T )jdx + jxj>r jxg(x)jdx: Using the convergence L 2 -weak of h w " (; T ) we have that the rst term of the right hand side converges to 0. The second term uniformly converges to 0 in h, when R! 1. This implies the convergence 1 ( h w " (; T ))! 1 (w 1(; " T )), when h! 0. In the same way, and using the convergences L 2 -strong of w f " (; T ), we have that h 1 (w f " (; T ))! h 1 (w f " (; T )), when h! 0. From these convergences of the moments, we deduce the result (30). Finally, we prove that w " 1 is the solution of (24). Using (29) and the compact injection Hloc 1 (IR),! L2 loc (IR), we have the following strong convergence (31) h w " (; t)! w " 1(; t) in L 2 loc(ir)? strong; when h! 0: We multiply the equation (28) by a test function ' 2 C 1 0 ([0; +1) IR) and we integrate by parts. Hence, by passing to the limit in (28), we obtain that w " 1 is a weak solution of (24). Therefore, by the existence and uniqueness of the solution of the linear parabolic problem (24), we have that function w " 1 is the strong solution in L 2 (0; T; L 2 (IR)) \ C 1 (IR (0; +1)) of this equation Viscous adjoint problem We showed that the Gateaux derivative of J e " (f) is well-dened for cost function J dened by (5). Now we shall use the Lagrangian formulation in order to give a characterization of this derivative. First we dene the weak form associated to (20) by T +1 E " (w " ; p " def ; f) =? (w t p " + f(w " )@ x p "? "@ x w x p " ) + 0 t=t?1 w " p "? t=0 w " p " ; and the Lagrangian by L " (w " ; p " ; f) = J (w " )? E " (w " ; p " ; f), where p " is a regular function in (x; t). We take p " equal to the solution of the following

12 FLUX IDENTIFICATION IN CONSERVATION LAWS 12 backward parabolic equation (called viscous adjoint problem): (32) 8 < t p " + f 0 (w " )@ x p " =?"@ 2 xxp " ; p " (x; T ) = p T (x); x 2 IR; t < T where the nal condition p T is dened by (14). In this case, the derivative of the Lagrangian with respect to w " is equal to zero, and the Gateaux derivative of the cost function is characterized by (33) D e J " (f)f = T +1 x p " f(w " f ): Equation (32) is a parabolic linear equation, and it is known that it only admits one solution in L 2 (0; T; L 2 (IR)) \ C 1 (IR (0; 1)) which depends on w ". On the other hand, if f 00 > 0, we can prove the following (OSLC) estimate (see [19]) by a maximum principle argument applied on the equation (20): (34) L + (f 0 (w " (x; t))) L+? w tl + (w 0 ) = m(t) 2 L1 (0; T ): From this result, we shall deduce BV and W 1;1 estimates on the adjoint state. Theorem 3.3. We consider the solution p " (x; t) of linear parabolic problem (32), with p T 2 W 1;1 (IR) \ BV (IR). We suppose f 00 > 0, with, constants independent of w. Then (x; t) 7! f 0 (w f (x; t)) veries the (OSLC) condition, and (35) (36) kp " (; t)k L 1 (IR) kp T k L 1 (IR) ; k@ x p " (; t)k L 1 (IR) kp T k BV (IR) ; k@ x p " (; t)k L 1 kp T k W 1;1 exp ( T t ) m(t? )d ; (37) for all t T, where m 2 L 1 (0; T ) is the function dened in (34). Proof. Estimates (35) and (36) are classical results of the theory of nonlinear hyperbolic equations, and the proofs can be found in [12] and [6]. The proof of the estimate (37) is very similar to the arguments used by Tadmor [19]. Let us recall them briey. We consider p and p " solutions to the problems (13) and (32) respectively. Let q " (x; t) = p " (?x; T? t) and " (x; t) x q " (x; t). We dierentiate equation (32), and we notice that the function " veries t " + f 0 (w " (?x; T? t))@ x " =? (@ x f 0 (w " (?x; T? t))) " + "@ 2 xx "

13 FLUX IDENTIFICATION IN CONSERVATION LAWS 13 Let 2 be an even integer. We multiply (38) by "?1 and we integrate by parts. We obtain (39) d dt k "(; t)k L x2ir =? (? 1)? (w" (?x; T? t)) x2ir (@ x " ) 2?2 " dx: " dx Using the (OSLC) inequality (34) and the Gronwall lemma in the equation (39), we deduce 8t T (40) k " (; t)k L k " (; 0)k L exp (?? 1 T t m(t? )d; We pass to the limit when tends to +1 in (40). By the denition of ", we deduce (37). ) : 3.3. Convergence of the method Now we prove that the articial viscosity method converges in the sense that the sequences w " and p " converge respectively to the entropy solution of (1) { which is logical { and to the reversible solution of the adjoint equation. Moreover the sequence of the derivatives of e J " also converges towards an element of the subdierential of e J. Using these BV and W 1;1 estimates we have the following convergence result concerning the adjoint state: Theorem 3.4. We consider the solution p " (x; t) of the linear parabolic problem (32). We suppose that the ux f satises f 00 > 0, and p T a function of W 1;1 (IR) \ BV (IR). Then (41) p "! p uniformly in ; where =! (0; T ),! is a compact of IR, and p is the reversible solution of (13) given by the theorem 2.4. Proof. The functions p " x p " are bounded in L 1 (by (35) and (37)). We can extract a subsequence, still denoted by p ", and we have (42) p " (; t) * p(; t) in W 1;1 (IR)? weak; By the Rellich-Kondrachov theorem (see R. Adams [1]) we deduce the strong convergence in C 0; (!) for all 0 < < 1, and! compact set of IR, and using a classical diagonalization argument (see for instance [6]), we obtain the uniform convergence in. Now, in order to prove that p is a Lipschitz continuous solution of (13), we multiply the backward parabolic equation (32) by a test function ' 2 C 1 0 (IR

14 FLUX IDENTIFICATION IN CONSERVATION LAWS 14 (0; +1)), and we integrate by parts. We have (43) 0 =?? 1 0 IR IR fp t '? f 0 (w " )@ x p " '? "@ x p x 'g dxdt p T (x)'(x; 0)dx: On the other hand, multiplying equation (32) by p(; t), integrating by parts, and using Gronwall's lemma, we deduce that (44) " 1 2 k@x p " k L 2 (IR(0;T )) C; where C is a constant independent of ". That implies (45) "@ 2 xx p"! 0; in L 2 (0; T; H?1 (IR))? strong: Now, we know by Theorem 3.1 that w "! w in L 1 loc (IR (0; +1))-strong, and by Lebesgue's dominated convergence theorem we have f 0 (w " )! f 0 (w), in L 1 loc (IR (0; +1))? strong. Using the convergence W 1;1 -weak of p " (42), we deduce (46) f 0 (w " )@ x p " * f 0 (w)@ x p in D 0 (IR (0; +1)): Using convergence results (45), (46), and the uniform convergence of p ", we let "! 0 in (43). We obtain (47) 0 =? 1 0 IR fp@ t '? f 0 (w)@ x p'g dxdt? IR p T (x)'(x; 0)dx: Thus the limit of p " is solution of the backward transport equation (13) in the sense of distributions. In keeping with the W 1;1?weak convergence (42), we obtain that the limit p veries p(; t) 2 W 1;1 (IR), 8t T. On the other hand, by letting "! 0 in (37), we obtain inequality (19). In order to prove the convergence of the whole sequence p ", we will prove that the limit of any converging subsequence is the unique reversible solution, which we note p r. For that, we suppose that p "! p, and using the denition (18) of the support of exceptional solutions, at rst we have (48) p = p r a:e: (x; t) 2 IR (0; T ) n V; We set " (x; t) x p " (?x; T? t) and r (x; t) x p r (?x; T? t). Then, we substract the equation (13) from (32) and we dierentiate with respect to x. We t ( "? r ) =?@ x (a " ( "? r x ((a "? a) r ) + "@ xx " ; where a " = f 0 (w " (?x; T? t)) and a = f 0 (w(?x; T? t)). We multiply this last equation by 2( "? r ), and we integrate by parts. Using the (OSLC) inequality

15 FLUX IDENTIFICATION IN CONSERVATION LAWS 15 (34), we deduce (49) d dt n k " (; t)? r (; t)k 2 L 2 (IR) o + "k@ x " (; t)k 2 L 2 (IR) m(t? t)k " (; t)? r (; t)k 2 L 2 (IR)???" +1?1 +1 (a "? a)( "? r )@ x r dx (@ x a x a)( "? r ) r dx?1 xx 2 "?1 rdx From (42), (41), (45) and (46), we have that the 2 nd and 4 th terms on the right hand side converge to 0 when "! 0. On the other hand, from property (i) of theorem 2.4 we have that r = 0 in V t = fx j (?x; T? t) 2 Vg, and using the equality (48), we deduce +1?1 (@ x a x a)( "? r ) r dx = x2irnv t (@ x a x a)( "? r ) r dx! 0; when "! 0. Then, the 3 rd term on the right hand side converges to 0. We can pass to the limit in (49) and using Gronwall's lemma, we obtain lim "!0 " = r and consequently p = p r. Remark. Another way to prove this last result is to make use of the characterization (ii) of theorem 2.4. Indeed standard arguments allow us to prove that x p T 0, x p " 0. Now, for any nal data p we rewrite p T = p T 1? p T 2, x p T 1 = (@ x p T ) + x p T 2 = (@ x p T )?. We denote by p " i the solution of (32) with nal data p T i. Therefore we have p" = p " 1? p" 2, x p " i 0. We know that p "! p, p " i! p i x p i 0, and p i solution of (13). Thus p is reversible. A similar monotonicity argument will be used for numerical schemes. Now we turn to the convergence of the derivative. A fundamental consequence of the convergence of the solution of the adjoint problem with articial viscosity is the convergence of D e J " when "! 0. More precisely, we have Theorem 3.5. Let e J " : f 7! J(w " f ) be the cost function (21) dened for all Lipschitz continuous function f, with w f " solution of the parabolic problem (32), and assume that w 0 has compact support. We suppose that the ux f satises f 00 > 0, and p T a function of W 1;1 (IR) \ BV (IR). Then we have, if w f is the entropy solution of (1), and p f the reversible solution of (13), (50) D e J " (f)f! T +1 x p f f(w f )dxdt; when "! 0; for all Lipschitz direction f.

16 FLUX IDENTIFICATION IN CONSERVATION LAWS 16 Proof. From theorem 3.4, and the continuity of f, we obtain the following result 8 < (51) : f(w " f )! f(w f ); kf(w " f )k L 1 (IR(0;+1)) C; in L 1 loc(ir (0; +1))? strong; for some constant C independent of ". Let 2 C 1 0 (IR (0; +1)). In keeping with the convergence W 1;1 -weak of p " (42), and using (51), we obtain (52) T +1 x p " f f(w" f )dxdt! T +1 x p f f(w f )dxdt By hypothesis, w 0 has a compact support. It is known that for nite propagation velocity, the function [@ x p f ]f(w f ) stays in a compact support, for T < 1 (see Kruzkov [12]). We have (53) jd e J " (f)f? T +1 0?1 T R 0?R T + x pf(w f )j j@ x p " f f(w" f x p f f(w f )j jxj>r j@ x p " f f(w " f )j for R large enough. Hence, by (52), the rst term of the right hand side in (53) converges to 0 when "! 0, for all R large enough. Let us prove the convergence of the second term. We deduce from the compact support of w 0 and from the maximum principle applied to the linear parabolic equation (32) that jp " (x; t)j Ce?rjxj, where C; r are constants which depend only on T and kw 0 k L 1, and thus are independent of ". We deduce that the second term of the right hand side of (53) uniformly converges to 0 in ", when R! 1. This concludes the proof of (50). If the derivative of e J(f) exists, then D e J(f) is characterized by (15), and we deduce from the theorem 3.5 that D e J " (f)f! D e J(f)f, when "! 0. The trouble is that we have no result concerning the dierentiability of e J. Nevertheless, since we are interested in the behaviour of e J near a minimum, we can assume that e J is convex in a neighborhood of this point. Therefore, we can dene its e J(f). Corollary 3.6. Assume that e J is a minimum at f, and that e J and e J " are convex in a neighborhood of f for all ". Then, under the hypotheses of theorem 3.5, D e J " (f)f converges to an element of the ~ J(f) of e J(f), when "! 0, that e J(f) 3 T 0 x p f f(w f )dxdt:

17 FLUX IDENTIFICATION IN CONSERVATION LAWS 17 Proof. From the denition of convexity we have D e J " (f)f? e J " (f) D e J " (f)? e J " (); for all Lipschitz continuous function. We apply theorem 3.5, and we pass to the limit when "! 0. We obtain T +1 0?1 [@ x p f ] [(f? )(w f )] dxdt e J(f)? e J(): That is a characterization of the subgradient for convex functions (see Rockafellar [15]). Thus, the limit of sequence D e J " (f)f is an element of the 4. Numerical approximation Now we shall give similar convergence results for discretization of the identi- cation problem, and we will remark that at the limit, both approximations (articial viscosity and discretisation) reach the same element of the characterized by the reversible solution of the adjoint problem. We shall prove these results for a large class of numerical schemes which contains the schemes used to resolve the identication problem in [9]. First we discretize the cost function (denoted J ) and the direct problem (1). Next we compute a discrete Lagrangian, which will lead to a discrete adjoint state, and nally to a discrete gradient of J. This method of computing the exact gradient of the discretized problem seems to have better properties (concerning stability, for instance) than discretizing the exact adjoint state. Moreover notice that we have no natural way to discretize it since the adjoint equation is ill-posed Discretization and convergence for the direct problem Let z (resp. t) be a positive space (resp. time) step. These parameters will tend to 0, the ratio = t=z remaining constant. For n = 0; : : : ; N, j = 0; : : : ; J, the sequence wj n is an approximation of solution w at the point (z j = jz; t n = nt). In the same way, we discretize w 0 (z), w obs (z), by wj 0, wj obs respectively. We consider a conservative (2K + 1)-points scheme for the hyperbolic equation (1) (54) w n+1 j n o = wj n? g n j+ 2 1 (f)? gj? n 2 1 (f) ; where g n (f) = g j+ 2 1 f (w n j?k+1 ; : : : ; wn j+k ), g f being the numerical ux of the scheme, consistent with f: g f (w; : : : ; w) = f(w). Then the discretized identication problem becomes the following minimization problem (55)? min J wf f ;

18 FLUX IDENTIFICATION IN CONSERVATION LAWS 18 where wf is the piecewise constant function dened by the sequence wj n(f) ; j = 0; : : : ; J? 1, n = 0; : : : ; N? 1 2 IR M, which was built out of (54). To obtain the exact gradient of the discretized cost function, we follow exactly the same lines as in the formal computation and viscous regularization, that is we build up a discrete Lagrangian L using a \discrete weak form" of the direct scheme (54), then we dierentiate with respect to wj n, and we choose the sequence p n j in order to =@wj n for all j and n. This denes the adjoint scheme. Finally, for the discrete gradient, the computations give: (56) D e J (f)f =? t X n;j? p n+1 j? p n+1 j+1 Dg n j+ 2 1 (f)f where p is solution of the adjoint scheme (57) 8 >< >: p n j = p n+1 j p N j?? N : n k=?k j g n j+k+ 2 1 (p n+1 j+k? pn+1 j+k+1 ); The complete computations are rather tedious, and we refer to [9] or [17] for greater detail and for some examples as well. First we show a convergence result of our discretized cost function, which allow us to say that the continuous identication problem can be approximated by the discretized identication problem. We consider a suitable set of functions F, which we suppose bounded and closed for the Lipschitz continuous norm k k Lip. Next we suppose that the numerical ux g introduced in (54) is Lipschitz continuous with respect to w?k ; : : : ; w k, and independent of f 2 F, i.e. that there exists a constant C Lip independent of f, such that jgf (w)? g f (v)j (58) sup w;v0 jw? vj We also suppose that the scheme (54) satises C Lip ; 8f 2 F: (59) jw n j (f)j C 1 ; for all j; n; and for any f 2 F; where C 1 is a constant independent of f. Condition (59) is veried for the Lax-Friedrichs, Godunov [7], and Van-Leer [20] schemes, when the following CFL-condition is satised (60) sup w 0 f 2 F jf 0 (w)j < 1: This leads us to our convergence result:

19 FLUX IDENTIFICATION IN CONSERVATION LAWS 19 Proposition 4.1. Let wf built out of the conservative scheme (54) which we suppose consistent with equation (1), and verifying hypotheses (58) and (59), f being the solution of (55). If the initial condition w 0 is bounded in L1 (IR)\ BV (IR), then wf is bounded in L 1 (IR (0; +1)) and in L 1 (0; T; BV (IR)), for all T > 0. Furthermore, if f k! f 2 F, for the Lipschitz continuous norm k k Lip, then (61) (62) w k f k! w f ; in L 1 (0; T; L 1 (0; L))? strong; w k f k * w f ; in L 1 ()? weak; Remarks. We can prove this proposition thanks to the continuity result of Lucier [13], and by copying the proof of a convergence result for schemes approximating scalar conservation laws in [6]. We omit the detail of this proof. For instance, we consider the identication problem arising from the chromatographic model (see [9]), and we take a bounded subset of parameters K 2 IR N. Then we deduce the CFL-condition (60), and we have at least one accumulation point of the sequence ff g x;t. We suppose that w obs and w 0 have a compact support, so that for 0 < t < T, the support of the solution w f to equation (1), with f 2 F, is in a compact set = (0; L) (0; T ). Proposition 4.1 and a convergence result of [9] imply Corollary 4.2. Let w f be the solution of a conservative and TVD scheme (54), consistent with equation (1). Then any accumulation point f 2 F of the sequence ff g x;t for the Lipschitz continuous norm, is solution of J(w f ) = min g2f J(w g); where w f is the entropy solution of equation (1), with f 2 F Monotone and TVD adjoint schemes Here we study some properties of monotonicity and BV estimates for adjoint schemes associated to the schemes in conservative form (54). We notice that adjoint schemes cannot be put in conservative form, which is not surprising, since the adjoint equation is not conservative. However, we shall prove that a family of TVD dierence schemes, including the Godunov and the Van-Leer schemes, is associated to TVD adjoint schemes. First we dene the function x p by x p (x; t) = p (x; t)? p (x + x; t) ; x where x p is the piecewise constant function with value p n j+ 1 2 =x in the rectangle n j = ((j? 1 2 )x; (j + 1 )x) (nt; (n + 1)t): 2

20 FLUX IDENTIFICATION IN CONSERVATION LAWS 20 On the other hand, we dene the dierences p n = p n j+ 1 j? pn j+1, for all j 2, 2 n N. Using the linearity of the scheme (57), we deduce the following scheme for p n : j+ 1 2 (63) p n j+ 1 2 = KX k=?k where the coecients A k j are dened by 8? j =?? (64) >< >: A?K n j?k A 0 j n n j?k+1 A k j+kp n+1 ; j+k+ 1 2 g n j+ 1 ; for k 62 f?k; 0; Kg; n g n j+ 1 ; A j = n g n j+ 1 ; g n : j+1 n j+ 2 1 Notice that by construction the coecients A k j satisfy (65) KX k=?k A k j = 1; 8j 2 Now we wish the adjoint scheme to have the same property of monotonicity preservation as the continuous equation. Thus, in view of (63), it is natural to impose (66) A k j 0; for? K k K; j 2 : This is somehow the discrete analogous to the (OSLC) condition (34) used for the convergence of the viscous perturbation. Example. In the model of chromatography, the Godunov scheme is very simple (it is just an upwind scheme). The adjoint scheme is given by (see [9]) (67) p n j = p n+1 j? f 0 (w n j )? p n+1 j? p n+1 j+1 : In this case, the coecients A k j are A?1 j = f 0 (w n j ); A 0 j = 1? f 0 (w n j ); A 1 j = 0: The CFL condition (60) implies that these coecients are positive. Similarly, the adjoint scheme associated to the Van-Leer dierence scheme veries the hypothesis A k j 0 when we have CFL condition (60), and it is a monotone and TVD scheme, whereas we can see that the adjoint scheme associated to the Lax-Friedrichs dierence scheme does not verify this hypothesis of positivity and that it is an unstable scheme in BV (IR). We have the following a priori estimates

21 FLUX IDENTIFICATION IN CONSERVATION LAWS 21 Proposition 4.3. We consider a linear scheme in form (57), with its coecients dened by (64) and verifying the monotonicity property (66). Then we have for all 0 < s; t < T the following estimates (68) T V (p (:; t)) T V (p (:; T )); (69) kp (:; t)k L 1 (IR) kp (:; T )k L 1 (IR) + CT T V (p (:; T )); (70) k x p (; t)k L 1 (IR) k x p (; T )k L 1 (IR) ; (71) kp (:; t)? p (:; s)k Lp (!) C jt? sj + t k jtj p?1 x p (; T )k L 1 (IR) ; p where! is a compact set in IR, 1 p +1, and C > 0 is a constant independent of the discretization. Remark. Estimates (68) and (69) are classical results of the theory of approximation of non-linear conservation laws. They rely upon monotonicity properties of the schemes, and are in particular independent of the conservative form. The estimate (71) is a technical result allowing us to use a diagonalization argument in order to pass to the limit, when t; z! 0. Notice that the result of estimate (70) bears some resemblance to that of the W 1;1 estimate proved in theorem 3.3. Proof. of proposition 4.3. Let R K. Summing up (63) from j =?R to j = R, and using the positivity of A k j and property (65), and passing to the limit R! +1, we obtain the estimate (68). On the other hand, using this BV estimate, the L 1 estimate (69) is obtained by an analysis that is similar to the one made by Godlewski & Raviart ([6], chapter III) for the dierent schemes in conservative form. Now, using the condition A k j+k 0, the estimate (70) results from the monotonicity of (H (p)) j = KX k=?k A k j+kp n+1 : j+k+ 2 1 In order to obtain the next estimate (71), we write the scheme (57) in the form K?1 X p n pn+1 j =? g n j+k+ 2 1 p n+1 ; j+k+ 2 n k=?k j and by the Lipschitz condition of g, we have jp n j? pn+1 j j C sup j2 jpn+1 j+ 2 1 with a constant C independent of the discretization. That implies kp (; nt)? p (; (n + 1)t)k Lp (!) p?1 p Cj!j 1 p jtj 1 p sup j; j2 jpn+1 j+ 2 1 j:

22 FLUX IDENTIFICATION IN CONSERVATION LAWS 22 Let m > n. Applying successively this same result for n; n + 1; : : : ; m? 1, and using the triangular inequality of the norm k k L p, we obtain (72) kp (; nt)? p (; mt)k Lp (!) Cjtj 1 p (n? m) sup j2 jpn j+ 2 1 j: Let s; t such as nt s (n + 1)t, mt t (m + 1)t. We notice that (m? n)t jt? sj + t. Inequality (72) gives the desired result (71) Convergence for adjoint schemes and derivatives Now we shall evidence that the sequence of discrete gradients converges to the same element of the subdierential of the cost function given by the limit of the viscous perturbation. First we have the following convergence result for the solution of the adjoint scheme Theorem 4.4. We consider the linear dierence scheme in the form (57), where the coecients are dened in (64) and verify the monotonicity property (66). We suppose p T 2 W 1;1 (IR) \ BV (IR), and the (OSLC) condition (34). Then, p! p in L 1 (0; T; L q loc (IR))? strong; 1 q < +1; where p is the reversible solution of (13). Proof. In accordance with the estimate (70) and in keeping with the hypothesis p T 2 W 1;1 (IR), we have (73) sup j x p (x; t)j C: x Applying the theorem of Riesz-Frechet-Kolmogorov (see Adams [1]) in the last inequality, we deduce that we can take a subsequence k x; k t! 0 which we still denote with x; t, such as p (; t)! p(; t) in L q loc (IR))? strong; 8t 2 (0; T ); for all 1 q < +1. Using the estimate (71) and a classical diagonalization argument (see [6]), we deduce the convergence in L 1 (0; T; L q loc (IR)). In order to prove that the limit p is a Lipschitz continuous solution of (13), we proceed by similar arguments to the proof of theorem 3.4. Next we notice that the scheme in the form (57) preserves monotonicity as soon as A k j 0 (which is satised for instance by the adjoints of Godunov and Van-Leer schemes). Finally, we use the same arguments of the rst remark following the proof of theorem 3.4. That is, 8 < : p! p in L 1 (0; T; L q loc (IR)) p = p 1? p2 ; with (pn j )i (p n j+1 )i ; and p i solution of the scheme (57)

23 FLUX IDENTIFICATION IN CONSERVATION LAWS 23 implies that p is reversible. It is easy to verify that J e : f 7! J(wf ) have a continuous derivative if the numerical ux g is of class C 1 with respect to (w n j?k+1 ; : : : ; wn j+k ). As a consequence of theorem 4.4, we have Corollary 4.5. Let J e : f 7! J(wf ) be a discretized cost function dened for all Lipschitz continuous function f, with wf solution of the scheme in the conservation form (54). We suppose that the coecients A k j in (64) satisfy Ak j 0, for?k k K, j 2, and that w 0 has a compact support. Furthermore we suppose that p T 2 W 1;1 (IR)\BV (IR). Let w f be the entropy solution of (1), and p f be the reversible solution of (13). Then we have (74) D e J (f)f! T +1 x p f f(w f )dxdt; when x; t! 0; for all Lipschitz direction f. Remark. We note that the theorem 3.5 and the corollary 4.5 make it clear that a viscous perturbation or a numerical approximation of the gradient give the same result at the limit. We clarify this result by another corollary which reads as follows: Corollary 4.6. Assume that J e is a minimum at f, and that J e and J e are convex for all (x; t) in a neighborhood of f. Then, under the hypotheses of theorem 4.5, D J e (f)f converges to the same element of the of J(f) which we have obtained by a viscous perturbation in Corollary 3.6, when x; t! Numerical results We illustrate in this section our results by a numerical application on real experimental data. We consider the propagation of a single, pure compound in a column, which leads under several physical assumptions to a scalar conservation law of the form (1). More precisely, the experimental data are concentration proles obtained from the adsorption of gaseous n-hexane on graphite carbon with helium vector gas. We refer to Rouchon et al. [16] for the complete description of the experiment, the discussion of the model and the original results. A remarkable feature of this experiment is that we have an experimentally identied ux to compare with. The observation here is a prole of concentration vs time, at a xed L > 0, where L is the length of the column. This is not quite the context of the previous analysis, but since in this kind of models the ux satises f 0 (w) > 0, a slight modication is only needed. The direct problem is discretized through a Godunov scheme, and we compute the exact gradient of the discrete functional as indicated in section 4.1 (see also [9] for details). At this discrete level, all the quantities are well-dened, so we are able to apply a gradient-based minimization algorithm. Thus we are left with the problem of non uniqueness for f which