Hamburger Beiträge zur Angewandten Mathematik

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1 Hamburger Beiträge zur Angewandten Matematik Identification of matrix parameters in elliptic PDEs Klaus Deckelnick and Micael Hinze Nr. 2-5 February 2

3 Identification of matrix parameters in elliptic PDEs Klaus Deckelnick & Micael Hinze Abstract: In te present work we treat te inverse problem of identifying te matrix-valued diffusion coefficient of an elliptic PDE from measurements wit te elp of tecniques from PDE constrained optimization. We prove existence of solutions using te concept of H convergence and employ variational discretization for te discrete approximation of solutions. Using a discrete version of H convergence we are able to establis te strong convergence of te discrete solutions. Finally we present some numerical results. Matematics Subject Classification (2): 49J2, 49K2, 35B37 Keywords: Parameter identification, elliptic optimal control problem, control constraints, H convergence, variational discretization. Introduction In tis work we consider te inverse problem of identifying te diffusion matrix A = A(x) in an elliptic PDE div (A(x) y) = g in, y = on (.) from measurements of data. Here, R n is a bounded open set wit a Lipscitz boundary. Furtermore, we assume tat A(x) =(a ij (x)) n i,j= satisfies a ij L () and tat tere exists a> suc tat n i,j= a ij(x)ξ i ξ j a ξ 2 for all ξ R n and a.a. x. Given g H (), te boundary value problem (.) ten as a unique weak solution y H () in te sense tat A y vdx = g, v for all v H (), (.2) were, denotes te duality pairing between H () and H (). Furtermore, y H C g H, (.3) wit a constant C wic only depends on a. We sall denote tis solution by y = T (A, g) in order to also empasize its dependence on A. Institut für Analysis und Numerik, Otto von Guericke Universität Magdeburg, Universitätsplatz 2, 396 Magdeburg, Germany Scwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55, 246 Hamburg, Germany.

4 In wat follows we assume tat measurements (z (i),f (i) ) Z H (), i N (Z = L 2 () or Z = H ()) are available, from wic we would like to reconstruct te diffusion matrix A. To do so, we employ a least squares approac togeter wit a Tikonov regularization, i.e. we consider (P ) min A M J(A) := 2 y (i) z (i) 2 Z + γ 2 A 2 s.t. y (i) = T (A, f (i) ), i N. (.4) i= Here, γ>and we use te symbol for te L 2 norm on spaces of scalar, vector or matrix valued functions, wile te admissible set M will be specified in Section 2. Our coice, motivated by te concept of H convergence, guarantees te existence of a minimum of J. By discretizing (.) wit te elp of linear finite elements we obtain an approximation J of J. Our main result, Teorem 3.4, says tat eac sequence of minimizers (A ) > of J as a subsequence tat converges strongly in L 2 to a minimum of J. In order to establis tis result we sall adapt a discrete version of H convergence, introduced by Eymard and Gallouët in [3] for finite volume scemes, to our setting. Te above convergence result justifies te use of J in solving te identification problem. In practice we employ a projected steepest descent algoritm for minimizing J, see Section 4. Let us review related work wic is concerned wit te identification of matrix valued parameters in elliptic PDEs. In [], Alt, Hoffmann and Sprekels obtain a reconstructed matrix by investigating te long time beaviour of a suitable dynamical system, see also [6]. In [], Kon and Lowe introduce a variational metod tat is based on a convex functional involving te variables y and A y and investigate its stability properties. Stability results for te reconstruction of matrices of te form A = p y can be found in [8]. In [3], Rannacer and Vexler prove a priori estimates for a matrix identification problem in wic a finite number of unknown parameters is estimated from finitely many pointwise observations. A lot of work as been devoted to te parameter estimation problem for a scalar diffusion coefficient. Identifiability results can e.g. be found in [2], [4] and [6]. A survey of numerical metods for parameter estimation problems can be found in []. Error estimates for a least squares approac ave been obtained by Falk in [4] and more recently by Wang and Zou [7] for a functional involving a Tikonov regularization. Tat paper also contains a long list of furter contributions. Let us finally note tat te concept of H convergence as recently been used by Leugering and Stingl in [2] in order to treat problems in material design, in particular to identify strain tensors from displacements in linear elasticity. 2 Existence of a minimum Let us denote by S n te set of all symmetric n n matrices endowed wit te inner product A B =trace(ab). We consider te subset K := {A S n a λ i (A) b, i =,...,n} were < a < b < are given constants and λ (A),...,λ n (A) denote te eigenvalues of A. SinceK is a convex and closed subset of S n we may introduce te ortogonal 2

5 projection P K : S n K for wic we can derive a formula as follows: given A S n, let S be an ortogonal matrix suc tat SAS t =diag(λ (A),...,λ n (A)) =: D. Ifwelet D =diag ( P [a,b] (λ (A)),...,P [a,b] (λ n (A)) ),werep [a,b] (x) :=max{a, min{x, b}},x R, ten clearly S t DS K and we ave for every B K (A S t DS) (B S t DS) =(D D) (SBS t D) = n (λ i (A) P [a,b] (λ i (A)))( b ii P [a,b] (λ i (A))), i= were B = SBS t K. Hence b ii [a, b],i =,...,n wic immediately yields (A S t DS) (B S t DS), for all B K and terefore P K (A) =S t diag ( P [a,b] (λ (A)),...,P [a,b] (λ n (A)) ) S. Next, let us introduce te set M := {A L () n,n A(x) K a.e. in }. In proving te existence to problem (P ) te following compactness result is crucial, see e.g. [5]. Teorem 2.. Let (A k ) k N be a sequence in M. Ten tere exists a subsequence (A k ) k N and an element A Msuc tat for every g H () T (A k,g) T(A, g) in H () and A k T (A k,g) A T (A, g) in L 2 () n. (2.) Te sequence (A k ) k M is ten said to be H convergent to A and one writes A k H A. Lemma 2.2. Suppose tat (A k ) k N is a sequence in M wit A k H A and Ak A in L () n,n.tena(x) A (x) a.e. in and A 2 A 2 lim inf A k 2. (2.2) k Proof. Te proof of Corollary 3.3 below will include an argument wic sows in a similar setting tat A A a.e. in. Furtermore, from te Courant Fiscer minmax teorem we infer tat λ i (A(x)) λ i (A (x)),i =,...,nand ence taking into account tat λ i (A(x)) A(x) 2 = n n λ i (A(x)) 2 λ i (A (x)) 2 = A (x) 2 a.e. in. i= i= Integration over togeter wit te weak lower semicontinuity of te L 2 -norm ten implies (2.2). We are now in position to establis te existence of a solution to te minimization problem (.4). Teorem 2.3. Problem (P ) as a solution A M. 3

6 Proof. Let (A k ) k N M be a minimizing sequence for problem (P ) so tat J(A k ) inf A M J(A) ask. Combining Teorem 2. wit te fact tat (A k ) k N is bounded in L () n,n we deduce tat tere exist A M,A L () n,n suc tat A k H A and A k A in L () n,n for some suitable subsequence. Letting y (i) k = T (A k,f (i) ),y (i) = T (A, f (i) ), i N, we terefore ave y (i) k y (i) in H (). Hence, J(A) = 2 i= y (i) z (i) 2 Z + γ 2 A 2 lim inf k lim inf J(A k k )= inf J(A), A M 2 i= y (i) k z (i) 2 Z + γ 2 lim inf k A k 2 were we also used (2.2). Let us next derive a suitable form of te necessary first order optimality conditions for a solution of (P ). To begin, it is not difficult to verify tat J is Frécet differentiable on M wit J (A)H = (y (i) z (i),w (i) ) Z + γ(a, H) L 2, H L () n,n (2.3) i= were y (i) = T (A, f (i) ) and w (i) = D A T (A, f (i) )H H (), i N is te partial derivative of T wit respect to A in direction H wic is given as te unique solution of A w (i) vdx = H y (i) vdx for all v H (). (2.4) In order to rewrite (2.3) we introduce te functions p (i) H (),i =,...,N as te unique solutions of te following adjoint problems: A v p (i) dx =(y (i) z (i),v) Z for all v H (). (2.5) Abbreviating (a b) kl = 2 (a kb l + a l b k ),k,l =,...,n for a, b R n we ten ave J ( N (A)H = y (i) p (i) + γa ) Hdx, H L () n,n. (2.6) i= Note tat te above integral exists since y (i) p (i) L () n,n.inconclusion Teorem 2.4. Let A Mbe a solution of (P ). Ten for every λ> ( )) A(x) =P K (A λ γa y (i) (x) p (i) (x) a.e. in. i= Proof. Te optimality of A implies tat J (A)(Ã A) for all Ã Mwic can be rewritten wit te elp of (2.6) as follows: ( ( ) ) ) A λ γa y (i) (x) p (i) (x) A (Ã A )dx for all Ã M. i= 4

7 A localization argument sows tat A(x) is te ortogonal projection of ( ) A(x) λ γa(x) y (i) (x) p (i) (x) onto K a.e. in wic implies te result. i= Let us note tat te particular coice λ = γ gives ( ( N )) A(x) =P K y (i) (x) p (i) (x) γ i= 3 Finite element discretization a.e. in. (2.7) Let T be a regular triangulation of wit maximum mes size := max T T diam(t )and suppose tat is te union of te elements of T ; boundary elements are allowed to ave one curved face. We define te space of linear finite elements, X := {v H () v is a linear polynomial on eac T T }. It is well known tat tere exists an interpolation operator Π : H () X suc tat Π w w in H () as for every w H (). (3.) For given A Mand g H (), te problem A y v dx = g, v for all v X as a unique solution y = T (A, g) X. Furtermore, a standard argument yields te error bound y y H b a inf y v H v X, were y = T (A, g). (3.2) In order to set up an approximation of (P ) we use variational discretization as in [5] and consider (P ) min A M J (A) := 2 i= y (i) z(i) 2 + γ 2 A 2 s.t.y (i) = T (A, f (i) ), i N. (3.3) Similar arguments as in Section 2 sow tat J is Frécet differentiable and tat for A M ( ) J (A)H = + γa Hdx, H L () n,n. (3.4) i= y (i) p(i) Since dimx < it is straigtforward to see tat (P ) as a solution A M.Furtermore, every solution A of (P )satisfies ( ) A (x) =P K y (i) γ (x) p(i) (x) a.e. in, (3.5) i= 5

8 cf. (2.7). Here, y (i) = T (A,f (i) )andp (i) X are te solutions of te adjoint problems A v p (i) dx =(y(i) z(i),v ) Z for all v X, i N. (3.6) Remark 3.. Let us note tat in view of (3.5) A is piecewise constant so tat a discretization of te set M is not required. Variational discretization automatically yields solutions to (3.3) wic allow a finite-dimensional representation. In order to investigate te convergence of te approximate solutions we sall employ a discrete version of Teorem 2.. Teorem 3.2. Let (A ) > be a sequence in M. Ten tere exists a subsequence (A ) > and A Msuc tat for every g H () T (A,g) T(A, g) in H () and A T (A,g) A T (A, g) in L2 () n. (3.7) We ten say tat te sequence (A ) M Hd converges to A and write A Hd A. Proof. Te line of argument follows te corresponding proof in te continuous case (see [5]) and a similar result for a finite volume sceme, see [3]. We terefore only sketc te main steps. Step : One first sows tat tere exists a subsequence, for ease of notation again denoted by (A ) >, and continuous linear operators S : H () H (), R : H () L 2 () n suc tat for every g H () T (A,g) S(g) inh (), A T (A,g) R(g) inl 2 () n as. (3.8) Step 2: We sow tat S is invertible. For g H () denote by w H (),w X te solutions of w vdx = g, v, v H (), w v dx = g, v, v X. Clearly, w H = g H and w w in H () in view of (3.). Setting y = T (A,g)we ave in addition tat A y v dx = g, v = w v dx, v X, from wic we infer tat w H b y H recalling te definition of M. Combining tis bound wit (3.8) and using again te properties of M we deduce tat g 2 H = w 2 H b2 a lim inf = lim w 2 H b 2 lim inf y 2 H A y y dx = b2 a lim inf g, y = b2 g, S(g), (3.9) a wic implies tat S is invertible. Step 3: Let C : H () L2 () n be defined by Cv := RS v.foragiveng H () te function y = T (A,g) satisfies by definition A y v dx = g, v for all v X. (3.) 6

9 Sending and taking into account (3.8) and (3.) we infer Cy vdx = g, v for all v H (), were y = S(g). (3.) Next, let g, g H () be arbitrary and define y = S(g), ỹ = S( g) as well as y = T (A,g), ỹ = T (A, g). Recalling (3.) we ave for every ϕ C () ϕa y ỹ dx = A y r dx + g, ϕỹ g, r A y ϕỹ dx, were we ave abbreviated r = ϕỹ I (ϕỹ )andi denotes te Lagrange interpolation operator. A standard interpolation estimate implies ϕỹ I (ϕỹ ) H (T ) C D 2 (ϕỹ ) L 2 (T ) C ϕ H 2, (T ) ỹ H (T ), T T, so tat r inh () as since ỹ H C. Observing in addition tat A y Cy in L 2 () n, ϕỹ ϕỹ in H () and ỹ ỹ in L 2 () we obtain ϕa y ỹ dx = g, ϕỹ Cy ϕ ỹdx, lim wic, combined wit (3.), yields ϕa y ỹ dx = lim ϕcy ỹdx. (3.2) Similarly as in [3, Proof of Teorem 2] one now deduces from (3.) and (3.2) tat tere exists A Msuc tat (Cy)(x) =A(x) y(x) a.e. in. (3.3) Tis completes te proof of te teorem. Hd Corollary 3.3. Suppose tat (A ) > is a sequence in M wit A A and A L () n,n.tena A a.e. in and A 2 A 2 lim inf A 2. A in Proof. We use te same notation as in te proof of Teorem 3.2. By Step 2 above tere exists for every y H () an element g H () suc tat y = S(g). Defining y = T (A,g) tere olds y yin H (), A y A y in L 2 () n.furtermore,weaveforany ϕ C (),ϕ tat ϕa (y y) (y y)dx = ϕa y y dx 2 ϕa y ydx + ϕa y ydx. Recalling (3.2), (3.3) and te fact tat A A in L () n,n we obtain upon sending ϕa y ydx + ϕa y ydx, 7

10 from wic we infer tat A y y A y y a.e. in. Since y H () is arbitrary we deduce tat A A a.e. in. Te remaining estimate is obtained in te same way as in te proof of Lemma 2.2. We are now in position to prove a convergence result for a sequence (A ) > of solutions of (P ). Teorem 3.4. Let A Mbe a solution of (P ). Ten tere exists a subsequence (A ) > and A Msuc tat A A in L 2 () n,n, T (A,f (i) ) T (A, f (i) ) in Z, i N and A is a solution of (P ). Proof. In view of Teorem 3.2 and Corollary 3.3 tere exists a subsequence, again denoted Hd by (A ) >,anda M suc tat A A and A A in L () n,n wit A A a.e. in. Let y (i) = T (A,f (i) ),y (i) = T (A, f (i) ), i N. Teny (i) y (i) in H (), A y (i) A y(i) in L 2 () n, so tat we may deduce similarly as in te proof of Teorem 2.3 tat J(A) lim inf J (A ). Next, Teorem 2.3 implies tat (P ) as a solution Ā M. Ten we ave J(Ā) J(A) lim inf J (A ) lim sup J (A ) lim sup J (Ā) =J(Ā), were te last equality follows from (3.2) and (3.). We deduce tat lim J (A )=J(A) =J(Ā), (3.4) in particular, A is a minimum of J. Furtermore,weave 2 i= y (i) y(i) 2 Z + γ 2 A A 2 = 2 = 2 i= y (i) z(i) 2 Z N i= i= (y (i) + γ 2 A 2 γ(a,a)+ γ 2 A 2 = J (A )+J(A) 2J(A) 2J(A) i= (y (i) y (i) z (i) 2 Z γ(a,a) i= (y (i) z(i) ) (y (i) z (i) ) 2 Z + γ 2 A A 2 z(i),y (i) z (i) ) Z + 2 y (i) z (i) 2 Z γ A 2 =, i= y (i) z (i) 2 Z i= z(i),y (i) z (i) ) Z γ(a,a) were we ave used (3.4) and te fact tat A A a.e. in. Te teorem is proved. 4 Numerical examples Let := (, ) 2 R 2. We consider a finite element approximation wit piecewise linear, continuous functions defined on a triangulation containing 52 triangles, constructed wit 8

11 te POIMESH environment of MATLAB. We take N = wit data (z,f) givenbyz = I y were y(x,x 2 )=( x 2 )( x 2 2)andf(x,x 2 )=( x 2 2)(6x 2 +2)+2( x 2 ). Note tat y is te solution of (.) wen A(x,x 2 )= +x2 In te definition of K we ave cosen a =.5 andb =. Te discrete problem (3.3) is solved using te projected steepest descent metod wit Armijo step size rule, see e.g. [9]. In view of Remark 3. it is sufficient to iterate witin te class of matrices in M tat are piecewise constant. Given suc an A te new iterate is computed according to A + = A(τ) witτ =max l N {βl ; J (A(β l )) J (A) σ β l A(βl ) A 2 } were β (, ) and. A(τ) T := P K (A T + τ ( y T p T γa T ) ), T T. Here, y = T (A, f) andp is te solution of te adjoint problem (3.6). In our calculations we cose γ =., σ = 4, β =.5 and as initial matrix A := 2 2 Te iteration was stopped if A + A() τ a + τ r A A () or te maximum number of 5 iterations was reaced. For τ a = 3 and τ r = 2 we ave A A () = ,J (A )=2.8 and te algoritm terminated after 4 iterations wit Ã and ỹ = T (Ã, f) suc tat ỹ z =.2 2, A Ã =2.5 and J (Ã) = Note tat we cannot expect te difference A Ã to become small since te diffusion matrix will not be determined uniquely by just one set of data. Performing 5 iterations we obtained Ã and ỹ suc tat ỹ z =8.22 3, A Ã =.53 and J (Ã) = Fig.fromlefttorigtsowsỹ,z and ỹ z after 4 iterations. By combining te projected gradient metod wit a omotopy in te parameter γ we were also able to treat te case γ =. We started wit γ = and reduced γ by a factor of.8 after every ten iterations. Using te same notation as above we obtained after 5 iterations. ỹ z =9.6 4, A Ã =.4 and te corresponding results are displayed in Fig. 2. One observes tat te difference between ỹ and z is comparatively large in regions were y is small wic is in agreement wit classical results on te identifiability of scalar diffusion coefficients, see e.g. [4]. 9

12 Figure : Numerical solution, desired state, and error y z for γ =. 3 after te stopping criterion of te projected steepest descent metod is met x 8 3 x Figure 2: Numerical solution, desired state, error y z for γ = after 5 iterations of te steepest descent metod. Acknowledgements Te presentation of te numerical results is partly based on a MATLAB code developed by Ronny Hoﬀmann in is diploma tesis [7] written under te supervision of te second autor. Te autors gratefully acknowledge te support of te DFG Priority Program 253 entitled Optimization Wit Partial Diﬀerential Equations. References [] Alt, H.W., Hoﬀmann, K.H., Sprekels, J.: A numerical procedure to solve certain identiﬁcation problems, Intern. Ser. Numer. Mat. 68, 43 (984). [2] Cicone, C., Gerlac, J.: A note on te identiﬁability of distributed parameters in elliptic equations, SIAM J. Mat. Anal. 8, (987). [3] Eymard, R., Galloue t, T.: H convergence and numerical scemes for elliptic problems, Siam J. Numer. Anal. 4, (23). [4] Falk, R.S.: Error estimates for te numerical identiﬁcation of a variable coeﬃcient, Mat. Comput. 4, (983). [5] Hinze, M.: A variational discretization concept in control constrained optimization: te linear-quadratic case, Comput. Optim. Appl. 3, 45 6 (25).

13 [6] Hoffmann, K.H., Sprekels, J.: On te identification of coefficients of elliptic problems by asymptotic regularization. Numer. Funct. Anal. Optim. 7, (984/85). [7] Hoffmann, R.: Entwicklung numeriscer Metoden zur Scätzung matrixwertiger verteilter Parameter bei elliptiscen Differentialgleicungen, Diploma tesis, TU Dresden, 25. [8] Hsiao, G.C., Sprekels, J.: A stability result for distributed parameter identification in bilinear systems, Mat. Met. Appl. Sciences, (988). [9] Kelley, C.T.: Iterative Metods for Optimization. SIAM, 999. [] Kon, R.V., Lowe, B.D.: A variational metod for parameter identification, RAIRO Modél. Mat. Anal. Numér. 22, 9 58 (988). [] Kunisc, K.: Numerical metods for parameter estimation problems, Inverse problems in diffusion processes (Lake St. Wolfgang, 994), 99-26, SIAM, Piladelpia, PA, 995. [2] Leugering, G., Stingl, M.: PDE-constrained optimization for advanced materials. In Constrained Optimization and Optimal Control for Partial Differential Equations. Birkäuser, G. Leugering et al. Eds., 2. [3] 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, (25). [4] Ricter, G.R.: An inverse problem for te steady state diffusion equation, SIAM J. Appl. Mat. 4, 2 22 (98). [5] Tartar, L.: Estimation of omegenized coefficients. Topics in te matematical modelling of composite materials, page 9-2, Andrej Carkaev, Robert Kon Eds., 997. [6] Vainikko, G., Kunisc, K.: Identifiability of te transmissivity coefficient in an elliptic boundary value problem, Z. Anal. Anwendungen 2, (993). [7] 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 4, (2).

Priority Program 1253

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