Priority Program 1253

Transcription

1 Deutsce Forscungsgemeinscaft Priority Program 1253 Optimization wit Partial Differential Equations K. Deckelnick and M. Hinze Convergence and error analysis of a numerical metod for te identification of matrix parameters in elliptic PDEs April 2012 Preprint-Number SPP ttp://

2

3 Convergence and error analysis of a numerical metod for te identification of matrix parameters in elliptic PDEs Klaus Deckelnick & Micael Hinze Abstract: We analyze a numerical metod for solving te inverse problem of identifying te diffusion matrix in an elliptic PDE from distributed noisy measurements. We use a regularized least squares approac in wic te state equations are given by a finite element discretization of te elliptic PDE. Te unknown matrix parameters act as control variables and are andled wit te elp of variational discretization as introduced in [8]. For a suitable coupling of Tikonov regularization parameter, finite element grid size and noise level we are able to prove L 2 convergence of te discrete solutions to te unique norm minimal solution of te identification problem; corresponding convergence rates can be obtained provided tat a suitable projected source condition is fulfilled. Finally, we present a numerical experiment wic supports our teoretical findings. Matematics Subject Classification (2000): 49J20, 49K20, 35B37 Keywords: Parameter identification, elliptic optimal control problem, control constraints, H convergence, variational discretization, source condition. 1 Introduction In tis paper we are concerned wit a convergence analysis for a numerical metod tat identifies a diffusion matrix in te elliptic boundary value problem div (A y) = f in, y = 0 on (1.1) from distributed noisy measurements. Here, R n is a bounded, polyedral domain and f H 1 (). In addition we assume tat te diffusion matrix A(x) =(a ij (x)) n i,j=1 satisfies a ij L () and is uniformly elliptic. Te boundary value problem (1.1) ten as a unique weak solution y H0 1 () wic we denote by y = T (A, f). Our aim is to identify te unknown diffusion matrix from distributed noisy measurements (z (i) satisfying,f(i) ) L2 () H 1 (), 1 i N z (i) z (i), f (i) f (i) H 1. (1.2) Institut für Analysis und Numerik, Otto von Guericke Universität Magdeburg, Universitätsplatz 2, Magdeburg, Germany Scwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, Hamburg, Germany. 1

4 Here, z (i) = T (A,f (i) ), 1 i N (1.3) for some symmetric diffusion matrix A wit a ξ 2 A (x)ξ ξ b ξ 2 for all ξ R n, a.e. in, (1.4) were 0 <a<b<. We sall employ a least squares approac in order to reconstruct te diffusion matrix, more precisely we consider te following optimization problem: (P ) min A M J(A) :=1 2 z (i) 2 + γ 2 A 2 subject to y (i) = T (A, f (i) ), 1 i N, were γ>0and we use te symbol for te L 2 -norm of scalar, vector- or matrix-valued functions. Te set M of admissible diffusion matrices will be defined in Section 2 below. One of te main difficulties in analyzing (P) is tat te mapping A y = T (A, f) is not weakly (sequentially) closed in L 2 (, R n,n ). Tis can be seen wit te elp of te following example taken from [16, Section 3]: let = (0, 1) R and a k L () be defined by m a, k a k (x) := x< m+ 1 2 k m =0,...,k 1. m+ b, 1 2 k x< m+1 k If we let y k = T (a k,f) it can be sown tat y k y = T (â, f) in L 2 (), were â ( 1 2 ( 1 a + 1 b )) 1. On te oter and, ak ain L 2 () wit a 1 2 (a+b), so tat y T (a, f), since â a. One possibility to overcome tis problem is to use a stronger norm in te Tikonov regularization (suc as H 1, see e.g. [15], [20]), but tis approac can be numerically cumbersome especially wen one takes ellipticity constraints into account. In [4] we were able to prove te existence of a solution to (P) by applying te concept of H convergence ([16]). Furtermore, we considered an approximation of (P) by discretizing (1.1) wit te elp of finite elements and establised te convergence of corresponding minimizers to a minimum of (P) in te case tat γ>0isfixedandz (i) = z (i), 1 i N. Te goal of te present work is to extend te convergence analysis to te case tat te regularization parameter γ tends to zero, wile we also take into account noisy measurements satisfying (1.2). Denoting by (P ) our approximation of (P) wit corresponding minimum A we sall prove in Section 3 tat A Ā in L 2 as te mes size and te noise level tend to zero provided tat γ is coupled to tese parameters in a suitable way. Here, Ā Mdenotes te norm-minimal (in te L 2 sense) diffusion matrix satisfying (1.3). Under a suitable projected source condition and appropriate smootness conditions on te data we ten sow in Section 4 tat an error bound of te form A Ā C olds. Section 5 presents a numerical test calculation wic supports te rates obtained. Let us briefly refer to related publications tat ave been concerned wit te identification of matrix valued parameters. In [1] and [9] a reconstructed matrix is obtained as te large time limit of a suitable dynamical system. A stability result, wic can also be used for te 2

5 convergence analysis of numerical metods, is derived in [11] by Hsaio and Sprekels for te reconstruction of matrices of te form A = p p. In [13], Kon and Lowe introduce a variational metod involving a functional wic is convex in te matrix A and te conductivities A y (i). Rannacer and Vexler prove in [17] error estimates for a matrix identification problem in wic a finite number of unknown parameters is estimated from finitely many pointwise observations. Te problem of identifying a scalar diffusion coefficient as been investigated muc more intensively compared to te matrix case. Identifiability results ave been obtained e.g. in [3], [18] and [19]. A survey of numerical metods for parameter estimation problems can be found in [14]. Error estimates for a least squares approac ave been obtained by Falk in [6] and more recently by Wang and Zou [20] taking into account Tikonov regularization. Te latter paper also contains a long list of furter references. 2 Notation and preliminary results Let us denote by S n te set of all symmetric n n matrices equipped wit te inner product A B =trace(ab). We introduce te subset K := {A S n a ξ 2 Aξ ξ b ξ 2 for all ξ R n }. Since K is a convex and closed subset of S n we may define te ortogonal projection P K : S n K, wic is caracterised by te relation (A P K (A)) (B P K (A)) 0 for all B K. (2.1) We define te admissible set for te optimization problem (P) by M := {A L (, R n,n ) A(x) K a.e. in } L (, S n ). According to [4, Teorem 2.2], (P) as a solution A M. For te convenience of te reader we include te necessary first order conditions satisfied by A. To begin, consider te mappings F i : M L 2 (), F i (A) :=T (A, f (i) ), 1 i N. It is not difficult to verify tat F i (A)H = w (i), H L (, S n ), were w (i) H0 1 () is te unique weak solution of te elliptic equation A w (i) vdx = H y (i) vdx for all v H0 1 () (2.2) and y (i) = T (A, f (i) ). Denoting by p(i) H0 1 (),i =1,...,N te solutions of te adjoint problems A p (i) vdx = (y (i) z (i) )vdx for all v H1 0 (), 3

6 we ave for H L (, S n ) J (A)H = = (y (i) z (i) )w(i) dx + γ H y (i) p (i) dx + γ A Hdx = A Hdx = (2.3) A p (i) w (i) dx + γ A Hdx ( N y (i) p (i) + γa ) Hdx, were (p q) ij = 1 2 (p iq j + p j q i ),i,j =1,...,n for p, q R n.wenowave Lemma 2.1. Let γ>0 and A Mbeasolutionof(P).Ten ( ) 1 A(x) =P K y (i) (x) p (i) (x) a.e. in. γ Proof. Te optimality of A yields J (A)(B A) 0 for all B M, wic can be rewritten wit te elp of (2.3) as ( 1 y (i) p (i) A ) (B A ) dx 0, for all B M. γ A localization argument ten sows tat ( ) 1 y (i) (x) p (i) (x) A(x) (C A(x) ) 0 for all C K and a.a. x γ wic implies te result. Next, let T be a regular triangulation of wit mes size := max T T diam(t ). Let us denote by X := {v C 0 ( ) v is a linear polynomial on eac T T,v =0} H 1 0() te space of continuous, piecewise linear finite elements. For a given A Mand f H 1 (), te problem A y v dx = f,v for all v X as a unique solution y X wicwedenotebyy = T (A, f). Here,, is te duality between H 1 () and H0 1 (). Standard arguments yield te following bounds T (A, f) H 1 C f H 1, (2.4) T (A, f) T (A, f) H 1 C inf T (A, f) v H 1 (2.5) v X wit a constant C tat is independent of f H 1 (), A Mand te mes size. We are now in position to formulate our numerical metod wic is based on solving te following semi-discrete minimization problem: (P ) min A M J (A) :=1 2 z(i) 2 + γ 2 A 2 subject to y (i) = T (A, f (i) ), 1 i N. 4 (2.6)

7 Note tat following te idea of [8] te matrix parameters are not discretized, compare owever Remark 2.3 below. Lemma 2.2. Problem (P ) as a solution A M. Every solution A Mof (P ) satisfies ( ) 1 A(x) =P K y (i) γ (x) p(i) (x) a.e. in, (2.7) were y (i) = T (A, f (i) ) and p (i) A p (i) v dx = X are te solutions of te adjoint problems z(i) )v dx for all v X, 1 i N. (y (i) Proof. Let (A k ) k N Mbeaminimizing sequence for problem (P )sotatj (A k) inf A M J (A) as k. Since (A k) k N is bounded in L (, R n,n ) tere exists A L (, R n,n ) suc tat A k A in L (, R n,n ) for some subsequence. In addition one readily verifies tat A M. Te sequences y(i) k = T (A k,f (i) ), 1 i N are uniformly bounded in te finite-dimensional space X so tat we may assume tat y (i) k y (i) in H 1 (), 1 i N. Clearly y (i) = T (A,f(i) ) and terefore J (A ) = 1 2 z(i) 2 + γ 1 2 A 2 lim k 2 k lim inf J k (A k )= inf J (A). A M Te relation (2.7) is obtained exactly as in te continuous case. z (i) 2 + γ lim inf 2 A k k 2 Remark 2.3. Let us note tat in view of (2.7) A is piecewise constant on T so tat a discretization of te set M is not required. Variational discretization automatically yields solutions to (2.6) wic allow a finite-dimensional representation. In order to analyze te convergence of te above metod we sall make use of te concept of Hd-convergence. Tis concept was introduced in [5] in te context of finite volume discretizations of elliptic boundary value problems wit te aim of adapting H convergence results to te discrete setting. Te following teorem is a finite element version of te result obtained in [5]. Teorem 2.4. Let (A k ) k N be a sequence in M and (T k ) k N a sequence of triangulations wit lim k k =0. Ten tere exists a subsequence (A k ) k N and A Msuc tat for every f H 1 () T k (A k,f) T(A, f) in H 1 0 () and A k T k (A k,f) A T (A, f) in L 2 () n. We ten say tat te sequence (A k ) k N Hd converges to A and write A k Hd A. Proof. See [4, Teorem 3.1]. Please note tat te Hd limit in general will depend on te sequence (T k ) k N. 5

8 Corollary 2.5. Let (A k ) k N be a sequence in M and (T k ) k N asequenceoftriangulations Hd wit lim k k =0. Suppose tat A k A 0 and A k A1 in L (, R n,n ).Ten A 0 A 1 a.e. in and A 0 2 A 1 2 lim inf A k 2. k Proof. See [4, Corollary 3.1]. 3 Convergence To begin, note tat (1.3) and (1.4) imply tat te set M := {B M z (i) = T (B,f (i) ), 1 i N} (3.1) is not empty. Since M is a closed, convex subset of L 2 (, R n,n ), tere exists a uniquely determined Ā M suc tat Ā =min B L2 L2. (3.2) B M Te following lemma gives a sufficient criterion for a matrix function Ā to be norm-minimal. Lemma 3.1. Let Ā Mand z(i) H0 1() wit z(i) = T (Ā, f (i) ), 1 i N. Suppose tat tere exist ψ (i) H0 1 (), 1 i N suc tat ( N ) Ā(x) =P K z (i) (x) ψ (i) (x) a.e. in. Ten Ā L 2 =min B M B L 2. Proof. Let B M be arbitrary. Ten B 2 Ā 2 = B Ā 2 +2 In view of our assumption on Ā and (2.1) we ave so tat Ā (B Ā)dx. ( N z (i) (x) ψ (i) (x) Ā(x)) (B(x) Ā(x)) 0, a.e. in, B 2 Ā 2 2 = 2 wic finises te proof. ( z (i) ψ (i)) (B Ā) dx B z (i) ψ (i) dx 2 Ā z (i) ψ (i) dx =0, In order to formulate our first main result we introduce ρ (i) Note tat ρ 0as 0. := inf v X z (i) v H 1, 1 i N and ρ := max 1iN ρ(i). 6

9 Teorem 3.2. Let A Mbe a solution of (P ) and suppose tat γ 0, γ 0, ρ γ 0, as 0, 0. (3.3) Ten A Ā in L2 (, R n,n ) as 0, 0, wereā Mis as in (3.2). Proof. Given sequences ( k ) k N, ( k ) k N wit lim k k =0, lim k k =0,cooseγ k > 0 suc tat lim k γ k =0and k γk 0, ρ k γk 0ask. (3.4) Furtermore, let A k = A k k Mbe a solution of (P k k ). We deduce from Teorem 2.4 tat tere exists a subsequence (A k ) k N and A 0 M, A 1 L (, R n,n ) suc tat Corollary 2.5 implies tat A k Hd A 0, A k A 1 in L (, R n,n ). A 0 A 1 a.e. in, A 0 2 lim inf k A k 2. (3.5) For better readability we write again (A k ) k N instead of (A k ) k N. We first claim tat A 0 M. SinceA k is a minimum of (P k k )weinfertatj k k (A k ) J k (Ā), k so tat 1 2 k z (i) k 2 + γ k 2 A k 2 1 z (i) 2 k z (i) k 2 + γ k 2 Ā 2. Here, y (i) k = T k (A k,f (i) k ), z (i) k te elp of (1.2) k z (i) 2 2 (i) = T k (Ā, f k ), 1 i N. As a consequence we obtain wit 2 k z (i) k 2 +2 z (i) k z (i) 2 z (i) k z (i) k 2 +2γ k Ā 2 +2N 2 k. (3.6) In order to estimate te first term on te rigt and side we use (1.3), (1.2), (2.4), (2.5) and te definition of ρ and obtain for i =1,...,N z (i) k z (i) k L 2 z (i) k z (i) H 1 + z (i) z (i) k L 2 (3.7) (i) T k (Ā, f k f (i) ) H 1 + T (Ā, f (i) k ) T (Ā, f (i) ) H 1 + k C f (i) k f (i) H 1 + Cρ k + k C( k + ρ k ). Inserting (3.7) into (3.6) we infer tat y (i) k oter and we ave tat z (i) in L 2 () as k, 1 i N. Onte y (i) k = T k (A k,f (i) )+T k (f (i) k f (i) ) T(A 0,f (i) ) in H 1 0 () 7

10 recalling (2.4), (1.2) and te fact tat A k Hd A 0. As a result we infer tat z (i) = T (A 0,f (i) ), 1 i N and terefore A 0 M. Let us sow next tat A 0 = Ā. To see tis, let B M be arbitrary. Since J k k (A k ) J k k (B) we ave 1 2 k z (i) k 2 + γ k 2 A k 2 1 z (i) 2 k z (i) k 2 + γ k 2 B 2, (3.8) were again y (i) k = T k (A k,f (i) k ), wile z (i) k = T k (B,f (i) k ), 1 i N. Similarly as in (3.7) we obtain z (i) k z (i) k C( k + ρ k ) so tat (3.8) yields A k 2 C ( k 2 + ρ2 k ) + B 2. (3.9) γ k γ k Sending k we infer from (3.5) and (3.4) tat A 0 B for every B M, sotatwe deduce tat A 0 = Ā. Finally, A k Ā 2 = A k A 0 2 = A k 2 + A 0 2 2(A k,a 0 ) L 2 from wic we infer wit te elp of (3.9) (wit B = A 0 )andtefacttata k A1 in L (, R n,n ) lim sup A k Ā 2 2 A 0 2 2(A 1,A 0 ) L 2 0, k since A 0 A 1 a.e. in. In conclusion, A k Ā, k in L2 (, R n,n ). Since te limit is unique, te wole sequence converges to Ā and te teorem is proved. 4 Error bound In order to obtain an error estimate we require stronger conditions on te data of our problem. In wat follows we sall assume tat is a bounded, polygonal subset of R 2 and tat z (i) H 1 0() W 2,p (),f (i) L p (), 1 i N for some p>2. (4.1) Furtermore, we suppose tat tere exist ψ (i) H0 1() W 2,p (), 1 i N suc tat z (i) = T (Ā, f (i) ), 1 i N, were te matrix function Ā satisfies ( N ) Ā(x) =P K z (i) (x) ψ (i) (x) a.e. in. (4.2) In addition, we assume tat tere exists μ>2suctat T (Ā, f) W 2,q C f L q, f Lq (), 1 <q<μ. (4.3) Note tat Ā Mand satisfies Ā =min B in view of Lemma 3.1. Let us write Ā(x) = B M ( ) P K E(x),wereE(x) = N z(i) (x) ψ (i) (x). Since z (i), ψ (i) W 1,p (, R n )and 8

11 te embedding W 1,p () C 0 ( ) is continuous we ave tat E W 1,p (, S n ), wic we mayextendtoafunctione W 1,p (R n, S n ). Furtermore, te projection P K is Lipscitz continuous wit Lipscitz constant 1, so tat Rn ā kl (x + ) ā kl (x) p ( ( ) ( )) PK E(x + ) PK E(x) ek e l p p dx = p dx wic implies tat R n Rn E(x + ) E(x) p p dx C, ā kl W 1,p (), 1 k, l 2. (4.4) Let us next interpret (4.2) as a projected source condition (see e.g. [7], wic also contains furter references). To do so, we assume for a moment tat f (i) = f (i) and write te objective functional in (P) in te form J(A) = 1 2 F (A) Z 2 + γ 2 A 2, were F : M L 2 () N is given by F i (A) =T (A, f (i) )andz i = z (i), 1 i N. We claim: Lemma 4.1. Tere exists Θ L 2 () N suc tat Ā = P M ( F (Ā) Θ), were P M denotes te ortogonal projection onto M in L 2 (, S n ). Proof. Recalling (2.2) we see tat were w (i) H0 1 () are te solutions of Ā w (i) vdx = F (Ā)H =( w(i) ) 1iN, H L (, S n ), H z (i) vdx for all v H 1 0 (). (4.5) In view of (4.4) te functions θ (i) := (Ā ψ (i)) belong to L 2 (). Hence Θ = (θ (i) ) 1iN L 2 () N and we ave for H L (, S n ) F (Ā) Θ,H = ( Θ,F (Ā)H) L 2 = = Ā w (i) ψ (i) dx = = ( N z (i) ψ (i),h ). L 2 Here we ave used (4.5). As a consequence we may identify (Ā ψ (i)) w (i) dx H z (i) ψ (i) dx F (Ā) Θ= z (i) ψ (i) L 2 (, S n ) 9

12 and terefore we obtain for every B M ( F (Ā) Θ Ā, B Ā) L 2 ( N = z (i) ψ (i) ( N P K z (i) ψ (i))) ( (B N P K z (i) ψ (i))) dx 0, by (2.1), wic implies te result. Next, let us suppose tat te sequence of triangulations (T ) >0 is quasiuniform. We introduce teritzprojectionr : H0 1() X associated wit Ā by Ā R z v dx = Ā z v dx for all v X. In view of (4.3) and (4.4) we may apply [2, Teorem ] and [2, Teorem 8.5.3] togeter wit inequality (8.5.5) and deduce tat tere exists 0 > 0 so tat for all 0 < 0 R z W 1, C z W 1,, z W 1, () (4.6) z R z + (z R z) C 2 z H 2, z H 2 () H0 1 (). (4.7) In particular we obtain for all f L 2 () T (Ā, f) T (Ā, f) C2 f. (4.8) Teorem 4.2. Let te conditions (4.1) (4.3) be satisfied and A Mbe a solution of (P ). If < 0 and γ = ρ 2 for some suitable ρ>0, ten were y (i) A Ā C, z(i) + (y (i) z(i) ) C 2, 1 i N, = T (A,f(i) ), 1 i N. Proof. Clearly, γ 2 A Ā 2 = γ 2 A 2 γ(a, Ā) L 2 + γ 2 Ā 2 (4.9) = J (A ) 1 2 z(i) 2 + γ(ā A, Ā) L 2 γ 2 Ā 2. Since A is a solution of (P )weinfer J (A ) J (Ā) =γ 2 Ā z (i) z(i) 2, were z (i) (i) = T(Ā, f ). Furtermore, recalling (4.2) and (2.1) we ave N (Ā A, Ā) L 2 (Ā A ) z(i) ψ (i) dx = 10 (Ā A ) z(i) ψ (i) dx.

13 Inserting te above estimates into (4.9) we obtain γ 2 A Ā z (i) z(i) z(i) 2 + γ In order to estimate te first term we employ (2.4), (4.8) and (1.2) z (i) z(i) (i) T(Ā, f If we use tis bound in (4.10) we deduce (Ā A ) z(i) ψ (i) dx. f (i) ) + T (Ā, f (i) ) T (Ā, f (i) ) + z (i) z (i) C f (i) f (i) H 1 + C 2 f (i) + C + C 2. γ 2 A Ā z(i) 2 C 2 + C 4 + γ (4.10) S (i), (4.11) were S (i) = (Ā A ) z(i) ψ (i) dx = (Ā A ) z(i) (ψ (i) R ψ (i) )dx + (Ā A ) (z(i) y (i) ) R ψ (i) dx + (Ā A ) y(i) R ψ (i) dx S (i) 1 + S (i) 2 + S (i) 3. Using te embedding W 2,p () C 1 ( ) togeter wit (4.7) we ave Next, (4.6) implies S (i) 1 A Ā z(i) L (ψ (i) R ψ (i) ) (4.12) C ψ (i) H 2 z (i) W 2,p A Ā C A Ā. S (i) 2 A Ā (z(i) y (i) ) R ψ (i) L C ψ (i) W 1, A Ā (z(i) y (i) ) (4.13) C A Ā (z(i) y (i) ). Abbreviating z (i) = T (Ā, f (i) ) we deduce from (2.5), an inverse estimate, (1.2) and (4.8) tat (z (i) y (i) ) (z(i) z (i) ) + ( z(i) C z (i) H 2 + C 1 z (i) C + C 1( z (i) C + C 1 + C 1 z (i) Inserting tis bound into (4.13) we obtain y(i) ) y(i) ) z(i) + z (i) z (i) + z(i) y (i) y (i). ( S (i) 2 C A Ā z (i) y (i) ). (4.14) 11

14 Finally, te definition of R togeter wit te fact tat y (i) = T (A,f(i) ),z(i) = T (Ā, f (i) ) imply S (i) 3 = = = = = Ā y (i) Ā y (i) R ψ (i) dx A y(i) R ψ (i) dx ψ(i) dx f (i),r ψ (i) (Ā ψ (i)) y (i) dx (Ā ψ (i)) y (i) dx Ā z (i) R ψ (i) dx + f (i) f (i),r ψ (i) Ā z (i) (R ψ (i) ψ (i) )dx Ā z (i) ψ (i) dx + f (i) f (i),r ψ (i) (Ā ψ (i)) (z (i) y (i) )dx + Ā (z (i) R z (i) ) (ψ (i) R ψ (i) )dx + f (i) f (i),r ψ (i). Hence, we may estimate wit te elp of (4.7) and (1.2) S (i) 3 (Ā ψ (i)) z (i) y (i) + f (i) f (i) H 1 R ψ (i) H 1 + Ā L (z(i) R z (i) ) (ψ (i) R ψ (i) ) C ( z (i) z (i) + z(i) y (i) ) + C ψ (i) H 1 + C 2 z (i) H 2 ψ (i) H 2 C z (i) y (i) + C + C2. (4.15) Inserting (4.12), (4.14) and (4.15) into (4.11) we deduce tat γ 2 A Ā z(i) 2 (4.16) C 2 + C 4 + Cγ A Ā ( N +Cγ z (i) y (i) + Cγ + Cγ2 γ 4 A Ā 2 +(Cγ If we coose γ = ρ 2 wit ρ = 1 8C ) N +C ( γ 2 + γ γ + γ 2). ( A 2 Ā 2 C we finally obtain z(i) 2 γ + 4 γ γ since 2. Returning to (4.16) we infer N y(i) z(i) z (i) ) C 2 y (i) ) z(i) 2 C 4,sotat z(i) + z(i) z (i) C 2 + C 2, 1 i N (4.17) 12

15 in view of te above relations betweeen γ, and. Finally, (4.7), an inverse estimate and (4.17) yield wic finises te proof. (y (i) z(i) ) (y (i) R z (i) ) + (R z (i) z (i) ) C 1 R z (i) + C z (i) H 2 C 1( z(i) + z (i) R z (i) ) + C C, 5 Numerical example Let := ( 1, 1) 2 R 2 and consider for N =1tedata(z,f) H 1 0 () H 1 () wit z(x) := (1 x 2 1)(1 x 2 2), f(x) := (Ā(x) z(x) ) for x =(x 1,x 2 ). Te diffusion matrix is given by Ā(x) :=P K ( z(x) z(x) ), were K = {A S 2 a ξ 2 Aξ ξ b ξ 2 for all ξ R 2 }.Byconstructionweavez = T (Ā, f), wile Lemma 3.1 implies tat Ā L 2 =min B M B L 2. It is not difficult to verify tat for a given matrix A S 2 we ave P K (A) =S t diag ( P [a,b] (λ 1 (A)),P [a,b] (λ 2 (A)) ) S, were λ 1 (A),λ 2 (A) denote te eigenvalues of A, S is an ortogonal matrix suc tat S t AS = diag ( λ 1 (A),λ 2 (A) ) and P [a,b] (κ) :=max(a, min(κ, b)). A calculation sows tat Ā(x) =ai 2 + P [a,b](η(x)) a η(x) were η(x) =4 ( x 2 1 (1 x2 2 )2 + x 2 2 (1 x2 1 )2). z(x) z(x), For te numerical verification of te estimates in Teorem 4.2 we coose a sequence of quasiuniform triangulations {T l } generated wit te POIMESH and REFINEMESH environment of MATLAB, were l =2 l,l N denotes te gridsize of T l. We consider te minimization problem (P )for = l, γ = γ l = l and take z = z l as te Lagrange interpolant of z, andf = f l as piecewise constant approximation to f on defined troug te function values in te barycenters of te triangles in T l. We note, tat since f is discontinuous (1.2) in general can not be satisfied wit 2 for tis coice of f. However, tis fact seems not to ave significant effects on te convergence istory of our numerical solutions presented in Tab. 1. Te constants a and b in te definition of K are cosen as a =0.5 andb = 10. Te 13

16 discrete problems (P ) are solved using te projected steepest descent metod wit Armijo step size rule (see e.g. [12]), wic we briefly describe for te convenience of te reader. In view of Remark 2.3 it is sufficient to iterate witin te class of matrices in M tat are piecewise constant on T.GivensucanA te new iterate is computed according to A + = A(τ) witτ =max r N {βr ; J l (A(β r )) J l (A) σ β r A(βr ) A 2 } were we ave abbreviated J l = J l l.furtermore,β (0, 1) and A(τ) T := P K (A T + τ ( y T p T γa T ) ), T T. Here, y = T (A, f l )andp X is te solution of te adjoint problem A v p dx = (y z l )v dx for all v X. In our calculations we cose as initial matrix A 0 := diag(1.01, 1.01)P L2 (A), were P L2 denotes te L 2 -projection onto te space of piecewise constant functions over te grid T. Te iteration on refinement level l was stopped if A + A(1) τ a + τ r A 0 A 0 (1) or te maximum number of l 100 iterations was reaced, were τ a =10 3 l and τ r = 10 2 l.furtermore,wecooseβ := 0.5 andσ := 10 4.For = l te numerical solution is denoted by Ā wit associated optimal state ȳ. Te numerical results are summarized in Tab. 1, were we display te refinement level l, te number of iterations, te value of γ l, te final L 2 -errors in te parameter, te final L 2 -errors in te states, bot togeter wit teir experimental order of convergence (EOC), and te convergence istory of te steepest descent algoritm. As predicted by Teorem 4.2, we observe quadratic convergence for te states. Te matrix parameters in te present example seem to converge faster tan predicted by te numerical analysis. Fig.1fromlefttorigtsowsȳ, te Lagrange interpolant of I z and ȳ I z for refinement level 5. As in te numerical experiments of [4] one observes tat te difference between ȳ and I z is comparatively large in regions were z (and tus I z) is small wic is in agreement wit classical results on te identifiability of scalar diffusion coefficients, see e.g. [18]. Acknowledgements Te presentation of te numerical results is partly based on a MATLAB code developed by Ronny Hoffmann in is diploma tesis [10] written under te supervision of te second autor. Te autors gratefully acknowledge te support of te DFG Priority Program 1253 entitled Optimization Wit Partial Differential Equations. 14

17 l it γ l Ā Ā EOC ȳ z EOC Ā A (1) τ a + τ r r e e e e e e e e e e e e e e e e e e e e e e e e-5 1/5: /5: 2.10 mean 1.71 mean 2.10 Table 1: Mes parameters, errors, experimental order of convergence, and convergence istory of te solution algoritm. Te table is supplemented wit te EOC between finest and coarsest level (1/5) and wit te mean value of te EOC over te refinement levels. Figure 1: Numerical solution, desired state, and error ȳ I z on refinement level 5. References [1] Alt, H.W., Hoffmann, K.H., Sprekels, J.: A numerical procedure to solve certain identification problems, Intern. Ser. Numer. Mat. 68, (1984). [2] Brenner, S. C., Scott, L. R.: Te Matematical Teory of Finite Element Metods. Tird edition. Texts in Applied Matematics, 15. Springer, New York, [3] Cicone, C., Gerlac, J.: A note on te identifiability of distributed parameters in elliptic equations, SIAM J. Mat. Anal. 18, (1987). [4] Deckelnick, K., Hinze, M.: Identification of matrix parameters in elliptic PDEs,toappear in Control & Cybernetics 40 (2011). [5] Eymard, R., Gallouët, T.: H convergence and numerical scemes for elliptic problems, Siam J. Numer. Anal. 41, (2003). [6] Falk, R.S.: Error estimates for te numerical identification of a variable coefficient, Mat. Comput. 40, (1983). 15

18 [7] Flemming, J., Hofmann, B.: Convergence rates in constrained Tikonov regularization: equivalence of projected source conditions and variational inequalities, Inverse Problems 27 (2011), doi:1088/ /27/8/ [8] Hinze, M.: A variational discretization concept in control constrained optimization: te linear-quadratic case, Comput. Optim. Appl. 30, (2005). [9] Hoffmann, K.H., Sprekels, J.: On te identification of coefficients of elliptic problems by asymptotic regularization. Numer. Funct. Anal. Optim. 7, (1984/85). [10] Hoffmann, R.: Entwicklung numeriscer Metoden zur Scätzung matrixwertiger verteilter Parameter bei elliptiscen Differentialgleicungen, Diploma tesis, TU Dresden, [11] Hsiao, G.C., Sprekels, J.: A stability result for distributed parameter identification in bilinear systems, Mat. Met. Appl. Sciences 10, (1988). [12] Kelley, C.T.: Iterative Metods for Optimization. SIAM, [13] Kon, R.V., Lowe, B.D.: A variational metod for parameter identification, RAIRO Modél. Mat. Anal. Numér. 22, (1988). [14] Kunisc, K.: Numerical metods for parameter estimation problems, Inverse problems in diffusion processes (Lake St. Wolfgang, 1994), , SIAM, Piladelpia, PA, [15] Kunisc, K., Sacs, E.W.: Reduced SQP metods for parameter identification problems, SIAM J. Numer. Anal. 29, (1992). [16] Murat, F., Tartar, L.: H-convergence, Topics in te matematical modelling of composite materials, page 21 43, Andrej Carkaev, Robert Kon Eds., [17] Rannacer, R., Vexler, B.: A priori estimates for te finite element discretization of elliptic parameter identification problems wit pointwise measurements, SIAM J. Cont. Optim. 44, (2005). [18] Ricter, G.R.: An inverse problem for te steady state diffusion equation, SIAM J. Appl. Mat. 41, (1981). [19] Vainikko, G., Kunisc, K.: Identifiability of te transmissivity coefficient in an elliptic boundary value problem, Z. Anal. Anwendungen 12, (1993). [20] Wang, L., Zou, J.: Error estimates of finite element metods for parameter identification problems in elliptic and parabolic systems, Discrete Contin. Dyn. Syst. Ser. B 14, (2010). 16