carrilo gomez fraguela optimization 21102013

8/12/2019 Carrilo Gomez Fraguela Optimization 21102013

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ber 22, 2013 Optimization carrilogomezfraguelaoptimization 21102013

Optimization Vol. 00, No. 00, Month 200x, 121

RESEARCH ARTICLE

A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem

Mauricio Carrillo Oliveros, Juan Alfredo G omez Fern andez a and Andres FraguelaCollar b

a Universidad de La Frontera, Temuco,Chile ; b Universidad Aut onoma de Puebla, Mexico

(October 18th, 2013 )

We study the convergence properties of an algorithm for the inverse problem of electricalimpedance tomography, which can be reduced to a PDE constrained optimization problem.The direct problem consists of the potential equation div( u) = 0 in a circle with Neumanncondition describing the behavior of the electrostatic potential in a medium with conductivitygiven by the function (x, y ). We suppose that at each time a current i is applied to theboundary of the circle (Neumanns data) and it is possible to measure the correspondingpotential i (Dirichlet data). The inverse problem is nding (x, y ) given a nite number of Cauchy pairs measurements ( i , i ), i = 1 ,...,N . The problem formulated as a least squareproblem and the developed algorithm solve the continuous problem using descent iterationsin its corresponding nite element approximations. Wolfes conditions are used to ensure theglobal convergence of the optimization algorithm for the continuous problem. Although exactdata are assumed, measurement errors in data and regularization methods shall be consideredin a future work.

Keywords: Inverse problem, elliptic equations, least square method, nite elementdiscretization, Wolfes conditions, global convergence algorithm.

AMS Subject Classication :

1. Introduction

Electrical impedance tomography (EIT) is the recovery of the conductivity at theinterior of a conducting body from known currents and voltages applied to itssurface. This problem is a well known inverse problem and it has many applicationsin different elds.

In Geophysics, where the method is used in prospecting and Arqueology, it isknown as electrical resistivity tomography. In industrial processes, tomography itis known as electrical resistance tomography or electrical capacitance tomography

[8],[25],[10],[29]. In medical imaging, it is still a experimental technique rather thana common clinical practice [28]. The formulation of the inverse problem is thefollowing:

Given 0 = B (0, R ), the ball centered at the origin and with radius R > 0, N pairs of functions ( i , i ) representing applied currents i and the correspondingmeasured potentials i on the boundary 0 = 0 , with i H

1

2 (0) , i H

1

2 (0), nd the function (x, y ) such that the potential functions ui (x, y ) satisfythe following Neumann elliptic problems:

Corresponding author. Email: [email protected]

ISSN: 0233-1934 print/ISSN 1029-4945 onlinec 200x Taylor & Francis

DOI: 10.1080/0233193YYxxxxxxxxhttp://www.informaworld.com


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2 Carrillo, G omez & Fraguela

div( ui) = 0 , in 0 u in = i , on 0

(1)

where i = 1 ,...,N.We assumed that L (0) satises:

(x, y) K > 0, a.e in 0, (2)

and i H 1

2 (0) satises the compatibility condition:

0 i dS = 0 . (3)It is known that the Neumann problem (1)-(3) has a unique solution except from

an additive constant, which we x requiring that:

0 u i dS = 0 i dS. (4)The variational formulation of the direct problem is:

Find u i H 1() satisfying (4) such that: (5)

a(u i , v) = F i (v), v H 1(), (6)

0

i dS = 0 , (7)

where

a(u, v) = 0 u vdx, (8)F i (v) = 0 i vdS, (9)

and i H 1

2 (0) satises (3). This problem has a unique solution ui H 1(),1 i N.

Remark 1 : Indeed, the problem is formulated in the quotient space V =H 1()/ R dened by the equivalence classes with respect to the relation:

u v u v = constant, u, v H 1().

V is a Hilbert space with norm:

u V = u L 2 () , u u, u V,

where u is the equivalence class represented by u.


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A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem 3

The bilinear form a(, ) dened in V V is coercive and the Lax-Milgram lemmaensures that the unique solution u i H 1() of (7)-(9) satises the inequality:

u i L 2 () C ()

i H 12 ( 0 ) ,

where > 0 is the elipticity constant and C () > 0 is the constant associated tothe trace operator .

With respect to the numerical aproximation of the inverse problem, the most ef-cient methods are the iterative methods. They are classied into variational andoutput least square methods [3]. Both methods reduce the problem to a quadraticminimization problem (PDE constrained problems). The rst one uses variationalprinciples and the minimization is implicitly performed in H

1

2 ( ) norm, where is the boundary of the domain . In case of output least squares, the H

1

2 ( )norm is changed to the L2( ) norm and the minimization is done over the dis-cretization problem by Newton-type optimization algorithms [16],[18],[30]. The dis-cretization can be done by nite diferences or nite elements. There are differ-ences between methods, both approximate the solution of the discretized problem[4],[5],[15],[16],[18],[28].

Here we propose a different approach: we approximate the continuous solution with solutions of approximated problems using nite element method, but ensuringthe fulllment of continuous Wolfes global convergent conditions at each iterationof the discrete problems. In section 2 we formulate the least square problem and itsnite element discretization, in section 3 the general global convergence results formulti-directional algorithms, given in [10], are recalled and it is shown its depen-dence of gradient convergence; in section 4 we compute the continuous and discrete

gradient and in section 5 we prove the convergence property between gradients. Insection 6 the relations between Wolfes conditions of both, continuous and discreteproblems, are examined. In section 7 the algorithm is described and in section 8some conclusions and future work are explained.

Additional hypothesis: As usual in nite element discretization, we assumethat 0 is replaced by a near enough convex polygonal domain , which shall beconsidered xed in the rest of the article. It is known that if the boundary of 0 issmooth enough then, the solution of the variational problem on 0 is a goodapproximation of the solution on 0[24]. In our case, we have 0 C .

We denote := the boundary of and we assume i , i continuous andpiecewise linear functions on .

2. Least Square Problem and Discretization

Given Cauchy pairs ( i , i ), i = 1 ,...,N , consider the following least square prob-lem:

minL ()

J () = 12

N

i=1 |u i i |2 dS, (10)


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s.t. for i = 1 ,...,N

a(, u i ) = u i vdx = F i (v) = i vdS, v V, (11) u i dS = i dS, (12)

with u i H 1().We want to nd L () minimizing the residual between the potential data

i on and the potential ui obtained by the model. In fact, the minimizationshould be done over the closed convex set M Q, where:

M := { L () : K > 0, a.e} ,

and Q is some class of uniqueness of the inverse problem. It is assumed to beclosed and convex with nonempty interior in L (). Our aim is to use rst orderoptimality conditions and, although in a rst approach we consider the problem(10)-(12) without constraints for , we later shall return to this restriction.

Discretization of problem (10)-(11)-(12) is done by the nite element method[26]. We denote by h the family of regular triangulations of domain , whereh > 0 is the diameter of h [27]. We have:

=N T h

k=1

T k ,

for any triangulation h , where N T h is the number of elements and T k is the k thtriangle of the mesh.

We dene the subspace of H 1():

V h = vh C () H 1() : vh |T P 1(T ),T h (13)

as the space of solutions of the discretized problem, where P 1(T ) is the set of polynomials of degree less or equal to one dened on T . We also denote by:

N h : number of h nodes,

N h : number of h nodes in ,

N h : number of h nodes in ,

therefore: N h = N h + N h . We will assume a node numbering of the form:

{1, 2, ...N h , N h + 1 , ...N h } ,

that is, we rst consider the inner nodes.We also give a basis B h = {1,..., N h } of V h (see [19]) satisfying:

i (x j ) = ij =1, i = j0, i = j

where x j is the j th node of , 1 j N h . Then, the discrete solution u i h V h


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can be written as follows:

u i h (x) =N h

j =1

u i h (x j ) j (x), x . (14)

R N T h denotes the space of discretized coefficients, i.e., L () shall be dis-

cretized by a N T h vector h = ( 1 , 2 ,...., N T h ) and considered constant = h

j ateach triangle T j of h .We also consider h as a function of L () by its canonical extension:

h (x, y) = j , if (x, y)

T j , j = 1 ,...,N T h ,

dened a.e. on . We denote h () the canonical extension of h to L (). More-over, we denote R N T h () the set of canonical extensions to L () of all possibleN T h vectors h dened on a triangulation h . Hence, R N T h () is a subspace of L () and its elements h () R N T h () shall be called h piecewise functions.

The double discretization of the variational problem (10)-(12) in both, equation

and parameter, generates the discretized problem:

Find u i h V h , such that: (15)

a(u i h , vh ) = h u ih vh dx = F i (vh ) = i vh dS, v V h , (16) u ih dS = i dS, (17)

for 1 i N, with V h dened in (13). System (16) gives a N h N h system of linear equations:

K (h ) (i ) = b(i ) (18)

for each i = 1 , 2, ...N , where:

(i )k = ui h (xk ), K (h )

jk = h jk dx, b(i ) j = i j dS, (19)and 1 j, k N h . This is a consistent and undetermined system. On the otherhand, condition (17) generates the additional equation:

N h

j = N h +1K (h ) j (i ) j = b(i ) , (20)

where

K (h ) j = j dS and b(i ) = i dS.Finally, substracting equation (20) to each one of the equations of the system

(18), we obtain a consistent and unique solvable N h N h system which can besolved by usual methods.


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Remark 1 : The previous step is a consequence of replacing:

u i h (x) =N h

j = N h +1

u i h (x j ) j (x),

for u i h restricted to the boundary for all nodes j = N h + 1 , ....N h into (17).

The discretized version of (10)-(12) in functional form is given by:

min h R

N T h

J h (h ) = 12

N

i=1 |u i h i |2 dS, (21)a(u i h , vh ) = h u i h vh dx = F i (vh ) = i vh dS, vh V h , (22)

u i h dS = i dS. (23)From nite element theory, it is well known (see [24]) that we have convergence

conditions, i.e., if L (), K , and if ui h is the solution of (16)-(17) for xedh = , h > 0, then:

limh 0

u i ui h H 1 () = 0 , (24)

for all 1 i N , where u i V is the solution of (11)-(12) corresponding to andV = H 1()/ R . Moreover, in section 5 we shall prove that:

limh 0

h L () = 0 = limh 0u i u i h H 1 () = 0,

where ui h is the solution of (16)-(17) corresponding to h .In addition, if u i H 2(), then:

u i ui h H 1 () Ch u i H 2 () . (25)

3. Descent Algorithm and Global Convergence

Let be X a normed space and J : X R . Denote X the topological dual spaceof X . We recall that a general descent algorithm searching local minima of the

functional J C 1, is given by the following steps:Descent Algorithm:A1) Choose 0 X and set k = 0 .A2) Stopping test: If J (k ) = 0 , take k as a solution.A3) Descent direction search: Find dk X, such that J (k ), dk X ,X < 0.A4) Step length search: Find k > 0 such that J k + k dk < J (k ).A5) Iteration: Take k+1 = k + k dk , k = k + 1 and go to step A2).

For each 0, this algorithm generates a nite or innite sequence {k }. We say itis globally convergent if for all 0 X , the sequence is nite and stops in A2) or itis innite and any accumulation point of {k } satises J () = .


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When dim X = n < , step A3) is implemented looking for a direction withinthe non-orthogonal directions set:

G (k ) = d : J (k ), d J (k ) d , where (0, 1). (26)

All the descent directions used in common algorithms belong to this set, forexample, in Quasi-Newtons versions (see [20],[22]). In step A4) an inexact linesearch is done, i.e., for xed (0, 1) and (, 1) , a real k > 0 is found suchthat:

w1) J (k + k dk ) J (k ) + k J (k ), dkw2) J (k + k dk ), dk J (k ), dk

(27)

w1) and w2) are called the and Wolfes conditions . According to classicalresults (see [20]), using (26) and (27) the resulting algorithm is globally convergentto the set:

= {x X : J (x) = } .

The search criteria (26) and (27) can be extended to a nite number of directions.Given , , (0, 1), > , p 1, in steps A3) and A4) we look for p directionsDk = ( dk1 ,...,d k p) X p and p step lengths k = ( k1,..., kp ) R

p+ , such that:

J (k ), k Dk J (k )X

k D k X , (28)

and

w1) J (k+1 ) J (k ) + J (yk ), k D k X ,X

w2) J (k+1

), k Dk J (k

), k D k X ,X (29)

where:

k+1 = k + k Dk ,

k Dk = p

j =1

kj dk j .

For p > 1, a multi-directional and multi-step search algorithm is obtained. Notethat this approach allows the case when one or several directions in Dk do not

belongs to G (k

) and/or the intermediate points

k+1r = k +

r

j =1

kj dk j , r < p,

do not satisfy w1) and w2); but the overall step k D k does. Furthermore, thenumber p of nite directions can be different at each iteration k and the algorithmbecomes with variable multi-directional search.

Theorem 3.1 : If J C 1 and it is bounded below. Then, the variable multi-directional and multi-step search (VMdMs) algorithm is globally convergent to the


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set

= {x X : J (x) = } .

Proof : See [10].

The algorithm ( VMdMs) can be applied to the continuous problem (10)-(12)through its discrete approximations (21)-(23) as follows:

At the beginning of k + 1 step, a current point kh R N T h is given at the dis-crete problem and its corresponding canonical extension kh () L

() is implicitlygiven at the continuous problem. Working in the discrete problem, a descent di-rection kh R

N T h is sought and an inexact line search is perfomed to nd a steplength k and a new point k+1h =

kh + k

kh R

N T h satisfying the , Wolfeconditions w1),w2) for J h . At the same time, we verify if the corresponding canon-ical extensions kh (),

k+1h () L

() satisfy Wolfes conditions for J . If it isnot,new iterations are done in the discrete problem (say pk iterations) until J h = or Wolfes conditions are fullled in the continuous one, i.e. k D k () and k+1h ()satisfy (28) and (29). Finally, the step h is decreased (for example to h = h/ 2)and a new discrete problem is dened with a given initial solution k+1h .

Actually, this process is a variable multi-directional and multi-step algorithm forproblem (10)-(11). The number pk of iterations performed in the k th discreteproblem corresponds to the number of directions taken in Dk for a single iterationof the multi-directional algorithm in the continuous problem. The number pk canbe different at each iteration, but it remains bounded. This approach was proposedby Gomez [10] in the context of nonlinear optimal control algorithms.

In section 6 we shall see when the fullling of non orthogonal and Wolfes condi-tions in the discrete problem imply the same properties in the continuous problem.It depends on the relations between , and conditions in both problems andtherefore it depends on J (), J h (h ) and on the continuous J () and discrete J h (h ) gradients of the objective functions.

Obviously, we cannot expect both conditions are satised for the same parameters, and , but some criteria for its appropiate selection shall be given also in section6. These are similar results than those given in [10] and the only difference shallbe the specic form adopted for the gradients at each respective problem, and itsconvergence property as h 0.

4. Computation of Gradients

Dene J i () = 12 u i 2i L 2 () and J () = N i=1 J i (). Then J () =N i=1 J i (). We will work in the quotient space V = H 1()/ R to avoid xing

the constant and we will skip the coset notation.We dene the Lagrangean L i : L () V V R as follows:

L i (, u i , pi ) = 12

u i i 2L 2 () + Ai (, u i ), pi V ,V (30)

where A i (, u i ) V is the linear operator dened by:

Ai (, u i ), v V ,V = u i v dx i v dS, v V.If u i = u i () V is the solution of (11) corresponding to , then Ai (, u i ) = ,


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and:

L i (, u i , pi ) = J i ().

Hence,

J i () = L i (, u i , pi ) + u i L i (, u i , pi ) u i (). (31)

Since ui = u i () V is arbitrary, we choose pi V such that:

u i L i (, u i , pi )u i = 0 ,u i V.

Differentiating (30) with respect to u i :

u i L i (, u i , pi )u i = u i i , u i L 2 () + u i A(, u i )u i , pi = 0 ,u i V,

which implies

(u i i )u i dS + u i pi dx = 0 ,u i V.Then, the i th adjoint equation for pi V is:

pi u i dx = (i u i )u i dS, u i V, (32)which corresponds to a variational formulation of the elliptic problem:

div( pi) = 0 in , p in | = (i u i ) on ,

where pi V, i = 1 , 2,...,N T h , and compatibility condition:

(i u i )ds = 0 i ds = 0 . (33)Problem (32)-(33) has a unique solution in V by the same reasons given above

for the problem (7). From (30) and (31 ) we have:

J i () = L i (, u i , pi ) (34)

= A(, u i ), pi (35)

= u i pi dx, i = 1 , 2, ...N. (36)Hence:

J () =N

i=1 lim u i pi dx, L (). (37)


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We denote by:

J () =N

i=1 u i pi , (38)

the function of L1() representing the integral linear functional (37).

4.1. Discrete Gradient

As in the previous case, we will work on V h = V h / R , where V h is given in (13) andwe will simplify notation writing V h as V h and vh as vh . Dene J i h (h ) : RN T h R

such that:

J i h (h ) = 12

u i h i2L 2 () ,

where:

Ai (h , u i h ), vh V h ,V h = h ()u i h vh dx i vh dS = 0 , vh V h , 1 i N,(39)then:

J h =N

i=1

J i h .

Taking into account that h = ( k )1 k N T h RN T h , integration in (39) over each

mesh triangle gives:

Ai (h , u i h ), vh V h ,V h =N T h

k=1

k T k u i h vh dx i vh dS (40)=

N T h

k=1

k ( u i h vh ) |T k i vh dS, (41)where in the last equality we use that the gradients are constant over each triangle,since functions u i h and vh are supposed to be piecewise linear on h .

We dene the Lagrangean L : R N T h V h V h by:

L(h , u ih , pi h ) = 12 u i h i 2L 2 () + Ai (h , u i h ), pi h V h ,V h . (42)

If u i h = ui h (h ) V h is the unique solution of the equation (39) correspondingto h , then:

L(h , u i h , pi h ) = J ih (h ), pih V h . (43)

Differentiating with respect to h :

J i h (h )h = h L(h , u i h , pi h )h + u i h L(h , u i h , pi h ) u ih h , (44)


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h R N T h . Dene ui h = u i h h V h , then the operator u i h L(h , u ih , pi h ) V h and we choose pi h V h such that:

u i h L(h , u i h , pi h )u i h = 0 , u ih V h . (45)

Computing differential u i h in (42), we obtain:

u i h L(h , u i h , pi h )u i h = u i h i , u ih L 2 () + u i h Ai (h , u i h )u i h , pi h V h ,V h = 0 ,

or equivalently:

(u i h i )u ih dS + h () u i h pi h dx = 0 .Then, pi h should satisfy the equation:

h () pih u i h dx =

(i u i h )u i h dS, u i h V h (46)

for 1 i N and pi h V h . This is the adjoint equation for pi h , which gives theN h N h linear system:

K (h ) (i ) = c(i) ,

for each i = 1 , 2, ...N , where:

(i)k = pi h (xk ),

K (h )

jk = lim h jk dx,

c(i) j = lim (i u i h ) j ds,and 1 j, k N h .

From (44), (45) and (41) we have:

J i h (h )h = h L(h , u i h , pi h )h (47)

=N T h

k=1

k |T k | ( u i h , pi h ) |T k (48)

for all h = ( k )1 k N T h RN T h and 1 i N .

Finally:

J h (h )h =N

i=1

N T h

k=1

k |T k | ( u i h pi h ) |T k (49)

=N T h

k=1

kN

i=1|T k | ( u i h pi h ) |T k (50)


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Therefore, the partial derivatives of J h are given by:

J h (h ) k

=N

i=1|T k | ( u i h pi h ) |T k (51)

for k = 1 , 2....N T h . We use the same notation J h (h )( ) R N T h () for the canonicalextension of the gradient vector J h (h ) = ( J h

k)

1k

N

T hR N T h to L (), which

equals to the constant (51) at each open triangle T k of h .Note that (50) can be written as:

J h (h )h = J h (h )( ), h () V ,V =N

i=1 h () u i h pi h dx (52)for all h R N T h . Therefore, we can consider the gradient J h (h ) as an integrallinear functional dened on the subspace R N T h () L (), which can be extendedto L () with the same integral formula:

J h (h ) [] =N

i=1 u i h pi h dx, L (). (53)We denote by:

J h (h )[.] =N

i=1 u ih pih L1() (54)

the function representing this integral linear functional over L ().In what follows, we shall use the notations J h (h ), J h (h )( ) or J h (h )[]

according to our needs. Resume: J h (h ) is a vector with N T h components, J h (h )( ) is a h piecewise function on , which is the constant

J h ( h ) k over

each open triangle T k of h , J h (h )[] is this same function as J h (h )( ), but now considered as the repre-

sentation in L1() of the integral linear functional (53).

5. Convergence of Gradients

Lemma 5.1: Let be {h } RN

T h such that {h ()} RN

T h () satises:

h (x, y) K > 0, a.e. on ,

and suppose:

h () in L (), as h 0.

Then:

J h (h ) [] J () in L1(), as h 0.


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Remark 1 : From Lemma 5.1, we conclude that if {h k } , k = 1 , 2, ..., is a se-quence of stationary solutions of the discrete problems (21)-(23) corresponding toa triangulation h k , with hk 0, i.e.:

J h k (h k )[] = , k = 1 , 2, ...,

and if is an accumulation point of {h k } then:

J () = ,

and is an stationary solution of the continuous problem.

Moreover, if we now add the convex constraint:

M = { L () : K > 0, a.e. } Q, (55)

for both, the continuous (10)-(12) and discrete (21)-(23) problems, by Lemma5.1 and convexity, we infer from the necessary optimality conditions of discreteproblems that

J h k (h k )[ h k ] 0, M Q,

and taking k + the fulllment of the necessary optimality condition in forthe continuous problem:

J ()( ) 0, M Q.

A more delicate situation appears if we consider a discretization M h of the con-straint set M in (21)-(23). For example:

h () M h = () R N T h () : |T k = nk K, k = 1 ,...,N T h Q, (56)

Seemingly M h M and since M and Q are supposed to be closed, any accumu-lation point of the sequence h k () M h k Q belongs to M Q. But the existenceof optimal solution for the continuous problem with constraint (55) does not im-ply the existence of optimal solution for the discrete problem with constraint (56).This situation can be avoided with a smart selection of the set Q ensuring existenceand uniqueness of the solution for the continuous inverse problems, but Q beingcomposed of piecewise constant functions in L () plus their sup-limits.

Proof : We use the L1() representations (38),(54), and the notations introducedin section 4 about the quotient spaces. Let

J () =N

i=1 u i pi

J h (h )[] =N

i=1 u i h pi h

where:


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1. u i V is the solution of:

u i vdx = i vdS, v V,2. pi V is the solution of:

pi vdx = (i u i )vdS, v V,3. ui h V h is the solution of:

h () u i h vh dx = i vh dS, vh V h4. pi h V h is the solution of:

h () pi h vh dx =

(i u i h )vh dS, vh V h .

We also dene:

5. ui h V h as the solution of:

ui h vh dx = i vh dS, vh V h .From (24) we know that uih u i in V as h 0.Dene wi h := u i h ui h V h . From above denitions 3. and 5. we have:

ui h vh dx = h u i h vih dx, vh V h ,and replacing u i h = wi h + ui h , we obtain that wi h V h is the solution of:

hwi h vh dx = ( h ) ui h vh dx, vh V h ,with the continuous and coercive bilinear form:

a(wi h , vh ) = hwi h vh dx.

Furthermore, F h : V h R dened by:

F h (vh ) = ( h ) ui h vh dx,is linear and satisfy:

|F h (vh )| h L () uih H 1 () vh H 1 () ,vh V h ,


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hence, F h is continuous and:

F h (H 1 ()) h L () ui h H 1 () ,h > 0,

where ui h H 1 () is uniformly bounded since uih converges to u i in H

1(). Then:

F h(H 1

()) 0, as h 0.

From Lax-Milgram Lemma:

wi h H 1 () 1

F h (H 1 ()) 0, as h 0,

where > 0 is the coercivity constant of a, which is independent of h.We have proved that:

u ih uih H 1 () 0, as h 0,

and as a consequence:

u i u i h H 1 () u i uih H 1 () + u i h u

i h H 1 () 0, as h 0.

Analogously, we dene pi h V h satisfying:

pi h vh dx = (i u i )vh dS, vh V h ,and zi h = pi h pi h V h , as the solution of:

h zi h vh dx = ( h ) pi h vh ,vh V h .With the same arguments as before, we obtain:

pi h pi h H 1 () = zi h H 1 () 0, as h 0,

then:

pi pi h H 1 () pi pi h H 1 () + pih p

i h H 1 () 0, as h 0.

Therefore we have:

u i h u i in L2(),

pi h pi in L2(),

and using triangular inequality, we obtain:

u i h pi h u i pi L 1 () u i h pi h u i h pi L 1 () + u i h pi u i pi L 1 () u i h L 2 () pi h pi L 2 () + pi L 2 () u i h u i L 2 ()


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converges to zero as h 0 and then:

u i h pi h u i pi in L1(),

for all i = 1 , 2, ...N. Finally:

J h (h )[] =

N

i=1 u i h pi h J () =

N

i=1 u i pi

in L1() as h 0.

Definition 5.2: Let h 1 be a regular triangulation of , with diameter h1 > 0,and h 1 () R

N T h 1 () be a h 1 piecewise function on . We say that h () R N T h ()is a reduction of h 1 () to h if and only if h is a regular triangulation of , whichis ner than h 1 , with diameter h > 0, h h1 and h () is a h -piecewise functionon , satisfying:

h (x) = h 1 (x), x .

Lemma 5.3: Let h1 > 0 and h 1 () RN T h 1 () be a h 1 piecewise function

dened on and suppose h () R N T h () is a reduction of h 1 () to h , where his any regular triangulation, ner than h 1 . Then:

J h (h )[.] J (h ()) L 1 () 0, as h 0.

Proof : Clearly we can write:

h () h 1 () in L (), as h 0.

Then, by Lemma 2:

J h (h )[] J (h 1 ()) in L1(), as h 0. (57)

In addition:

J (h 1 ()) = J (h ()) , (58)

for all h h1 , since J is continuous and h () and h 1 () are the same function,piecewise dened in different triangulations. The result follows from (57) and (58).

6. Relations between Wolfes Conditions

In what follows, we rewrite in our notations the general results given in [10]. Weconsider the discrete problem (21)-(23) dened in the space R N T h , i.e., the set R N T hprovided with the norm:

vR

N T h

= max1 j N T h

|v j | .


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It is known that the topological dual of R N T h is RN T h1 , i.e., R

N T h provided withthe norm:

vR

N T h1

=N T h

j =1|v j | .

If h () L () is the canonical extension of h RN T h , it is easy to see that:

h R N T h = h () L () .

We use the two representations of the discrete gradient obtained in section 4:- the vectorial form:

J h (h ) =N e

i=1|T k | ( u ih pi h ) |T k

1 k N T h

RN T h , where =

N T h

k=1

T k ,

- and the functional form:

J h (h )[] =N e

i=1

u i h pi h L1().

Then, we have:

J h (h )[] L 1 () = N i=1 u i h pi h dx =N

i=1

N T h

k=1 T k u ih pih dx=

N T h

k=1

N

i=1 |T k | ( u ih pi h ) |T k = J h (h ) R

N T h

1 ,

which implies that:

J h (h )[] L 1 () = J h (h ) R N T h1 . (59)

In the same way, it can be inferred the equality:

J h (h )[.], h () L 1 ,L = J h (h ), h R N T h . (60)

Definition 6.1: For the continuous and discrete funtional J and J h introducedin section 2, we dene the following error functions:

J (1, 2) := J (1) J (2) (61)

J h (1h , 2h ) := J h (1h ) J h (2h ) (62)

h = h (1h , 2h ) := J (1h (), 2h ()) J h (1h , 2h ) (63)

h = h (h ) := J (h ()) J h (h )[.] L 1 () (64)

where 1, 2 L () and 1h , 2h RN T h . Note that h 0 as h 0 by Lemma

5.3.


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Theorem 6.2 : ([10]) Let be h 1 RN T h 1 and , , (0, 1) with < . Choose

(0, 1) satisfying:

< min

2 + ,

(1 )2 + (1 )

,

1 + (2 + ).

Let h h1, be a regular triangulation h and suppose the reduction h () of h 1 ()to h , are such that:

h (h ) = J (h ()) J h (h )( ) L 1 () J (h ()) L 1 () .

Let be h = h + h with h RN T h for which the following inequalities hold:

h (h ) = J (h ()) J h (

h )( ) L 1 () J (h ()) L 1 ()

h (h , h ) = | J (h (), h ()) J h (

h , h )| J (h ()) L 1 () h () L ()

If h RN T h satises , , conditions for the discrete problem, then there exist

numbers 1, 1, 1 with 1 < 1 such that its canonical extension

h () L

()satisfy 1, 1, 1 conditions for the continuous problem. Furthermore, 1, 1, 1can be chosen inside the following intervals:

1 ( 1, (1 + )) , 1 ( + (1 + ), 1), 1 (0, (1 + )) ,where:

1 = max

(1 + )2 + (1 )

, (1 + )1 + (2 + )

,

= 11

11 +

.

Remark 1 : Note that given intervals do not depend on h and only depend onthe parameters , , .

7. Algorithm

1. Choose h inic > 0, inic RN T h inic

2. Choose ,,, , (0, 1) satisfying: > , < min

2 + ,

(1 )2 + (1 )

, (1 + )1 + (2 + )

,

max (1 + )

2 + (1 ),

(1 + )1 + (2 + )

3. Choose 0 (0, ), 1 ( , 0(1 + )) , 1 ( + (1 + ), 1), 1 (0, (1 + )) with = 1 1 11+


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4. Set h0 = hinic , 0() = h 0 () L (), l = 0 , k = 0 .5. If J (l()) L 1 () = 0, stop, function l() is a local minima for the continuous

problem.If J (l()) L 1 () = 0 , go to step 6 .

6. Set k k + 1 , k = k 1

2 , hk = hk 1

27. Dene h k = l at the new triangulation h k .8. Verify if

J (h k ()) J h k (h k )( ) L 1 () k J (h k ()) L 1 ()

If it holds, go to step 9 ,If it does not hold, take hk = hk2 and go to step 7 .

9. Choose h k RN T h k and k > 0 satisfying , and conditions for the

discrete problem.10. Dene h k = h k + k h k and verify:

J (h k ()) J h k (h k )[.] L 1 () k J (h k ()) L 1 ()

J (

h k (), h k ()) J h k (

h k , h k ) k J (h k ()) L 1 () h k L ()

If inequalities hold or if J h k (h k ) = 0 , take l+1 () = h k (), l = l + 1 ,

and go to step 5 .If they do not and J h k (h k ) = 0, take h k =

h k and go to step 9 .

Remark 1 : Quasi-Newton methods with inexact line search or trust region rou-tines can be used at step 9.

Remark 2 : Estimations for the continuous gradients J (h ()) , J (h ()) andfor the continuous increment J (h (), h ()) can be obtained using smaller stepsin the discrete problems.

8. Conclusions

We have obtained a globally convergent descent algorithm for the innite dimen-sional optimization problem (10)-(12), which interacts step by step with approxima-tions of the discrete problems (21)-(23) using Wolfes conditions. In our approach,we have assumed no error on data. However, we know that with measurement errorsthe inverse problem is ill conditioned and it requires regularization techniques.

In particular, considering Tikhonovs regularization, the problem is dened as:

minL ()

J () = 12

N

i=1 u i i2 dS + B ( ) 2L 2 ()

s.a. a (, u i ) = u i vdx = F i (v) = i vdS, v V,where B is a linear bounded operator, for example, the identity or the gradient, is a function representing apriori information on , > 0 is the regularizationparameter and i , i represent data with error such that i i and

i i , where i , i are the exact data. The above convergence resultscan be applied to the regularized problem for a xed regularization parameter


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20 REFERENCES

(the same for both problems). For example, if B is the identity, the gradientsassume the form:

J () =N

i=1 u i pi + ( ), J h (h ) =

N

i=1 u ih pi h + (h ),

with

( ) (h ) L 1 () = | h | dx || h L () 0,as h 0. It follows from section 5 that:

J () J h (h ) L 1 () 0,

when h as h 0.A more interesting question is if the variable selection of = h , regularizing the

discrete problem at each triangulation h , can be done to regularize the continuous

problem in some limit sense.

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