Primal-Dual Formulations for Parameter Estimation

Primal-Dual Formulations for Parameter Estimation Problems Guy Chavent, Karl Kunisch and Jean E. Roberts October 22, 1997

Abstract

A new method for formulating and solving parameter estimation problems based on Fenchel duality is presented. The partial dierential equation is considered as a contraint in a least squares type formulation and is realized as a penalty term involving the primal and dual energy functionals associated with the dierential equation. Splitting algorithms and mixed nite element discretizations are discussed and some numerical examples are given.

tion

Key-words: parameter estimation, Fenchel duality, inverse problems, regulariza-

1

1 Introduction In this paper, we describe a general framework for a new approach to the formulation of parameter estimation problems. This approach is based on Fenchel duality and is applicable to linear as well as nonlinear problems of monotone type. Let us rst describe some concepts involved in formulating parameter estimation problems in a manner that is suitable for computations. Many of the currently used methods are modi cations and/or combinations of the equation error and the least squares formulations. Within the class of least squares formulations, we can distinguish those which treat the partial dierential equation as an implicit constraint and others which impose the dierential equation as an explicit constraint. To be more concrete, let e : Q  X ! X~ , with Q; X; and X~ Banach spaces, describe the partial dierential equation

e(a; u) = 0;

(1.1)

where a denotes the parameter and u the state variable of the dierential equation. Let z denote the observation (or data) in the data space Z , and let B : X ! Z be the observation operator. With the equation error approach one replaces u in (1.1) by z and solves

e(a; z) = 0;

(1.2)

for a. The least squares approach is based on the optimization problem minimize 21 jBu ? zj2Z over Q (1.3) (or over some appropriately de ned subset of Q), where u is a solution to (1.1). If (1.1) is treated as an implicit constraint, then u = u(a) is a dependent variable in (1.3). Alternatively, (1.1) can be treated as an explicit constraint, and a as well as u becomes an independent variable in this case. We refer to [A; BK; C 1; C 2; KL; IK 1; IK 2] and the references given therein for a more detailed discussion. Concerning the equation error approach, (1.2), here we only mention that this approach requires that distributed data be available and further that the data be dierentiable. However, for linear dierential equations with e ane in a and in u, (1.2) has the advantage of being ane with respect to the unknown a. The output least squares approach on the other hand is versatile with respect to the type of data required. If, for example, only point-wise or boundary data are available one can easily nd an appropriate observation B and choose an appropriate norm for Z . But, the simple structure yielded by the equation error approach is lost when a least squares formulation is chosen. Even if e is ane in a and u, problem (1.3) is highly nonlinear if (1.1) is considered as an implicit constraint. If (1.1) is realized as an explicit constraint then (1.3) is quadratic in u with a bilinear constraint. 2

Because it requires the dierentiation of data, a pure equation error technique will not be the method of choice for most applications. The least squares approach with explicit constraints appears at present to be one of the most ecient methods for solving parameter estimation problems numerically. With this approach, for example, the gradient of the t-to-data function, which is required in every iterative technique for solving (1.3) numerically, may be calculated in a straightforward manner. Moreover, since a and u are independent variables, an equation error method can be used to generate an initial guess for the parameter a, thus combining the advantages of the equation error and the least squares formulations. It seems appropriate to mention here as well the adaptive control technique, also referred to as the asymptotic embedding method [AHS, BS]. The idea of this method is to introduce a dynamical system having the t-to-data-term Bu ? z as inhomogeneity and having the solution to e(u; a) = 0 as stationary solution. The adaptive control technique can also be interpreted as a continuous version of a gradient method for solving the least squares problem (1.3). The new framework that we propose in this paper belongs to the class of least squares formulations with explicit treatment of the dierential equation as constraint. In x2 we present the method and give several applications to second order dierential equations of elliptic and parabolic type. The subsequent sections are devoted to the analysis of a particular formulation for a problem proposed in x2. Basic properties are developed in x3, splitting algorithms are derived in x4 and a mixed nite element implementation is described in x5. Numerical results are given in x6.

Acknowledgement The authors would like to express their gratitude to Guillaume Vigo who very eciently and graciously carried out the numerical experiments reported in x6.

2 Primal-dual formulation based on Fenchel duality

We consider the case of a system whose equilibrium state u 2 X is obtained by minimization, over the Hilbert space X of states, of an energy functional Ea (u) :

min Ea (u) over u 2 X: (E ) We suppose that this energy functional depends on an unknown parameter a in a set C of admissible parameters in a Banach space Q:

a 2 C  Q: 3

In order to estimate this unknown parameter, we suppose that we have at our disposal a measurement z of the observation B (u). Here B 2 L(X; Z ) is the observation operator, and Z is the observation space which is assumed to be a Hilbert space. The classical least squares formulation for parameter identi cation problems is given by min 1 jz ? Buj2Z 2 where u = u(a) is a solution to (E ) with a 2 C .

over a 2 C;

(P )

Hence (E ) appears in (P ) as a constraint between a 2 C and u 2 X . In particular, each evaluation of the objective function in (P ) requires a full solution of (E ). However, solving (E ) precisely in the rst steps of an iterative technique for (P ), when the parameter a is still far from its converged value, may be inecient. Moreover, in situations where the observation operator B is rich enough, one can build up, by interpolation or smoothing of the available data z, a (possibly rough) estimate u~ of the state variable u, but the fact that u = u(a) is a hard constraint in (P ) makes it impossible to use this estimate. One has to chose an initial value a0 for the parameter a, which determines the rst approximation u(a0) to the state variable which may be quite far from z. The idea presented here is to relax, in (P ), the constraint that a and u satisfy the state equation by imposing it through penalization. A rst realization of this idea would consist in taking advantage of the fact that the state equation is de ned via the minimization problem (E ), and to replace (P ), for " > 0, by 



min 21 jz ? Buj2Z + 1" Ea(u) over (a; u) 2 C  X: (P~") We would expect this problem to be a perturbation of (P ) if the set of minimizers of the penalization function Ea(u) were made up of all couples (a; u) which satisfy the constraint a 2 C; u = u(a). But the energy min E (u) = Ea (u(a)) u2C a

of the state u(a) associated to a given parameter a depends, in general, on this parameter ! Since under appropriate hypotheses min

(a;u)2C X

Ea (u) = min E (u(a)) = Emin a2C a

the set of minimizers of Ea (u) over C  X is made up of only those couples (a; u) for which the constraint a 2 C; u = u(a) is satis ed and which produce states with minimum energy 4

Emin! So Ea (u) is not a penalization function for the constraint in (P ), and (P~") is not an appropriate perturbation of (P ). The above considerations suggest, however, that the penalization approach would work if we could replace, as characterization of the equilibrium state, Ea (u) by a dierent energy functional whose minimum with respect to u is independent of the parameter a. Fenchel duality provides us with a systematic way of doing this - at the price of an enlargement of the state space - provided that the energy functional Ea (u) can be written as the sum of two convex functionals. So we shall suppose from now on that

Ea (u) = Fa (u) + Ga(Au) for every u 2 X; (2.1) where for each a 2 C , Fa : X ! R and Ga : Y ! R are proper, convex and lower semicontinuous with R = R [ f+1g, Y is a Hilbert space, A 2 L(X ; Y ). We suppose as well

that

for each a 2 C , there exists uo = uo(a) such that (2.2) Fa (uo) < +1; Ga(Auo) < +1 and Ga is continuous at Auo: The Fenchel duality theorem [BP; ET ] asserts that, with these hypotheses, for every

a2C

inf fFa (u) + Ga (Au)g + min fFa(?A q) + Ga(q)g = 0; q2Y

u2X

(FD)

where

A 2 L(Y ; X ) is the adjoint of A,

Fa : X ! R; and Ga : Y ! R are the convex conjugates of Fa and Ga respectively, de ned by Fa (u) = sup f< u; v >X ?Fa (v)g ; for all u 2 X; v2X

and

Ga (p) = sup f< p; q >Y ?Ga (q)g ; for all p 2 Y: q2Y

If in addition to (2.1) and (2.2) we suppose that for every a 2 C;

lim fFa(u) + Ga(Au)g = +1;

juj!1

5

(2.3)

then the inf in (FD) becomes a min (which means that the original problem (E ) has at least one solution), and any pair of minimizers (u; q); u = u(a); q = q(a); of the left hand side of (FD) satis es the extremality conditions

?A q 2 @Fa (u) q 2 @Ga (Au)

(EC )

where @ denotes the sub-dierential operator for convex functions. We refer to u 2 X as the primal state variable, and to q 2 Y as the dual state variable. In view of (FD) it is natural to associate to the dual variable q a dual energy function Ea(q) de ned by

Ea(q) = Fa(?A q) + Ga (q) for every q 2 Y

(2.4)

and to consider the dual problem over q 2 Y:

min Ea(q)

(E )

To any couple (u; q) of primal/dual state variables, one associates the total energy Ea (u) + Ea (q), and one considers the corresponding minimization problem min fEa (u) + Ea(q)g

over (u; q) 2 X  Y

(EE )

which, due to the Fenchel duality formula (FD), unlike (E ), has the property that for each a 2 C;

min fEa (u) + Ea (q)g = 0:

(u;q)2X Y

(2.5)

We summarize the above discussion in the following proposition.

Proposition 2.1 Let (2.1) (2.2) and (2.3) hold. Then far all a 2 C one has i) the primal, dual and total energy minimization problems (E ); (E ) and (EE  ) admit solutions ii) solving (EE ) is equivalent to solving (E ) and (E  ) iii) any pair of solutions u and q to (E ) and (E ) satis es (EE ) iv) the minimum of the total energy is zero:

The problem of the estimation of a 2 C from the measurement z 2 Z of Bu, can therefore be approximated by penalizing (P ) by the total energy, which has a minimum (equal to zero) if and only if (a; u; q) satis es the constraint a 2 C; u = u(a) and q = q(a). This leads to the sought primal-dual formulation of the parameter estimation problem : 6





(P") min 21 jz ? Buj2Z + 1" (Ea (u) + Ea(q)) over (a; u; q) 2 C  X  Y; where " > 0 is the penalization parameter. Note that when the data z 2 Z are attainable - i.e. when there exists a 2 C such that z = Bu(a) - the problem (P") is an exact penalty formulation. That is to say that for any " > 0, the solutions to (P") and (P ) coincide. Moreover, the minimum value of the cost functional in (P") is zero in this case. We complete this section by giving several examples illustrating the general approach proposed here for solving parameter estimation problems by means of a primal-dual formulation. The later sections will then be devoted to a detailed analysis of one of these examples. It is assumed throughout that is a bounded domain in Rn with suciently smooth boundary @ , and that B 2 L(X ; Z ). Example 2.1 : Estimation of the diusion coecient a in

?r(aru) = f in

(2.6)

u = 0 on @

where f 2 L2 ( ) is given, and where the set C of admissible parameters satis es, for some  > 0: C  fa 2 L1( ) j 0 <   a(x)g :

(2.7)

Equation (2.6) corresponds to the minimization of

Z Z 1 2 Ea (u) = 2 a j ru j ? fu (2.8)



over X = H01( ). A rst way of casting Ea (u) in the form (2.1) consists in choosing :

Au = ?ruZfor all u 2 X; Ga (q) = 21 a j q j2Rn ;

Y = L2n( )Z; Fa (u) = ? fu;

which clearly satis es the hypotheses (2.2) and (2.3). Then

A q = (?
)?1 rq F (u) = a





for q 2 Y;

0 if ?
u + f = 0 ; +1 if ?
u + f 6= 0 7

G (q) = a

(2.9)

1 Z 1 j q j2 2 a Rn

(2.10)

so that the dual energy Ea(q), (2.4), is 1 Z 1 jqj2 n Ea(q) = : 2 a R +1 8
H01 ;H ?1 where Ac : H01( ) ! H ?1 is de ned by

Ac(u) = ?
u + cu: 10

(2.18)

The penalized problems (P") are 



min 12 j z ? Bu j2 + 1" 21 over (c; u) 2 C  H01:

Z



j ru j2

dx + 21

Z



cu2 dx ?

Z

fu dx + 12

Z



(A?1f )f dx



c

(2.19)

Example 2.5 : We turn to nonlinear potential problems and consider

?
u + @(u) 3 f in

(2.20)

uj@u = 0

associated with the energy

Z Z Z 1 2 E (u) = 2 j ru j ? fu + (u)



de ned on X = H01( ). For any parameter  2 C with C as in Example 2.3 but with n = 1, we cast E (u) in the form (2.1) by choosing Y = L2 ( ); A = H01 ! L2 embedding; f 2 L2 ; Z Z 1 2 F (u) = 2 j ru j ? fu for u 2 H01( )



Z G (u) = (u) for u 2 L2 ( ):



For the convex conjugate functions we nd

F  (u) = 21 j u + f j2H ?1

and

for u 2 H ?1( )

Z

1 (u)

2 Hence the penalized problems (P") are

G (u) =

for u 2 L2 ( ):







Z

min 12 j z ? Bu j2 + 1" 12 j ru j2 ?

1 2 over (; u; q) 2 C  H0  L :

Z



fu +

Z



(u) +

Z



 (q) + 1

2 j ?q + f

j2H ?1



(2.21) 11

Remark 2.2 For the above example, the extremality conditions (EC ) have the form q =
u + f q 2 @(u)

(2.22) (2.23)

Hence if we choose u 2 H01( ) with
u 2 L2 ( ) and if we use the value of q 2 L2 ( ) given by (2.22) in the expression of the total energy, we nd that, for  2 C inf

u2H01 ;q2L2

fE (u) + E (q)g = inf 1

Z

u2H0 ;
u2L2

j ru j2



Z

Z

? fu + (u) +



Z



(
u + f )



:

(2.24)

This shows that the functional on the right hand side can be used as a penalization function for equation (2.17), which gives rise to the following penalized least squares formulation : 



Z

  min 21 j z ? Bu j2Z + 1" j ru j2 ?fu + (u) +  (
u + f )

1 over (; u) 2 C  H0 ;
u 2 L2

(2.25)

which was analyzed in [BaK]. Example 2.6 : We revisit the nonlinear potential problem of Example 2.5 but with a dierent choice of duality: the dual variable q, instead of being linked to u, at the optimum, by q =
u + f as in the previous Example 2.5, will satisfy q = ?ru, in a way similar to the mixed nite element formulation of Example 2.1. This will be achieved by keeping the same X = H01( ) and C as in Example 2.5, and by choosing

Au = ?ru;

Y = L2n ( ); F (u) =

Z



((u) ? fu) ;

The convex conjugates are found to be 12

G (q) = 21

R



j q j2 :

F (u) =

8 Z < :

G = G:



1

(u + f ) if u 2 L2 ( ) if u 2 H ?1( ) but u is not in L2 ( )

The corresponding penalized least squares problems (P") are 



Z Z Z Z 1 1 1 1 2  2 min 2 j z ? Bu j + " ((u) ? fu) + 2 j ru j +  (?div q + f ) + 2 j q j2Rn





1 over (; u; q) 2 C  H0 ( )  H (div ; ): (2.26)



One checks easily that, according to the second extremality condition in (EC ), the relationship between primal and dual variables at the minimum is given by q = ?ru.

3 Convergence as " ! 0 and Convexity Properties

We return now to the estimation of the diusion coecient in an elliptic equation in the setting of Example 2.1, and consider its regularized version:  min 21 jz ? Buj2Z + 2 ja ? a] j2Q (P ) 1 over (a; u) 2 C  H0 which satisfy equation (2.6); where a] is an a-priori guess for the true parameter, and is a regularization parameter assumed to satisfy > 0, and where the set of admissible parameters C is de ned by: C = fa 2 Q : 0 <   a   a.e. on g; (3.1) with  and  known lower and upper bounds for a, j jQ is a semi-norm on an Hilbert space Q such that, for any B  Q bounded in the semi-norm j jQ; the se (3.2) This ensures the existence of a solution to (P ). As was seen in x2, the primal-dual formulation (P") of (P ) is, cf (2.13),   Z Z Z 1 1 1 1 1 2 2 ] 2 2 min 2 jz ? BujZ + 2 ja ? a jQ + " [ 2 ajruj ? fu + 2 a jqj ]



1 2 over (a; u; q) 2 C  H0  Hdiv ; rq = f 2 L ; 13

(P")

where .

Hdiv = fq 2 L2n( ) : div q 2 L2 ( )g

It will be convenient to introduce the following notation:

X = Q  H01( )  Hdiv ; and to call J1 : Q  H01( ) ?! R the least squares functional: J1(a; u) = 21 jz ? Buj2Z + 2 ja ? a] j2Q; (3.3) and J2 : Q  H01( )  Hdiv ?! R the total energy functional: Z Z Z 1 1 2 (3.4) J2(a; u; q) = 2 ajruj ? fu + 2 a1 jqj2:



Of course J1 is non-negative, and, as we saw in x2, J2 is non-negative whenever (a; u; q) satis es rq = f , and vanishes only when (a; u; q) also satis es q + aru = 0, i.e. when the elliptic equation (2.6) is satis ed. The functional in (P") may now be written J (a; u; q) = J (a; u) + 1 J (a; u; q): (3.5) "

1

"

2

Proposition 3.1 For every " > 0 there exists a solution (a"; u"; q") 2 X to (P"). Proof : Let f(an; un; qn)gn2N  X be a minimizing sequence such that for each n 2 N div qn = f

  J1 (an; un) + 1" J2 (an; un; qn)   + n1 ;

(3.6)

where  denotes the in mum of the cost in (P"). From (3.6) it is simple to argue that

f(an; un; qn)gn2N is bounded in X : Hence, there exists a subsequence still denoted f(an; un; qn)gn2N ; and (a"; u"; q") 2 X such

that

(an; un; qn) * (a"; u"; q") weakly inX : In particular this implies that (an; un) ! (a"; u") strongly in L1  L2 ; and moreover div q" = f . It also follows that

run * ru" weakly in L2n; 14

and hence as well. Similarly

pa ru * pa ru weakly in L2 ; n n " " n p1a qn * p1a q" weakly in L2n : n

We therefore nd that and

Z



"

Z

a"jru"  lim anjrunj2

j2

(3.7)



Z

1 jq j2  lim Z 1 jq j2: (3.8) p " p n

a"

an Using (3.7) and (3.8) in (3.6) we obtain J1(a"; u") + 1" J2 (a"; u"; q") Z Z Z a 1 1 1 1 n 2 ] 2 2  lim 2 jz ? BunjZ + lim 2 ja ? a jQ + lim " 2 jrunj ? lim " fun + lim " 2a1 jqnj2



n 1  lim(J1(an ; un) + J2(an; un; qn))  :

" This implies that (a"; u"; q") is a solution to P".

" ! 0+ :



We next turn to the convergence of (a"; u"; q") to a solution of P as Proposition 3.2 For every " > 0 let (a"; u"; q") denote a solution to (P"). Then f(a"; u"; q")g contains a weakly convergent subsequence as " ! 0+ , and every weak cluster point (a; u; q) is a solution to (P ). Proof : Let (a; u; q) 2 X satisfy div q = f and q = ?aru. Then, J2(a; u; q) = 0; and, for every " > 0; J (a ; u ) + 1 J (a ; u ; q )  J (a; u): (3.9) 1 "

"

1 " 2 " " " Since div q" = f we know that J2 (a"; u"; q")  0; and hence by (3.9) Z (3.10) 0  ( a2" jru"j2 + 21a jq"j2 ? fu")dx  "J1(a; u): "

It follows that f(a"; u"; q")g">0 is bounded in X : Hence there exists a weakly convergent subsequence (denoted by the same symbols) and (a; u; q) 2 X such that (a"; u"; q") * (a; u; q) 2 X :

15

As in the proof of Proposition 2.1 this implies that (a"; u") ! (a; u) strongly in L1  L2 ;

pa ru * paru weakly in L2 " " n and

p1a q" * p1a q weakly in L2n : "

Taking the lim inf in (3.10) we obtain Z Z a  1 1 jqj2 + qru) 2 2 0  ( 2 jruj + 2a jqj ? f u) = a2 (jruj2 + 2 a

Z

1 2 jq + aruj dx = 0; =

2a and hence q = ?aru. Consequently (a; u; q) satis es all constraints of (P ). Moreover by (3.9) we have

J1 (a; u)  J1(a; u) for all (a; u) such that there exists q 2 C with (a; u; q) admissible for (P ). It follows that (a; u; q) is a solution of (P ).

Proposition 3.3 Let > 0 and let (a; u; q) be a weak cluster point of f(a"; u"; q")g">0 as

" ! 0+. Then

(a"; Bu") ! (a; B u) strongly in Q  Z and





ru" + aq"  " jB (Bu" ? z)jH01 " L2n

where  is the lower bound of the elements of C .

In particular:  if Z = H01 and B is the identity, then u" ! u strongly in H01, and



ru" + aq"  " ju" ? zjH01 " L2n

16

(3.11) (3.12)



 if Z = L2 and B is the canonical injection from H01 into L2, then u" ! u strongly in L2 only and, as in this case B  = (?
)?1 , one has



ru" + aq"  " ju" ? zjH ?1 : " L2n

Proof : From the de nition of (a"; u"; q") we obtain

J1(a"; u") + 1" J2(a"; u"; q")  J1 (a; u) + 1" J2(a; u; q):

Proposition 2.1 implies that J2 (a"; u"; q")  0 and that J2 (a; u; q) = 0 as (a; u; q) satis es the elliptic equation. Hence J1(a"; u")  J1(a; u) and, taking the lim lim J1(a"; u")  J1(a; u): On the other hand, J1 is convex and (a"; Bu") is weakly convergent in Q  Z to (a; B u) so that J1(a; u)  lim J1 (a"; u"); and J1(a"; u") ! J (a; u) when " ! 0: This proves that the sequences ja" ? a] jQ and jBu" ? zjZ converge to ja ? a] jQ and jB u ? zjZ . Hence the weakly convergent sequence (a" ? a] ; Bu" ? z) converges strongly in Q  Z to (a ? a] ; B u ? z), and the rst result (3.11) of Proposition 3.3 is veri ed. In order to prove the second result (3.12), we write the rst order necessary conditions which are satis ed by the solution (a"; u"; q") of (P"): Z 1 j q " j2 ] 2 " (a" ? a ; h)Q + 2 [jru"j ? a2 ]h  0; a" + h 2 C and  small enough (3.13)

Z " Z "(Bu" ? z; Bv) + (a"ru"; rv) ? fv = 0; v 2 H01( ) (3.14)

Z



( q" ; p) = 0;

a"

p 2 Hdiv ; rp = 0 (3.15) rq" = f: (3.16) Transposing B in (3.14), substituting rq" for f , using (3.16) and integrating by parts, we obtain

"(B (Bu

" ? z ); v )H01 +

Z

Z



(a"ru"; rv) + (q"; rv) = 0;

17

v 2 H01( ):

R

As H01 is equipped with the scalar product (u; v)H01 = (ru; rv) we get Z



("r[B (Bu" ? z)] + a"ru" + q"; rv) = 0;

v 2 H01( ):

(3.17)

We use (3.17) in two dierent ways :  rst, we choose v = u": Z



("r[B (Bu" ? z)] + a"ru" + q"; ru") = 0:

(3.18)

 second, we see from (3.17) that the vector eld "r[B  (Bu" ? z)] + a"ru" + q" has zero divergence. Hence we can chose p in (3.15) equal to this vector eld, which gives Z



("r[B (Bu" ? z)] + a"ru" + q"; q" ) = 0:

a"

(3.19)

By adding (3.18) and (3.19) we obtain Z



i.e.

Z

("r[B (Bu" ? z)] + a"ru" + q"; ru" + aq" ) = 0; "



"

Z 2 q " a ru" + a + " (r[B  (Bu" ? z)]; ru" + aq" ) = 0; " "



and, as a"   > 0,





 ru" + aq"  "jB (Bu" ? z)jH01 ; " L2n

which is (3.12). This ends the proof of Proposition 3.3. We conclude this paragraph with the analysis of the convexity properties of the cost functional J (a; u; q) = J (a; u) + 1 J (a; u; q) "

1

"

2

in (P"). We give rst the second derivative of J2. Proposition 3.4 For any (a; u; q) 2 X such that rq = f and any (h; v; p) 2 X such that rp = 0 one has Z Z h 1 2 00 2 (3.20) J2 (a; u; q)(h; v; p) = a j ? a q + arv + pj + 2 h(ru + aq ; rv):



18



Proof : Dierentiating twice in (3.4), the de nition of J2 , we obtain Z Z Z Z Z 2 j q j ( q; p ) 00 2 2 2 J2 (a; u; q)(h; v; p) = a3 h + 2 (ru; rv)h ? 2 a2 h + ajrvj + a1 jpj2:

De ne









p0 = ? ha q + arv + p

and substitute for p in the third term of the right-hand side

J 00(a; u; q)(h; v; p)2 2

Z Z 2 j q j 2 = a3 h + 2 (ru; rv)h ? 2 ( aq2 ; p0 + ha q ? arv)h Z

Z

1 2 2 + ajrvj + a jpj : Z





Expanding and rearranging terms we obtain

J 00(a; u; q)(h; v; p)2 2

Z

Z Z 2 j q j h 2 = ? a3 h ? 2 (q; p0) a2 + 2 (ru + aq ; rv)h Z

Z



1 2 2 + ajrvj + a jpj :





Noting that the rst two terms of the right-hand side form part of a perfect square, we write 2 Z Z Z qh 1 1 00 2 2 J (a; u; q)(h; v; p) = ? +p + jp j + 2 h(ru + q ; ru)

a a Z 0 a + ajrvj2 + a1 jpj2:



2

Z



0



a

Using the de nition of p0, we can rewrite the rst term as Z 1 qh + p 2 = Z 1 jarv + pj2 = Z ajrvj2 + Z (rv; p) + Z 1 jpj2: 0

a a



a





a

But the central term vanishes, as rp = 0. Plugging the two remaining terms into the last formula for J200 produces the announced result. We remark that (3.20) is not unexpected given the properties of J2 (a; u; q) seen in x2: if (a; u; q) is a minimizer of J2, then q + aru = 0 (the equation is satis ed) and (3.20) reduces to Z 00 2 J (a; u; q)(h; v; p) = 1 jhru + arv + pj2 2



a

which is always positive and vanishes in the directions (h; v; p) in which the equation q + aru = 0 is satis ed up to the rst order (such directions are \tangent" to the set of minimizers of J2, on which J2 has the constant value zero). 19



Corollary 3.1 (partial convexity of J2 and hence of J")  for any xed a 2 C; J2 and J" are globally convex with respect to (u; q)  for any xed u 2 H01( ); J2 and J" are globally convex with respect to (a; q). Proof : These results follow immediately from (3.20) and from the fact that a function whose Hessian is positive everywhere is necessarily convex.

We can now investigate the coercivity of the primal-dual objective function J" at minimizers (a"; u"; q") of (P"). We shall need the following hypotheses: the observation space Z is H01; hence B = Id; the data z 2 H01 is attainable, i.e. there exists (a; u; q) 2 C  H01  Hdiv such that rq = f; q + aru = 0 and u = z:

(3.21) (3.22)

Before giving the uniform coercivity result for J", let us remark that one could replace hypothesis (3.21) by the inclusion of a regularization proportional to jruj2 in J" (state space regularization). Proposition 3.5 Let hypotheses (3.21) and (3.22) hold, and let f(a"; u"; q")g">0 be any sequence of minimizers of (P"). Then there exists  > 0 and for every 2]0; [ an "( ) > 0 such that for every " 2]0; "( )[ there exists a convex neighborhood V (a" )  V (u")  V (q" ) in Q  H01  L2n of (a" ; u"; q" ) and > 0 such that for all (a; u; q) 2 V (a" )  V (u")  V (q" ) and all (h; v; p) 2 Q  H01  L2n with rq = f and rp = 0 J"00 (a; u; q)(h; v; p)2  (jhj2Q + jrvj2L2n + jpj2L2n ):

(3.23)

Proof : Due to the attainability assumption, one has for any " > 0 and > 0 :

1 jr(u ? z)j2 + ja ? a]j2 + 1 Z a jru + q" j2  ja ? a] j2: " " " Q 2" L2n 2 " 2 a" 2

Thus it is simple to argue that there exists M > 0; such that for 0 <  1 and 0 < "  1; ja"jQ  M and jq"jL2n  M:

(3.24)

Now Proposition 3.4 implies that, for any a 2 BQ(a"; 1); u 2 H01; q 2 L2n ; " > 0 and > 0, Z 2 1 2 2 00 2 2 J" (a; u; q)(h; v; p)  jrvjL2n + jhjQ + "k (M + 1) jp0jL2n + " h(ru + aq ; rv); (3.25) 1

20



where k1 is the imbedding constant Q in L1( ), and where

p0 = ?h aq + arv + p:

(3.26)

Using the Cauchy-Schwarz inequality in the last term of (3.25) we obtain Z 2 2 j " h(ru + aq ; rv)j  2 jhj2Q + 2"k2 1 jru + aq j2jrvj2



so that (3.25) becomes 2 J 00(a; u; q)(h; v; p)2  (1 ? 2"k2 1 jru + aq j2L2n )jrvj2L2n + 2 jhj2Q + "k (M1 + 1) jp0j2L2n (3.27) 1 We now choose ; " and the neighborhood V (a")  V (u")  V (q") in such a way that 2k12 ru + q 2  1 ; "2 a L2n 2

i.e. such that

2k1 ru + q  1: 1 a 2

" 2

We have



Ln







2k1 ru + q  2k1 ru + q" + 2k1 r(u ? u ) + q ? q" ; " " 1 1 1 a 2 a a a

" 2

Ln

" 2

" L2n

" 2

" L2n

i.e., using Proposition 3.3, 2k1 ru + q  2k1 ju ? zj 1 + 2k1 r(u ? u ) + q ? q" : " " H0 1 1 1 a a a

" 2

 2

" 2

" L2n

(3.28)

Let (a ; u ) 2 C  H01 be the solution of the regularized problem: (3.29) min 21 jr(u ? z)j2L2n + 2 ja ? a] j2Q over (a; u) 2 C  H01 and r(aru) = f: Due to the attainability assumption (3.22), it is well-known [14] that there exists a ~ > 0 such that, for any 2]0; ~], the solution (a ; u ) of (3.29) is unique. Hence we see from Proposition 3.3 that the sequence (a"; u"), itself, converges to (a ; u ) in Q  H01. Hence for any 2]0; ~], there exists a function  (") > 0 with  (") ! 0 as " ! 0, such that

ju" ? u jH   ("): 1 0

21

Another classical consequence of the attainability assumption (3.22) is the following rate of convergence of u to z in H01( ):

ju ? zjH = 1=2( ); 1 0

where ( ) > 0 and ( ) ! 0 as ! 0. Knowing this, we can write, for any 2]0; ~[ and " > 0:

ju" ? zjH  ju" ? u jH + ju ? zjH   (") + 1=2( ): 1 0

1 0

1 0

Thus (3.28) becomes 2k1 jru + q j  2k1 ( ) + 2k1 ( )(") + 2k1 jr(u ? u ) + q ? q" j 2 : " Ln 1=2 1=2 1=2

"

a





"

a a"

(3.30)

We can now chose 0 <   ~ such that 2k1 ( )  1=3 for each 2]0; ]; 

and, for any such , choose "( ) such that 2k1  (")  1=3 for each " 2]0; "( )]:  1=2

Then, for and " chosen as above, one can choose the neighborhood V (a")  V (u")  V (q") in such a way that V (a")  BQ(a"; 1); V (q")  BL2n (q"; 1); and that



2k1 r(u ? u ) + q ? q"  1=3; " 1 a a

" 2

for each (a; u; q) 2 V (a")  V (u")  V (q"):

" L2n

Then for 0 <   and 0 < "  "( ), inequality (3.27) may be rewritten as: J"00(a; u; q)(h; v; p)2  2 jhj2Q + 12 jruj2L2n + "k (M1 + 1) jp0j2L2n 1

(3.31)

We now estimate the continuity constant of the mapping (h; v; p0) ; (h; v; p) when a 2 BQ(a"; 1) and q 2 BL2n (q"; 1). It will be convenient to de ne the following weighted norm:

j(h; v; p)j2 = jhj2Q + jrvj2Ln + "k (M1 + 1) jpj2Ln : 1 2

2

From (3.26) we obtain immediately that

jpj2Ln  3jh aq j2Ln + 3jarvj2Ln + 3jp0j2Ln ; 2

2

2

22

2

(3.32)

i.e., using (3.24) and the fact that a 2 C de ned in (3.1) and q 2 BL2n (q"; 1), 2 2 jpj2  3k1(M + 1) jhj2 + 3k2 (M + 1)2jrvj2 + 3jp j2 : L2n

2

1

Q

L2n

0 L2n

Plugging this estimate for jpj2L2n into (3.32), we get

j(h; v; p)j2 Thus

j(h; v; p)j2 or





2 2  1 + 3 k1(M 2+ 1) jhj2Q + (1 + 3k12 (M + 1)2)jrvj2L2n + 3jp0j2L2n :

 2 (M + 1)2 k 1 2 2  max [2 + 6  2 ]= ; 2 + 6k1(M + 1) ; 3"k1(M + 1) j(h; v; p0)j20; 

j(h; v; p0)j2  j(h; v; p)j2; where



 1 1

= min 2 + 6k2 (M + 1)2= 2 ; 2 + 6k2 (M + 1)2 ; 3"k (M + 1) : 1 1 1

Hence (3.31) becomes

J"00 (a; u; q)(h; v; p)2  (jhj2Q + jrvj2L2n + jpj2L2n );

which completes the proof of Proposition 3.5.

4 Approximation of problem (P")

Numerical realization of (P") requires its discretization. This section is therefore devoted to the de nition of a family of nite dimensional problems (Ph) and the analysis of the convergence of their solutions (ah; uh; qh) to (a"; u"; q"). We start here from (P") in the form of x3 under condition (3.21), that the observation space Z be H01. We consider, for a xed " > 0, the problem min J(a; u; q) (P") over (a; u; q) 2 C  H01  Hdiv ; divq = f; where  Z  Z Z 1 1 1 1 1 2 2 # 2 2 J(a; u; q) = 2 jr(u ? z)j + 2 ja ? a jQ +  2 a jr uj ? f u + 2 a jqj :



(4.1) 23



We choose in this paragraph the following regularization, (

@v = 0 on @ g Q = fv 2 H 2( )j @ jvjQ = j
vjL2( )

(4.2)

which satis es hypothesis (3.2) because of the regularity theorems for elliptic equations. The splitting algorithms to be developped in x5 will involve separate minimization of J" with respect to a; u and q. Minimization with respect to u amounts to solving the elliptic problem  ?r((a + ")ru") = f ? "
z in

(4.3) u" = 0 on @ ; Similarly, the minimization with respect to q amounts to solving a slightly dierent elliptic problem: 8 = f in

< ?r(ar" ) " = 0 on @

(4.4) : q" = ?ar": As we have chosen in x6 to implement there splitting algorithms by using a mixed nite element approach to solving both (4.3) and (4.4), we de ne the discretized functional J";h in such a way that its minimization with respect to uh (respectively qh) produces exactly a mixed nite element approximation to (4.3) (respectively to (4.4)). Following the exposition in [15], let fThgh>0 be a regular family of triangulations of

and let Hdiv;h and L2h denote Raviart-Thomas-Nedelec spaces in dimension two or three (say of order k = 1). Here h is the largest diameter of a triangle T 2 Th . By construction, Hdiv;h is a nite dimensional subspace of Hdiv = fq 2 L2 ( )njr
q 2 L2 ( )g; L2h is a nite dimensional subspace of L2 ( ), and the divergence of any vector eld of Hdiv;h is in L2h. Hence we can de ne an approximate divergence operator rh
simply as the restriction to Hdiv;h of the divergence operator r
de ned on Hdiv:  rh
: Hdiv;h ?! L2h (4.5)

7 r h
sh ! rh
= r
: sh

We equip now the nite dimensional spaces Hdiv;h and L2h with the scalar products induced respectively by L2n = L2( )n and L2 = L2 ( ). We need to de ne approximate gradient operators for the (in general discontinuous) functions of L2h. From the usual Green's formula 8 Z
0 is bounded in L2 ( ) as well (discrete Poincarre inequality), and there exists u 2 H01( ) and a subsequence also denoted fuhg such that, uh ! u in L2 weakly :

for

and

rDh uh
sh

=



rDh uh ! ru in L2n weakly:

Proof : The chosen Raviart-Thomas nite elements satisfy a uniform Babuska-Brezzi condition,

Z

v r
s  inf sup jvh jL2 =1 ksh kHdiv =1 h h h

> 0;

with independent of h. Let now vh be any element of L2h with jvhjL2 = 1:

jrDh vhjL2n

= sup

s 2 L2n s 6= 0



sup

R

D

rh vh
s jsjL2n

sh 2 Hdiv;h sh 6= 0 25

R

D

rh vh
sh jshjL2n

and, as jshjL2n  jshjHdiv and

R

D

rh vh
sh

jrDh vhjLn  2

R

= ? vhrh
sh: R

sup

sh 2 Hdiv;h sh 6= 0

vh rh
sh jshjHdiv

and, using the Babuska-Brezzi condition:

jrDh vhjLn  : 2

This proves the discrete Poincarre inequality:

jvhjL  C ( )jrDh vhj for all vh 2 L2h; where C ( ) = ?1 is independent of h. The hypothesis that rDh uhis bounded implies then that the sequence fuh 2 L2h gh>0 is also bounded. Consequently there exists p 2 L2n and u 2 L2 and a subsequence, still indexed by h, such that  uh ! u in L2 ; weakly (4.9) rDh uh ! p in L2n weakly: Let then s 2 Hdiv be given. From the properties of the Raviart-Thomas space, we know that there exists a sequence fsh 2 Hdiv;hgh>0 in Hdiv;h such that  sh ! s in L2n strongly (4.10) r
sh ! r
s in L2 strongly: If we plug the above choice of sh into the de nition (4.8) of rDh uh, we can pass to the limit 2

in (4.8) (for a subsequence) using (4.9) and (4.10), which gives: Z

Z



p
s = ? urs:

(4.11)

As (4.11) holds for any s 2 Hdiv, it implies (by choosing s 2 [D( )]n) that p = ru, hence u 2 H 1( ). Comparing then (4.11) with (5.4) we see that u 2 H01( ).

 Neumann Boundary Conditions We shall also need to take the gradient of functions ah 2 L2h corresponding to the discretization of the parameter a 2 C , which do not vanish on @ (recall that C is made up of functions bounded below by  > 0 - see (2.6)). Application of the previous discrete gradient rDh to ah would hence generate arti cially large values near the boundary of . Hence we introduce another discrete gradient rNh which will produce an approximation to ra whenever the normal derivative @a = ra
 @n vanishes on @ . So we are led to de ne 0 = fs 2 H Hdiv div;h : s
 = 0 on @ g ;h

26

(4.12)



equipped with the same scalar product as Hdiv;h, namely the one induced by L2n. Then we 0 can de ne a discrete Neumann gradient rNh : L2h ?! Hdiv ;h by (

rNh = ?(rh
) ; where 0 2 rh
: Hdiv ;h ; Lh ;

i.e., using the chosen scalar products 8 Z
0g of parameters satis es

ah 2 Ch for all h

(4.16)

j
Nh ahjL ( )  C independant of h;

(4.17)

2

there exists an a 2 C and a subsequence of ah, still denoted by ah, such that: ah ?! a uniformly on 

(4.18)


Nh ah *
a weakly in L2( )

(4.19)

fh = ?
Nh ah

(4.20)

Proof : For all h > 0, we set which satis es

Z



fh = 0

jfhjL ( )  C independant of h: 2

27

(4.21) (4.22)

Then we de ne, for all h > 0, bh as the solution of the elliptic problem: 8
>
> : R

*f *a

in L2 ( ) weakly; in H 2( ) weakly; ?! a uniformly on  ;

ah ?!  in R; with j j    j j:

(4.24)

Because of the weak convergence of fh in L2( ) and bh in H 2( ), f and a satisfy necessarily: 8


uh r
sh > " > :

for all sh 2 Hdiv;h:

(4.42)

Then (4.40) may be rewritten, using (4.41) and (4.8): 8 Z
0. There exists a subsequence of (ah; uh; qh) of solutions of (Ph) and a solution (a; u; q) of (P") such that (ah;
Nh ah; uh;
Dh uh; qh; rh
qh) converges to (a;
a; u; ru; q; r
q) strongly in L1  L2  L2  L2n  L2n  L2 . Moreover, every subsequence such that (
Nh ah; rDh uh; qh ) converges weakly in L2  L2n  L2n contains a subsequence converging strongly, in the above sense, to a solution of (P"). The theorem will follow from the following lemmas. Lemma 4.4 For every h > 0, there exists a solution (ah; uh; qh) to (Ph). 32



Proof : Let f(ai; ui; qi)g1i=1 denote a minimizing sequence for (Ph). Then there exists a

constant K independent of i such that

0  Jh(ai ; ui; qi)  K:

(4.49)

faig1i=1 is bounded in L1;

(4.50)

By de nition of C , we have and from (4.49) and (??) we see that

f
Nh ai g1i=1 is bounded in L2

(4.51)

frDh uig1i=1 is bounded in L2n

(4.52)

fMh? (a?i 1)rDh ui ? Mh (a?i 1)qi g1i=1 is bounded in L2n:

(4.53)

and 1 2

1 2

From (4.52) we deduce, using the discrete Poincarre inequality of Lemma 4.1, that

fuig1i=1 is bounded in L2 ;

(4.54)

and from (4.52) (4.53) we deduce, using twice the property (4.38) of Mh, that

fqig1i=1 is bounded in L2n:

(4.55)

As the spaces Qh; L2h and Hdiv;h are nite dimensional, there exists (ah; uh; qh) 2 Ch  L2h  Hdiv;h, with rhqh = fh and a subsequence such that

! ah N
h ai !
Nh ah rDh ui ! rDh uh ui ! uh qi ! qh ai

in L1 in L2 in L2n in L2 in L2n:

We can now pass to the limit in the form (4.48) of the cost functional Jh, which implies that (ah; uh; qh) is a solution to (Ph).



Lemma 4.5 Let (ah; uh; qh) be a solution of (Ph). Then the sequence f(ah;
Nh ah; uh; rDh qh)gh>0 is bounded in L1  L2  L2  L2n  Hdiv , and limh!0Jh(ah; uh; qh)  J (a"; u"; q"): (4.56) 33

Proof : Let (a"; u"; q") 2 (C \ Q)  L2  Hdiv be a solution of (P"). We choose rst a sequence f(^ah; u^h; q^h)gh>0 in Ch  L2h  Hdiv;h, with rq^h = fh, as follows:  choice of a^h: we proceed in two steps. Let  > 0 be given. Step 1: for any  > 0, let  be the ane function which maps [; ] onto [ + ;  ? ]

(where  and  are the lower and upper bounds de ning C and Ch). Then a" is in C \ Q (its range is in [ + ;  ? ], it is in H 2( ) and its normal derivative on @

is zero), and it converges towards a" strongly in L1 \ H 2. Hence there exists  > 0, such that, if we set a~ =  a": 8
? ^rs = 0 8s 2 Hdiv < Z a" Z

(4.63) > : vrq^ = fv 8v 2 L2 :



But the solution q" of (P") satis es also (4.63) for some " by construction, which implies, as (4.63) has a unique solution, that q^ = q" and ^ = ". Hence the convergence results for mixed approximations imply that:  ^h ! " in L2h (4.64) q^h ! q" in L2n: Comparing the second equation of (4.62) with the de nition (4.39) of the fh we see also that

rhq^h = fh 8h > 0: 35

(4.65)

With the above choices, the sequence f(^ah; u^h; q^h)gh>0 is admissible for (Ph) for every h > 0, hence: 0  Jh(ah; uh; qh)  Jh(^ah; u^h; q^h):

(4.66)

But from (4.60) (4.61) and (4.64) it follows that one can pass to the limit in Jh(^ah; u^h; q^h), so that: lim J (^a ; u^ ; q^ ) = J (a"; u"; q"): h!0 h h h h

(4.67)

which, together with (4.66) proves (4.56). In order to prove the announced a-priori estimations, we remark that (4.56) implies the existence of a constant K > 0, independent of h, such that, for all h:

Jh(ah; uh; qh)  K:

(4.68)

One can then proceed from (4.68) exactly as we have done in the rst part of the proof of lemma 4.4 to obtain the expected estimates.

Lemma 4.6 Let f(ah; uh; qh)gh>0 be a sequence of solutions of (Ph). Then there exists ((a; u; q) 2 C \ Q  H01  Hdiv such that, for a subsequence: ah ! a in L1 strongly; (4.69)
Nh ah !
a in L2 weakly;

(4.70)

uh ! u in L2 weakly;

(4.71)

rDh uh ! ru in L2n weakly;

(4.72)

qh ! q in L2n weakly;

(4.73)

rqh ! rq = f in L2 strongly; (4.74) and (a; u; q) is a solution to (P"). Proof : We use rst the a-priori estimates of lemma 4.5:  as (ah;
Nh ah) is bounded in L1  L2, we see from Lemma 4.2 that there exists a 2 C \ Q such that, for a subsequence, (4.69) and (4.70) hold.  as (uh; rDh uh) is bounded in L2  L2n, we deduce from lemma 4.1 the existence of u 2 H01 such that (4.71) and (4.72) hold for a subsequence. 36

 as qh is bounded in L2n, and r
qh = rh
qh = fh is bounded in L2 by de nition of fh, we see that qh is bounded in Hdiv. Hence there exists q 2 Hdiv such that, for a subsequence, qh * q weakly in Hdiv, which implies (4.73) and rqh * rq weakly in L2 . But rqh = fh which, by de nition, converges to f strongly in L2 , which proves (4.74). Then using the convergence (4.69) thus (4.74) and the weak lower semicontinuity of norms, we have

J (a; u; q)  limJh(ah; uh; qh)  limJh(ah; uh; qh)  J (a"; u"; q")

(4.75)

which shows that (a; u; q) is a solution of P". Lemma 4.6 proves the theorem 4.1, up to the strong convergence results for
Nh ah; uh; rDh uh and qh. But from (4.75) we obtain, as J (a; u; q) = J (a"; u"; q"): limJh(ah; uh; qh)  J (a; u; q)  limJh(ah; uh; qh): This implies that lim J (a ; u ; q ) = J (a; u; q) h!0 h h h h i.e., using the form (4.48) of Jh: 1 jrD (u ? z )j2 + j
N a j2 + 1 jM ? 12 (a?1)rD u ? M 21 (a?1 )q j2 lim h L2n h h h h h L2n h!0 2 2 h h L2 2" h h h h 1 jr(u ? z)j 2 + j
aj 2 + 1 ja 21 ra ? a? 12 qj2 Ln 2 L L2n 2 2"

Together with the fact that ah ! a strongly in L1 and that the w ? lim(
Nh ah; rDh uh; qh) = (
a; ru; q) in L2 L2nL2n , it follows that (
Nh ah; rDh uh; qh) converges strongly to (
a; ru; q) in L2  L2n  L2n. The discrete Poincarre inequality of Lemma 4.1 implies then that uh ?! u strongly in L2 . This ends the proof of theorem 4.1.

5 Splitting algorithms for the numerical resolution The uniform convexity of the cost functional J" in each of the variables separately suggests solving (P") by splitting algorithms. The functional J" is not jointly convex in the variables (a; u; q), however, and hence the convergence of the splitting algorithms must be considered locally. We shall carry out the analysis on the basis of Proposition 3.5 which asserts the local convexity of J" in some neighborhood of each solution (a"; u"; q") to (P"). Throughout this section it is assumed that (3.21) and (3.22) hold and that and " are chosen such that for a xed solution (a"; u"; q") the conclusion (3.23) of Proposition 3.5 37



holds. Further U (a")  U (u")  U (q" ), is a convex neighborhood of (a"; u"; q") satisfying U (a")  U (u")  U (q")  V (a")  V (u")  V (q"). In the rst algorithm that we analyze, minimization is carried out separately with respect to a; q and u. Algorithm 5.1 (i) (ii) (iii) (iv) (v)

Set u0 = z 2 U (u"); set n = 1 and choose q0 2 U (q"): an = argmin J"(a; un?1; qn?1) over a 2 C \ U (a"): qn = argmin J"(an; un?1; q) over q 2 U (q"); div q = f: un = argmin J"(an; u; qn) over u 2 U (u"): check convergence, stop or set n = n + 1 and go to (ii):

Note that the cost functions in (iii) and (iv) are quadratic. Also theRcost functional is separable with respect to u and q, the only coupling occuring in the ajr uj2-term. Hence (iii) and (iv) can be solved in parallel. We shall prove that (an ; un; qn) ! (a"; u"; q") in X so that after nitely many steps of the iteration the constraints a 2 U (a"); q 2 U (q") and u 2 U (u") become inactive. In the statement of the following theorem the notation of Proposition 3.5 is used. Theorem 5.1 Assume that (3.21) and (3.22) hold, and let 2]0; ]; and " 2]0; "( )]: Then the sequence (an; un; qn) generated by Algorithm 4.1 converges in X to (a"; u"; q"). Proof : It is simple to argue the existence of unique solutions (an; un; qn) in (ii); (iv) of the algorithm. Concerning the convergence of (an; un; qn); the proof relies on arguments that are similar to standard ones in the context of splitting algorithms [4]. The special structure of the problem, however, does not allow us to refer directly to known results. The solutions of (ii)-(iv) satisfy @ J (a ; u )(a ? a ) + 1 (J (a; u ; q ) ? J (a ; u ; q ))  0 n (5.1) @a 1 n n?1 " 2 n?1 n?1 2 n n?1 n?1 for all a 2 C \ U (a")

@ J (a ; u ; q )(q ? q )  0 n @q 2 n n?1 n

for all q 2 U (q") such that div q = f;

(5.2)

@ J (a ; u )(u ? u ) + 1 (J (a ; u; q ) ? J (a ; u ; q ))  0 n (5.3) @u 1 n n " 2 n n 2 n n n for all u 2 U (u"): Note that for every n = 2; 3; ::: J"(an?1; un?1; qn?1) ? J"(an; un; qn) = J"(an?1 ; un?1; qn?1) ? J"(an ; un?1; qn?1) + J"(an; un?1; qn?1) ?J"(an; un?1; qn) + J"(an; un?1; qn) ? J"(an; un; qn)  0; 38

and therefore J"(an; un; qn) is monotonically decreasing with respect to n. Since this sequence is also bounded from below it is necessarily convergent. Moreover we nd from (5.1) - (5.3) that

J"(an?1; un?1; qn?1) ? J"(an; un; qn) = J1 (an?1; un?1) ? J1(an; un?1) ? @ J1 (an; un?1)(an?1 ? an) + @ J1(an; un?1)(an?1 ? an ) @a @a 1 1 + (J2 (an?1; un?1; qn?1) ? J2 (an; un?1; qn?1)) + (J2(an; un?1; qn?1) ? J2 (an; un?1; qn)) " " @ +J1(an; un?1) ? J1 (an; un) ? J1(an; un)(un?1 ? un) + @ J1(an; un)(un?1 ? un) @u @u 1 + (J2 (an; un?1; qn) ? J2(an; un; qn)) " Z @ 1  J1(an?1; un?1) ? J1 (an; un?1) ? @a J1(an ; un?1)(an?1 ? an) + " a1 jqn?1 ? qn j2

n @ +J1(an; un?1) ? J1 (an; un) ? @u J1(an; un)(un?1 ? un) Z 1 1 2 2  2 jan ? an?1jQ + 2 jr(un?1 ? un)jL2 + " a1 jqn?1 ? qnj2: n In particular it follows that lim j(an; un; qn) ? (an?1; un?1; qn?1 )jX = 0:

n!1

(5.4)

In the following step we use (3.22) and the fact that by construction (an; un; qn) 2 V (a")  V (u")  V (q"): (J"0 (an; un; qn) ? J"0 (a"; u"; q"))(an ? a"; un ? u"; qn ? q") Z 1

J"00(t(an ; un; qn) + (1 ? t)(a"; u"; q"))((an; un; qn) ? (a"; u"; q"))2 dt 0  j(an; un; qn) ? (a"; u"; q")j2X : Since J"0 (a"; u"; q")(an ? a"; un ? u"; qn ? q")  0; this implies that ( @ J"(an; un; qn) ? @ J"(an ; un?1; qn?1))(an ? a") + @ J"(an; un?1; qn?1)(an ? a") @a @ @a @ @a@ +( @q J"(an ; un; qn) ? @q J"(an; un?1; qn))(qn ? q") + @q J"(an; un?1; qn)(qn ? q") + @ J"(an; un; qn)(un ? u")  j(an; un; qn) ? (a"; u"; q")j2X : @u =

(5.5)

@ J (a ; u ; q )(a ? a )  0; @ J (a ; u ; q ))(q ? q )  0 But @a " n n?1 n?1 n " " @q " n n?1 n n @ J (a ; u ; q )(u ? u )  0, so that @u " n n n n " 39

and

1 Z (jru j2 ? jru j2)(a ? a )dx ? 1 Z 1 (jq j2 ? jq j2)(a ? a )dx n n?1 n " n?1 n " 2"

2" a2n n  j(an; un; qn) ? (a"; u"; q")j2X : The boundedness of fang in Q together with (5.4) implies that lim (a ; u ; q ) = (a"; u"; q") in X : n!1 n n n The second splitting algorithm that we discuss requires only an initial guess for u0 and

z is a good choice. Algorithm 5.2 (i) (ii) (iii) (iv)

Set u0 = z 2 U (u") and let n = 1 (an ; qn) = argmin J"(a; un?1; q) over (a; q) 2 (C \ U (u"))  U (q"); div q = f; un = argmin J"(an; u; qn) over u 2 U (u"): check convergence, stop or set n = n + 1 and go to (ii):

Theorem 5.2 Under the hypotheses of Theorem 5.1 the sequence (an; un; qn) generated by Algorithm 4.2 converges in X to (a"; u"; q"). Proof : Necessary optimality conditions for (an; qn) and for un, respectively, are given by J"(a; un?1; q) ? J"(an; un?1; qn)  0 (5.6) for all (a; q) 2 (U (a") \ C )  U (q"); div q = f; and @ J (a ; u ; q )(u ? u ) + 1 (J (a ; u; q ) ? J (a ; u ; q ))  0 n (5.7) @u 1 n n n " 2 n n 2 n n n for all u 2 U (u"): Further, for all n = 1; 2; ::: we have J"(an?1; un?1; qn?1) ? J"(an; un; qn)  J"(an?1 ; un?1; qn?1) ? J"(an ; un?1; qn) + J"(an; un?1; qn) ? J"(an ; un; qn) @ J (a ; u )(u ? u ) + @ J (a ; u )(u ? u )  J1(an; un?1) ? J1 (an; un) ? @u 1 n n n?1 n 1 n n n?1 n @u + 1" [J2 (an; un?1; qn) ? J2(an; un; qn)]  21 jr(un?1 ? un)j2; and thus

lim ju ? un?1jH01 n!1 n 40

= 0:

(5.8)



The estimate corresponding to (5.5) is

( @ J"(an; un; qn) ? @ J"(an; un?1; qn)(an ? a") + @ J"(an; un?1; qn)(an ? a")

@a @ @a @ @a @ +( J"(an; un; qn) ? J"(an; un?1; qn))(qn ? q") + J"(an; un?1; qn)(qn ? q") @q @q @q @ 2 + @u J"(an; un; qn)  j(an; un; qn) ? (a"; u"; q")jX ;

and hence

1 Z (jru j2 ? jru j2)(a ? a )  j(a ; u ; q ) ? (a ; u ; q )j2 : (5.9) n n?1 n " n n n " " " X 2"

It is simple to argue that fang is bounded in Q and hence combining (5.8) and (5.9) it follows that limn!1(an; un; qn) = (a"; u"; q"):

6 Mixed nite element implementation We describe here the numerical discretization used for the implementation of the primaldual formulation for the estimation of the diusion coecient in the elliptic equation ?r(aru) = f in

(6.1) u = 0 on @ : More precisely, we construct a primal-dual formulation for a discretization of (6.1) rather than discretize one of the primal dual formulations given in the examples of x2. We have used for the discretization a mixed nite element scheme as with such a scheme both the primal and dual energy functionals are readily calculated. The primal-dual formulation that we shall use for the discretized problem follows the primal-dual formulation given in Example 2.1 of x2 and analyzed in x3: 





Z Z Z 1 1 1 1 2 2 ] 2 min 2 jz ? BujZ + 2 ja ? a jQ + " 2 ajruj ? fu + 2 a1 jqj2 (P")



1 2 over (a; u; q) 2 C  H0  Hdiv ; rq = f 2 L : We rst describe brie y the mixed nite element numerical scheme that we use to discretize (6.1). For more details concerning the numerical method see [9] for the implementation or [6, 15] for a more theoretical treatment. For simplicity we suppose that

 R2 . Extension to higher dimension is straightforward as is extension to other types of boundary conditions. Let Th be a triangulation of by triangles and/or rectangles, and let Eh be the set of edges of elements of Th. We shall denote by K , respectively E , the typical element of Th, respectively Eh, and by NT , respectively NE , its cardinality.

41



The mixed nite element method that we shall use for the discretization is based on the following mixed variational formulation of (6.1): (u; q) 2 L2 ( )  Hdiv Z 1 q q0 ? Z u div q0 = 0;

a Z



Z

div q u0 =



for q0 2 Hdiv

(6.2)

for u0 2 L2 ( ):

f u0 ;

We shall thus approximate the state variable u in a nite-dimensional subspace of The space of discretized state variables, Xh, will be the space of piecewise constant functions, functions constant on each element K of Th. The dual state variable q will be approximated in the lowest-order Raviart-Thomas space for the approximation of Hdiv functions, which in keeping with the notation of x2 we shall denote Yh. A basis for Xh is the set of characteristic functions K of the elements K of Th. Xh = spanfK ; K 2 Thg: To specify a basis for Yh we choose for each edge E 2 Eh a unit vector E normal to E and de ne the basis function !E to be the unique function satisfying  each component of !E is linear on each element K of Th  !E 2 Hdiv i.e. !E jK
E = !E jK 0
E if E is an edge of K and of K 0  for each edge E 0 2 Eh; !E
E0 is constant on E 0  !E has 0 ux across each edge in Eh other than E itself where it has ux equal to one:  Z = E0 : !E
E0 = 01 ifif EE 6= E0

L2 ( ).

E0

Now

Yh = spanf!E : E 2 Ehg:

Thus we may write

uh =

X

K 2Th

UK K

qh =

X

E 2Eh

QE !E ; R

(6.3)

with UK giving the constant value of uh on K , and QE = E qh
E the ux of qh across E in the direction E . Thus uh is identi ed with the vector fUK gK 2Th 2 RNT and qh with fQE gE2Eh 2 RNE. The source function f and the diusion coecient a will be assumed to be piecewise constant and thus can be written as

f=

X

K 2Th

fK K

a= 42

X

K 2Th

aK K :

The mixed nite element method yields an approximation (uh; qh) of (u; q) satisfying (uh; qh) 2 Xh  Yh Z Z 1 q q0 ? u div q0 = 0; for qh0 2 Yh h h h h (6.4) a



Z



div qh

u0

h=

Z



for u0h 2 Xh:

f u0h;

De ning the NE  NE matrix M and the NT  NE matrix D by Z Z 1 ! ! 0 D = div !  ; M 0= E;E

a

K;E

E E



E K

(6.5)

and letting F denote the vector in RNT with coordinates FK = fK jK j; we may write (6.4) as a linear system MQ ? DT U = 0 (6.6) DQ = F: The matrix M is symmetric and positive de nite. Thus we can use the rst equation of (6.6) to express Q in terms of U , and plug this expression into the second equation of (6.6) to obtain the problem DM ?1 DT U = F: (6.7) Remark 6.1 The (K; E ) entry of the divergence matrix D is 0 unless E is an edge of K in which case DK;E is 1 if E points outward from K and is -1 if E points inward. Remark 6.2 If Th is a set of triangles, the symmetric matrix M has ve nonzero entries in rows corresponding to interior edges and three for those corresponding to boundary edges. In case Th is made up exclusively of rectangles, M is tridiagonal (with any reasonable ordering of the edges). Further, if we calculate the integrals used to de ne the matrix M by the numerical quadrature formula that approximates an integral over an element K by the average of the values at the vertices multiplied by0 the area of the element, then M becomes a diagonal matrix with ME;E = 2 j1E j ( jaK j + jaK 0j ), for E an edge of K and of K 0 . K

K

Having described the discretization of (6.1), we turn now to the construction of the primal-dual formulation of the identi cation problem. The solution U of problem (6.7) is characterized as the solution of the minimization problem

U 2 Xh 0 Eah (U ) = U inf 0 2X Eah (U ); h

43

(6.8)

where the primal energy functional Eah : Xh ?! R is de ned by

Eah (U 0 ) = 12 M ?1 DT U 0
DT U 0 ? F
U 0 ;

(6.9)

and the solution Q of problem (6.6) is characterized as the solution of the minimization problem

Q 2 Yh Eah (Q) = Q02Xinf;DQ=F Eah (Q0);

(6.10)

h

 : Yh ?! R is de ned by where the dual energy functional Eah

Eah (Q0) = 21 MQ0
Q0 :

(6.11)

(Note that the U calculated in (6.6) is considered here simply as a multiplier used to solve the constrained problem (6.10) and does not appear in either energy functional.) In the notation of x2 the space Ch of permissible parameters is RNT, the state space Xh is RNT; the space of dual states Yh is RNE; and

AhU = DT U; Fah(U ) = ?F
U; Thus

Gah(Q) = 21 M ?1 Q
Q:

AhQ = ?DQ; 

+F =0 ; Gah(Q) = 21 MQ
Q: Fah (U ) = +01 ifif UU + F 6= 0 We remark that Fah and Gah are convex functions and that Q 2 @F (DT U ) i.e.

Q = M ?1 (DT U ):

Further, the Fenchel duality theorem guarantees that infNT Eah(U ) +

U 2R

Q2R

 (Q) = 0: E ah ;DQ=F

inf

NE

In the mixed formulations (6.2) and (6.4) the diusion coecient a appears only by means of its inverse a1 and in (6.7) and (6.6) only by means of the matrix M which is de ned in terms of 1 . We have thus, for the numerical experiments, chosen to identify not a =

P

a 1 K 2Th aK K itself but  = a =

44

X

K 2Th

K K , where K = 1 : The vector

aK

 = fK gK 2Th like faK gK 2Th is in RNT.

Remark 6.3 Identifying the reciprocal of a is in fact suggested by the linear structure in which a and u appear in (6.1) and (6.2). In particular, for the case in which a is constant, u depends linearly on a1 not on a.

To estimate the diusion coecient in (6.4) or (6.6) we must de ne the observation space Zh and the observation operator from the state space Xh = RNT to Zh. In the numerical examples of this paper, we have chosen the the observation operator to be the identity operator. In the case of noisy data, this leads to better reconstruction of the unknown coecients than the choice of B as the divergence operator. Consequently Zh is taken to be Xh = RNT. The space used for the regularization (denoted Q is x2 but not denoted Qh here for obvious notational reasons) will be taken to be RNT but with the semi-norm jj2 = G
G, where G is the gradient matrix obtained from DT by eliminating the rows corresponding to edges E contained in the boundary of . Thus the regularized least squares functional J1 : Ch  Xh ?! R is J1 (; U ) = 21 (Z ? U )
(Z ? U ) + 2 G( ? ])
G( ? ] ); and the total energy functional J2 : Ch  Xh  Yh ?! R is J2(; U; Q) = 21 M ?1 DT U
DT U ? F
U + 21 MQ
Q: The discretized version of (P") may now be given: 

min 1 (Z ? U )
(Z ? U ) + G( ? ])
G( ? ] ) 2  2  1 1 1 ? 1 T T + " 2 M D U
D U ? F
U + 2 MQ
Q over (; U; Q) 2 Ch  Xh  Yh; DQ = F: We write J" for the functional to be minimized in (P"h): J (; U; Q) = J (; U ) + 1 J (; U; Q): "

1

"

(P"h)

2

In all of the experiments that we consider in x6 ] is chosen to be 0; thus, the regularization penalizes oscillations in . To use one of the splitting algorithms of x4 we note that to minimize J" with respect to  for U and Q xed is to minimize   1 1 1 ] ] ? 1 T T J"B = 2 G( ?  )
G( ?  ) + " 2 M D U
D U + 2 MQ
Q 45

over  2 Ch: To minimize J" with respect to Q for  and U xed is to minimize J"Q = 21 MQ
Q over Q 2 Yh; DQ = F , which is equivalent to solving the dual problem (6.6). The linear system (6.6) may be solved by using a hybridization of the mixed method; see [9]. (Recall that the vector U produced by the resolution of (6.6) serves only as an auxiliary variable.) Finally, to minimize J" with respect to U for  and Q xed is to minimize   1 1 1 ? 1 T T J"U = 2 (Z ? U )
(Z ? U ) + " 2 M D U
D U ? F
U over U 2 Xh, which is equivalent to solving the following regularization of the primal problem (6.7): [DM ?1 DT + "I ]U = F + "Z

(6.12)

Thus in the discretized context, Algorithm 4.1, for example, becomes Algorithm

i) ii) iii) iv) v)

Set U0 = Z 2 Xh = RNT; choose Q0 2 Yh = RNE; set n = 1 n = argmin J"(; Un?1; Qn?1) Qn = solution to (6:6) with M formed using n Un = solution to (6:12) with M formed using n check convergence, stop or set n = n + 1 and go to (ii):

7 Numerical results In the numerical examples presented here we identify the diusion coecient, or rather its reciprocal  = 1 (see Remark 5.3), in the equation

a

?r(aru) = f in

u = 0 on @ :

(7.1)

In all of the experiments, is taken to be the unit square in R2 : = [0; 1]  [0; 1], and the source function is f =
(x(1 ? x)y(1 ? y) exp(x y). The same grid is used for the discretization of the reciprocal of the diusion coecient , the state variable u, and the dual state variable q. We take a regular grid of squares of side length 0.05, so Th contains 400 squares, (NT = 400), and Eh contains 420 vertical edges and 420 horizontal edges, (NE = 840). 46

In terms of the preceding section, x5, we solve the discretized minimization problem  min 12 (Z ? U )
(Z ? U ) + 2 G( ? ])
G( ? ] )   1 1 1 ? 1 T T (P"h) + " 2 M D U
D U ? F
U + 2 MQ
Q over (; U; Q) 2 RNT  RNT  RNE ; DQ = F;

where U 2 RNT is the state variable, Q 2 RNE is the dual state variable,  2 RNT is the reciprocal of the piecewise constant diusion coecient: K = a1 . The matrix D is the K divergence matrix described in x5 and DT is its transpose. The gradient matrix G used in the regularization term is also given in x5. As we have used a regular square grid, we use the numerical quadrature rule described in Remark 2.2 to obtain the matrix M . Thus M is a diagonal matrix with entry corresponding to the edge E between the rectangles K 0 and K 0 equal to jE j K +2 K and with entry corresponding to the edge E of K on the NT has component corresponding to the boundary of equal R to jE j K . The ]vector F 2 R element K 2 Th, K fdxdy. Here  is taken to be 0; the penalization parameter "2 is 10?1. What will change with the experiment are the observation Z , the amount of noise and the amount of regularization . The observation Z will be determined as follows:

Z = Uexact + 2N jUmax ? Umin j R; where Uexact is the solution of (6.7) for F as given above and M computed using the sought diusion coecient , the noise level N is taken to be either 0 or 0.1, Umax and Umin are the maximum and minimum values of Uexact, and R 2 RNT is a uniform random distribution of numbers in [?1; 1]. The sought coecient will be  = fK gK 2jTh given by K = 2 ? xK + yK ; where (xK ; yK ) is the coordinate of the center or K . We also consider the case K = 1 if XK < :5; K = 6 if XK > :5 We will show results of experiments with no regularization = 0, a small amount of regularization, = 2:5  10?5, for the case in which there is no noise and the sought  is ane, and larger amounts of regularization, = 10?2 or = 2:5  10?1, for cases in which there is noise or the sought  is highly discontinuous. The algorithm follows Algorithm 4.1 as well as that given at the end of x5. Algorithm

 initialization { choose ? arbitrarily { set U0 = Z 47

{ set Q0 = 1 DT U0 ? { calculate 0 using a minimization routine to minimize with respect to  with U = U0 and Q = Q0 xed. { set n = 1

 main loop { minimize with respect to  by using a minimization routine to obtain n { minimize with respect to U by solving (6.12) with matrix M calculated using

n to obtain Un { minimize with respect to Q by solving (6.6) with matrix M calculated using n to obtain Qn { check convergence, stop or set n = n + 1 and continue. For all of the numerical calculations we used the library SCILAB which is very convenient to use and produces an ecient code due to its ability to compile the entire code prior to beginning execution. The minimization routine used in the initialization and in the main loop to obtain  is the minimization routine optim of SCILAB. It is used with the quasi-Newton, low memory option qnm. The number of iterations of the minimization algorithm used before updating U and Q was xed and in the results reported here was taken to be 1. In the rst experiment we seek to identify an ane coecient function K = 2?xK +yK . The observation is from noiseless data, N = 0, so that Z is Uexact. In Figure 1 we see four graphs of the coecient function . The two graphs on the left show  after convergence, after 30 iterations of the main loop. The upper one is without regularization and we see oscillation around the singular point near the center of . For the lower one, the regularization coecient is = 2:5  10?5, and the oscillation has essentially disappeared. The graph on the lower right shows 0, i.e.  after initialization, (with regularization). Thirty iterations were allowed in the minimization routine to obtain 0. The graph on the upper right is obtained by a classical least squares method without regularization, starting from the same initial guess, ? as in the algorithm above. Again 30 iterations were used to obtain the result. The calculation time used to obtain each of the three results on a Digital 3000/900 was of the order of one minute. In the second experiment we have added noise. The coecient  which we seek to identify is the same as in the preceeding experiment so that Uexact is the same as before but here the noise level N is taken to be 0.1. The two upper graphs in Figure 2 show  after convergence, the one on the left obtained without regularization, the one on the right with regularization coecient = 10?2. The graph on the lower left shows the noisy observation Z while the graph on the lower right shows U , the pressure, calculated with the coecient  shown on the upper right; i.e. U after convergence. In the nal experiment, we have tried to identify a strongly discontinuous function , K = 1 if XK < :5; K = 6 if XK > :5. Both graphs show  after convergence. The graph 48

4

4

3

3

2

2

1 20

1 20

10 0 0

20 10 no regularization

0 0

20 10 least squares

0 0

20 10 initialization

10

3

4 3

2 2 1 20

1 20

20

10 0 0

10

10 regularized

Figure 7.1: Identi cation of an ane coecient function by the primal-dual method, with and without regularization, compared with identi cation by a least squares method.

49

10

3

5

2

0 20

1 20

10 0 0

20 10 unregularized

20

10

2

0 0

10 regularized

0 0

10 pressure

0

0 −2 −2 −4 20 10 0 0

−4 20

20 10 observation

20

10

Figure 7.2: Identi cation with noisy data.

50

on the left was obtained from an observation with no noise, N = 0, using a regularization coecient, = 10?2. The right-hand side corresponds to a coecient  obtained from noisy data, N = 0:1, but with more regularization; = 25  10?2. We see the eect of the greater amount of regularization in the latter case in that the discontinuity is not represented as well as in the former. 6

6

4

4

2

2

0 20

0 20

20

10 0 0

20

10

10 no noise

0 0

10 noise

Figure 7.3: Identi cation of a discontinuous coecient function. In conclusion, the primal-dual method works well for the experiments we have carried out on a model problem and in general is superior to the least squares method. The fact that information from the observation can be exploited for initialization as well as for regularization makes the primal-dual method ecient and robust. In particular, it works well even with noisy data.

References [1] R. Accar, Identi cation of coecients in elliptic equations, SIAM Journal on Control and Optimization, 31 (1993), pp. 1221{1244. [2] H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, in Systems and Control, Foundations and Applications, Basel, ed., Birkhauser, Boston, 1989. [3] V. Barbu and K. Kunisch, Identi cation of non linear elliptic equations, Applied Mathematics and Optimization. to appear. [4] V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Reidl Publishing, Dodrecht, 1986. [5] J. Baumeister and W. Scondo, Adaptive methods for parameter identi cation, in Methoden und Verfahren der Mathematischen Physik Vol. 34, Verlag P. Lang, 1987, pp. 87{116. 51

[6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer series in Computational Mathematics n 15, Springer-Verlag New York Inc., New York, 1991. [7] G. Chavent, Identi cation de Coecients Repartis dans les Equations aux Derivees Partielles, These de Doctorat d'Etat, Faculte des Sciences de Paris, 1971. [8] , On the theory and practice of nonlinear least squares, Advances in Water Resources, 14 (1991), pp. 55{63. [9] G. Chavent and J. E. Roberts, A uni ed physical presentation of mixed, mixedhybrid nite elements and standard nite dierence approximations for the determination of velocities in water ow problems, Advances in Water Ressources, 14 (1991), pp. 329{348. (Preprint in report INRIA n 1107, Oct. 89). [10] I. Ekeland and R. Temam, Analyse Convexe et Problemes Variationnels, Etudes Mathematiques, Dunod, Paris, 1974. [11] K. Ito and K. Kunisch, The augmented Lagrangian method for parameter estimation in elliptic systems, SIAM Journal on Control and Optimization, 28 (1990), pp. 113{136. , Sensitivity analysis to optimization problems in Hilbert spaces with application [12] to optimal control and estimation, J. Dierential Equations, 99 (1992), pp. 1{40. [13] R. Kohn and B. Lowe, A variationnal method for parameters estimation, RAIRO, M2AN, 22 (1988), pp. 119{158. [14] K. Kunisch and X. Tai, Sequential and parallel splitting methods for linear control problems in Hilbert spaces, SIAM J. Num. Analysis. to appear. [15] J. E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, eds., vol. 2 Finite Element Methods{ Part 1, Elsevier Science Publishers B.V. (North{Holland), Amsterdam, 1991.

52