Adaptive Finite Element Methods for Optimal Control of Partial Di erential Equations: Basic Concept R. Becker, H. Kapp, R. Rannacher 1
Summary. We develop a new approach towards error control and adaptivity for nite element
discretizations in optimization problems governed by partial di erential equations. Using the Lagrangian formalism the goal is to compute stationary points of the rst-order necessary optimality conditions. The mesh adaptation is driven by residual-based a posteriori error estimates derived by duality arguments. This approach facilitates control of the error with respect to any given quantity of physical interest. The speci c feature introduced by the optimization problem is the natural choice of the error-control functional to conincide with the cost functional of the optimization problem. In this case, the Lagrangian multiplier can directly be used in weighting the cell-residuals in the error estimator. This leads to a particularly simple and cost-ecient algorithm for adapting the mesh according to the particular needs of the optimization problem. This approach is developed and tested for simple model problems in optimal control of semiconductivity.
1 Introduction In this article we develop an adaptive nite element method for optimal control problems governed by elliptic partial di erential equations. The main goal is the derivation of a posteriori error estimates as basis for controlling the discretization error. Therefore, we begin our analysis for a simple model problem with a linear state equation. The control acts on a part of the boundary and aims to minimize a quadratic cost functional which involves observations on a possibly di erent part of the boundary. Dispite its simplicity, this problem represents the main structure of optimal control and is chosen in order to clarify the idea underlying the proposed procedure. The control problem is rst described in the continuous setting and the Euler-Lagrange equations are formulated using the classical framework of Lions [5]. This leads to a system of coupled partial di erential equations for the state variable u, the control variable q, and the Lagrangian multiplier . This system has the usual saddle point structure and admits a unique solution under natural conditions. We use a standard nite element discretization of the Euler-Lagrange system. This implies that the set of admissible solutions is also discretized and therefore di ers from the continuous one. As long as the discretization procedure uses pure Galerkin techniques the discrete problem actually corresponds to a formulation of the original minimization problem on the discrete space. Since discretization in partial di erential equations is expensive, at least for praxis-relevant models, the question of how they a ect the quality of the optimization result is crucial for an economical computation. The need for adaptive error control is therefore evident. For a posteriori error estimation, we apply the general error-control approach developed in [2] for nite-element Galerkin discretizations of partial di erential equations. In this method a This work has been supported by the German Research Foundation (DFG), SFB 359 Reactive Flows, Di usion and Transport, Institute of Applied Mathematics, University of Heidelberg, INF 293, D-69120 Heidelberg, Germany, E-mail:
[email protected],Internet: http://gaia.iwr.uni-heidelberg.de 1
1
posteriori estimates for the error with respect to arbitrary functional output are obtained via duality arguments. In these estimates local cell residuals of the computed solution are weighted by factors involving derivatives of the dual solution. These weights describe the dependence of the error functional on the local residuals. Evaluating them computationally results in a feed-back process by which successively more and more improved error bounds and economical meshes are generated. In applying this approach for optimization problems, the main question is that of the appropriate choice of an error-control functional according to which the mesh is optimized. The answer turns out to be particularly simple if we choose the given cost functional of the optimization problem as the error-control functional for mesh adaptation. In this case the corresponding a posteriori error estimator only involves the knowledge of the state variable and the Lagrangian multiplier de ned by the rst-order necessary optimality condition. The latter observation is easily understood if one thinks, e.g., of solving the rst boundary value problem of the Laplacian,
?u = f in ; u = 0 on @ ;
(1.1) by a nite element method with error control in the natural energy norm, krek . In this case, the corresponding \dual solution" z 2 H01 ( ) coincides with the error function, z = e . The natural variational formulation of this boundary value problem is equivalent to an optimization problem, namely the minimization of the energy functional J (u) := (ru; ru) ? 2(f; u)
over the Sobolev space H01 ( ). Hence, observing that krek2 = J (uh) ? J (u); energy-error control for the variational equation is equivalent to controlling the error in the cost functional, J (u) ? J (uh ) , of the optimization problem. The resulting a posteriori error estimates have the form X 2 jJ (u) ? J (uh )j hT T (uh) !T (z); (1.2) T 2Th
where T (uh ) are the cell residuals and !T (z ) kz ? ih z kT = ku ? ih ukT the local weights. This explains why this kind of error control can be based on the \forward solutions" u and uh alone and does not require the solution of an additional dual problem as is necessary in controlling general error functionals. We develop our approach to adaptivity in optimization problems within a general setting in order to abstract from inessential technicalities. At the end, we present some numerical computations for simple test problems which illustrate the adaptive algorithm and demonstrate the bene ts of the local mesh re nement. We compare our estimators with a more heuristic approach based on an energy estimator for the state variable alone. Our approach immediately carries over to cases with non-linear state equations and also to more complicated cost functionals.
2 A linear model situation We consider an abstract setting for optimal control: Let Q , V and H be Hilbert spaces for the control variable q 2 Q , the state variable u 2 V , and given observations c0 2 H . The 2
inner product and norm of H are (; ) and k k, respectively. The state equation is given in the form a(u; ') = (f; ') + b(q; ') 8' 2 V; (2.1) where the bilinear form a(; ) represents a linear elliptic operator and the bilinear form b(; ) expresses the action of the control. The goal is to minimize the cost functional J (q) = 21 kcu(q) ? c0 k2 + 12 n(q; q); (2.2) where c : V ! H is a linear bounded observation operator. We assume that each q 2 Q de nes a unique solution u = u(q) 2 V of (2.1) and that the resulting functional J () has the appropriate continuity and coercivity properties in order to apply the calculus of variations. This guarantees the existence of a unique solution of the optimal control problem and the classical regularity theory for elliptic equations applies (see, e.g., [5]). For simplicity, we suppose that a(; ) and n(; ) induce norms denoted by kka and kkn on the spaces V and Q , respectively, which will be used in the following. Introducing a Lagrangian parameter 2 V and the corresponding Lagrangian function L(u; q; ) , the rst order necessary conditions (Euler-Lagrange equations) of the optimization problem reads a(v; ) + (cu ? c0 ; cv) = 0 8v 2 V; a(u; ) ? b(q; ) = (f; ) 8 2 V; (2.3) b(r; ) + n(q; r) = 0 8r 2 Q: This system has the usual saddle point structure (cu; cv) + a(v; ) = (c0 ; cv) 8v 2 V; a(u; ) ? b(q; ) = (f; ) 8 2 V; (2.4) ?b(r; ) ? n(q; r) = 0 8r 2 Q: Introducing operators A; B; C; N which represent the corresponding bilinear forms, system (2.4) can also be written in matrix form as 2 32 3 2 3 C AT 0 u c0 4A 0 ?B 5 45 = 4 f 5 : (2.5) T 0 ?B ?N q 0 Below, we will consider the following realization of the foregoing abstract setting which represents the case of an elliptic linear state equation subjected to boundary control. Let
R2 be an open bounded domain with Lipschitz boundary @ which is decomposed into a Dirichlet part ?D and a control part ?Q on which the control acts, ?u + s(u) = f in ; (2.6) u = 0 on ?D ; @nu = q on ?Q; with s(u) := u . The observations are given on a part ?O of the boundary and the associated cost functional is (2.7) J (fu; qg) = 21 ku ? c0 k2?O + 12 kqk2?Q : 3
with a regularization parameter 0. In this case the natural function spaces are V = fv 2 H 1( ) : v = 0 on ?D g and Q = L2(?Q), whereas the operator c corresponds to the trace operator. The bilinear forms a(; ) , b(; ) and n(; ) are given by
a(u; v) = (ru; rv) + (u; v) ; b(q; v) = (q; v)?Q ; n(q; r) = (q; r)?Q ; where (; ) denotes the L2 -inner product on .
3 A priori error estimate For simplicity of notation, we introduce the space X = V V Q, with elements of the form x = fu; ; qg which is equipped with the product-space norm
kxkX := kuk2a + kk2a + kqk2n
1=2
:
Furthermore, we de ne a bilinear form A(; ) on X by
A(x; y) = A(fu; ; qg; fv; ; rg) := (cu; c) + a(u; v) ? b(q; v) + a(; ) ? b(r; ) ? n(q; r): Using this notation, system (2.4) can be written in compact form as
A(x; y) = F (y) 8y 2 X;
(3.1)
with the linear functional F () de ned by
F (y) = F (fv; ; rg) := (c0 ; c) + (f; v): In order to simplify the analysis, we impose the following conditions,
jA(x; y)j cAkxkX kykX ; jb(r; v)j cbkrkn kvka :
(3.2) (3.3)
The second condition which relies on the regularization term n(; ) is rather strong. It can be substituted by an 'inf-sup'-condition for b(; ) under which the regularization could be omitted. The bilinear form A(; ) satis es the following stability condition: Proposition 3.1. Under the assumptions (3.2) and (3.3), we have (3.4) inf sup A(x; y) > 0: x2X y2X
kxkX kykX
Proof. For any xed x = fu; ; qg , we choose the test triple y = fv; ; rg := fu; ; ?qg , in order to obtain
A(x; y) = kcuk2 + kuk2a + kk2a + kqk2n + b(q; ) ? b(q; u) kcuk2 + kuk2a + kk2a + kqk2n ? 14 kqk2n ? 43 kk2a ? 14 kqk2n ? 43 kuk2a
kcuk2 + 14 kuk2a + 41 kk2a + 12 kqk2n :
We conclude the asserted estimate by noting that kyk = kxk. 4
We consider the discretization of the variational equation (3.1) by a standard Galerkin method using trial spaces Xh := Vh Vh Qh X . For each x 2 X , there exists an \interpolation" ih x 2 Xh , such that kx ? ih xkX ! 0 (h ! 0) . The discrete problem reads
xh 2 Xh : A(xh; yh) = F (yh) 8yh 2 Xh:
(3.5)
This discretization is automatically stable since a discrete analogue of (3.4) is full lled by the same argument as used above, (3.6) inf sup A(xh ; yh ) > 0: xh 2Xh yh 2Xh
kxh kX kyhkX
Combining equations (3.5) and (3.1), we get the Galerkin orthogonality
A(x ? xh; yh) = 0; yh 2 Xh:
(3.7)
This leads us to the following abstract a priori error estimate. Proposition 3.2. For the Galerkin approximation on spaces Xh X , there holds
ku ? uhka +k ? hka + kq ? qhkn : k q ? p k k ? v k + inf k u ? v k + inf c v inf n a a h h h p 2Q v 2V 2V h h
h h
h
(3.8)
h
Proof. We note that
kih x ? xhk sup A(ihkxy?kxh; yh) = sup A(ihkxy ?k x; yh) cAkih x ? xkX : h X h X yh 2Xh yh 2Xh Here, we have used the Galerkin relation (3.7) and the continuity estimate (3.2). Of course, more precise error estimates can be given using re ned arguments. For instance, it would be interesting to equip the space Q with a di erent norm than the one induced by n(; ) in order to get robustness with respect to the regularization. This a ords replacing (3.3) by an appropriate inf-sup-condition. We note that for the model example with boundary control and boundary observations given above the conditions (3.2) and (3.3) are satis ed.
4 A posteriori error estimate In this section, we derive a posteriori error estimates for the control problem. Starting with a general error functional, we focus on the case where we wish to control the value of the cost functional itself. Of course, we are not so much interested in this value but expect the resulting local error indicators to induce a mesh re nement which is most suited for the special features of the optimization problem. We start with a general linear functional G() = fGu (); G (); Gq ()g de ned on X . In order to obtain an a posteriori estimator for G(x ? xh) , we consider the following dual problem:
z 2 X : A(y; z) = G(y) 8y 2 X: 5
(4.1)
Denoting the dual solution by z = (w; ; p) , (4.1) can be written in detailed form as (cw; cv) + a(v; ) = Gu (v) 8v 2 V; a(w; ) ? b(p; ) = G() 8 2 V; ?b(r; ) ? n(p; r) = Gq (r) 8r 2 Q:
(4.2)
This leads to the following general error representation
G(x ? xh) = A(x ? xh; z) = A(x ? xh; z ? ihz) = F (z ? ihz) ? A(xh; z ? ihz);
(4.3)
which can be used to derive an error estimator. Before doing so, we will rst consider the special case that the functional G() is related to the cost functional of the optimization problem. Proposition 4.1. Let the cost functional J () be quadratic and the state equation linear. Then, there holds
J (x) ? J (xh) = F (z ? ih z) ? A(xh ; z ? ih z);
(4.4)
with the \dual solution" z = fw; ; pg given by
z = ? 21 (u ? uh; ? h; q ? qh):
(4.5)
Proof. We denote by L(x) = J (u; q) + hAu ? Bq; i the Lagrangian functional, which is stationary at the continuous solution fu; ; qg. We therefore have
J (u; q) ? J (uh ; qh) = L(u; ; q) ? L(uh; h; qh) = ?rL(x)(x ? xh ) ? 12 r2 L(x ? xh )2 = ? 12 r2 L(x ? xh )2 : Choosing now the special error functional G(y) := r2 L(x ? xh )(y) and using the symmetry of the second derivative, we obtain the asserted form of the dual solution, since r2 L constitutes the left hand side of (4.2).
We now derive a precise form of the error estimator for our model optimization problem. In order to avoid unnecessary complications due to curved boundaries, we suppose the domain
to be polygonal. Based on a quasi-uniform triangulation Th = fT g of , we build a nite element space Vh V for the state variable and the Lagrange multiplier in the usual way. For simplicity, we assume that the space Qh of discrete controls is given by the traces of the nite element functions of Vh . This is not necessary for our results but simpli es notation. Proposition 4.2. For control of the cost functional J (), there holds the weighted a posteriori error estimate o X (1) (1) X n (2) (2) (3) ! ? !? + (3) ? !? + ? ? ??Q ??O o o X n (6) (6) X n (4) (4) (7) (7) (5) ! + ! + ! T !T + (5) + ? ? ; ? ? T T ?@T T 2Th
jJ (u; q) ? J (uh; qh)j
6
(4.6)
with the cell residuals and weights
!?(1) = ku ? ihuk? ; (1) ? = kuh ? c0 + @n h k? ; !?(2) = k ? ihk? ; (2) ? = k@n uh ? qh k? ; !?(3) = kq ? ihqk?; (3) ? = kh ? qh k? ; !T(4) = k ? ihkT ; (4) T = kuh ? s(uh ) + f kT ; 0 !T(5) = ku ? ihukT ; (5) T = kh ? h s (uh )kT ; !?(6) = k ? ihk? ; (6) ? = k[@n uh ]k? ; !?(7) = ku ? ihuk? (7) ? = k[@n h ]k? ; Proof. Proposition 4.1 implies that
J (u; q) ? J (uh; qh) = F (z ? ih z) ? A(xh ; z ? ih z) = (c0 ; c( ? ih )) + (f; ? ih ) ? (cuh ; c( ? ih )) ? a(h ; u ? ih u) ? a(uh; ? ih) + b(qh; ? ih) + b(q ? ihq; h) ? n(qh; q ? ihq): Then, the assertion follows by summation over all elements and element-wise integration by parts. Of course, the error estimator (4.6) presented in this form is of no direct practical use since it involves the continuous solutions f; u; qg . As proposed in [2], we therefore use local interpolation and a simple approximation of derivatives by di erence quotients. Especially, we can use the discrete solutions in order to approximate the weights, e.g., (4.7) !T(5) = ku ? ihukT CI hT kr2ukT CI hT kr2h uhkT ; where r2h uh is a suitable di erence quotient and CI is an interpolaton constant in the range CI 0:1 ? 1. This means for practical computation that error control with respect to the value of the cost functional J () can be done without much additional cost.
Remark 4.1. As follows from its derivation, the error estimator (4.6) also controls the error with respect to r2 J (x ? ih x)2 = (c(u ? uh ); u ? uh ) + n(q ? qh; q ? qh). Remark 4.2. We note that in the a posteriori error estimate (4.6), the residual of the state
equation is weighted by terms involving the Lagrangian multiplier from the original equation (2.4). This has a natural interpretation as it is well-known from sensitivity analysis that the Lagrangian multiplier measures the in uence of perturbations on the cost functional. Since discretization can be interpreted as a special perturbation, the appearance of in the estimator is not surprising. The special form of the weights involving the interpolation ih z is a characteristic feature of the Galerkin discretization (orthogonality of residuals with respect to the test space).
Remark 4.3. The a posteriori error estimate (4.6) is derived from the rst-order optimality
condition which is a system of partial di erential equations. We want to give an interpretation in terms of the original minimization problem. Indeed, the discretization of the state equation leads to numerical solutions which are not admissible (in the strict sense) for the original constrained minimization problem. The situation can be summarized as follows: Let s : Q ! V denote
7
the (linear) solution operator which associates the state variable to a given control function. The optimal control then minimizes the functional j (q) := J (s(q); q) without constraints over the space Q. Since the discretization changes the state equation, not only the space of possible controls is changed, but also the functional. Denoting by sh : Q ! Vh the discrete solution operator, the discrete optimal control qh minimizes the functional jh (q) := J (sh (q); q) over the space Qh . If we want to perform numerical computation, we have to substitute the notion of \admissible" solution by an error estimate for the state equation. Of course, the distance between the numerical and the continuous state should be measured with respect to the speci c needs of the optimization problem, i.e., the in uence on the functional to be minimized. This is exactly what the a posteriori estimator derived above is designed for.
5 Numerical results { linear case First, we present a linear model problem as described in (2.6), where is a T-shaped domain with side length one. The control acts along the lower boundary ?Q, whereas the observations are taken along the (longer) upper boundary ?O . The cost functional is chosen in the form
J (u; q) := ku ? c0 k2?O + kqk2?Q ; with c0 1 and = 1. In this case, kqk2?Q may be viewed as part of the cost functional with its own physical meaning. We perform computations on a series of locally re ned meshes. On each mesh, the system of the rst order necessary condition (2.4) is discretized by a standard nite element Galerkin method using piecewise bilinear shape functions for both the state and adjoint variables u and , while the traces on ?Q of the bilinear shape functions form the discrete control space Qh . Then, the resulting discrete systems are solved iteratively and new meshes are generated on the basis of the above error estimator. To facilitate local mesh re nement, hanging nodes are allowed (at most one per mesh cell). The weights in the error estimator (4.6) are evaluated using the strategy indicated in (4.7), while the interpolation constant is usually set to CI = 0:1 . Table 1 shows the quality of the error estimator for quantitative error control. The e ectivity index is de ned by Ieff := Eh =h , where Eh := jJ (u; q) ? J (uh ; qh )j is the error and h := (uh ; qh) the value of the error estimator. The reference value is obtained on a mesh with more than 200000 cells. We compare the weighted error estimator to a simple ad hoc approach based on the standard energy-error estimator applied only to the state equation. Figure 1 shows meshes generated by the two estimators. N
320
1376
4616
11816
23624
48716
Ieff
1.1
0.7
0.7
1.0
0.8
0.7
Eh 1:0e ? 3 3:5e ? 4 3:2e ? 5 1:6e ? 5 6:4e ? 6 2:8e ? 6
Table 1: Linear test ( = 1): E ectivity index of the weighted error estimator.
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Figure 1: Linear test ( = 1): Comparison between meshes obtained by the weighted error estimator (left) and the energy-error estimator (right), N 5000 cells in both cases. The di erence in the meshes can be explained as follows. Obviously, the energy-error estimator observes the irregularities introduced on the control boundary by the jump in the inhomogeneuous Neumann condition, but it tends to over-re ne near the reentrant corners. However, this re nement is apparently not necessary for the optimization process as is well observed by the more selective weighted error estimator. The quantitative e ects on the mesh eciency of these two di erent re nement mechanisms is shown in Figure 2 -5
-6
-7
log(err)
-8
-9
-10
-11
-12
-13 5.5
6
6.5
7
7.5
8 log(N)
8.5
9
9.5
10
10.5
Figure 2: Linear test ( = 1): Comparison of the eciency of the meshes generated by the two estimators, x error values for the energy estimator, 2 error values for the weighted estimator (log = log 0
0
0
scale).
9
0
6 Numerical Results { nonlinear case As non-linear test case we choose an example from superconductivity. The equations of state are again
?u + s(u) = f on ; (6.1) @nu = q on ?Q; @nu = 0 on @ n ?Q; now with the nonlinearity s(u) = u3 ? u , and the right hand side f = 0. The cost functional is the same as for the linear test case. The corresponding rst-order necessary condition to be solved reads
? + s0(u) = 0 in ; @n = u ? c0 on ?O ; @n = 0 on @ n ?O ; ? u + s(u) = f in ; @nu = q on ?Q; @nu = 0 on @ n ?Q; = (q ? q0) on ?Q (optimality condition):
(6.2)
Compared to the linear situation, the derivation of the weighted error estimator introduces an additional linearization error in the duality argument. Theory as well as practical experience show that, in the present case, this additional error is of higher order on well-adapted meshes and can therefore be neglected. Hence, we use the a posteriori error estimate derived in (4.6) for the linear case also for the non-linear problem. The discretization is the same as in the linear case combined with linearization by a Newton iteration. We note that the Newton iteration is always carried to the limit before the error estimator is applied for mesh re nement. The results of this process may signi cantly di er from those obtained if discretization and iteration error are mixed together (see the preceding publication [1]). We again compare the weighted error estimator with a simple ad hoc energy-error estimator. We consider two di erent choices for the boundaries of control and observation. First, we take the same boundary for control and observation, ?Q = ?O (lower boundary of the T-shaped domain). In this case, we have the main parts of the optimization problem at one boundary. Hence, we do not expect any need for strong mesh re nement 'far away' from this boundary if we only want to deal with the optimization problem. In the second case, we take the control and the observation on opposite boundaries, ?Q \ ?O = ; , as in the linear case (lower and upper boundary of the T-shaped domain). In this case, we expect better results for the energy-error estimator because the information must pass from the control to the observation boundary and the corner singularities will have a stronger e ect. Test case 1: The observations for this non-linear case are taken as c0 (x) = sin(0:19x). Table 2 shows the quality of the weighted error estimator for quantitative error control for this rst nonlinear test case. The reference value J (u; q) for the objective function is computed on a re ned mesh with about 131000 cells. Due to the special choice ?Q = ?O , the dual solution equals (1) zero almost everywhere away from ?Q , and the error indicators (1) ? !? in (4.6) dominate all the other terms in the estimator. The weighted error estimator considers only the neighborhood of the control boundary, whereas the energy-error estimator re ects too much the singularity in primal solution at the corners (see Figure 3). 10
N
596
1616
5084
8648
15512
Eh 2.56e-04 2.38e-04 8.22e-05 4.21e-05 3.99e-05 Ieff 0.34 0.81 0.46 0.29 0.43 Table 2: Nonlinear test 1 ( = 1): E ectivity index of the weighted error estimator in the case ?Q = ?O .
Figure 3: Nonlinear test 1 ( = 1): Comparison of discrete solutions obtained by the weighted error estimator (left, N 5000 cells) and the energy-error estimator (right, N 4800 cells). In Figure 4, we compare the eciency of the meshes generated by the two estimators in the rst nonlinear case with = 0. We see that in this "extreme" boundary layer example, we can approximate the solution of the optimization problem on a grid with much less cells using the weighted error estimator. 0.0001
"energy_ee" "opt3_ee"
E_h
1e-05
1e-06
1e-07 1000 10000 Number of elements N
100000
Figure 4: Nonlinear test 1 ( = 0): Comparison of eciency of meshes generated by the two estimators (log = log scale).
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Test case 2: We take the observations as c0 1 , as in the linear case, and set = 0:1. Now, depending on the nonlinearity s(u), there exist several stationary points of L(u; q; ), which can be obtained by varying the starting values for the Newton iteration. One solution (actually the global minimum) is a constant equal to c0 . For this example, we get an objective function value equal to zero (up to round-o error). Accordingly, we match these observations with our numerical solution already on a rather coarse mesh with N = 512 cells. The corresponding Newton residual and the Newton increment are both converged to zero. We do not show the trivial results of these computations. The two other stationary points obtained are symmetric to each other with respect to the plane fx = 0g in this case. Table 3 shows the quality of the weighted error estimator quantitative for error control for one of these local minima. The reference value 0:04888934625 for the objective function is obtained on an adaptive mesh with N = 545216 cells corresponding more than 106 unknowns.
N
512
15368
27800
57632
197408
Eh 9.29e-05 8.14e-07 4.86e-07 2.31e-07 4.58e-08 Ieff 1.32 0.56 0.35 0.42 0.32 Table 3: Nonlinear test 2 ( = 0:1): E ectivity index of the weighted error estimator for computing the second stationary point.
The numerical results demonstrate the correct qualitative behavior of the weighted error estimator. The e ectivity index indicates also a relatively good quantitative accuracy (with interpolation constant CI = 0:1), although the values produced are still too big. This defect is caused by taking the absolute signs under the sums thereby suppressing possible error cancellation. Furthermore, the error Eh is very small for = 0:1. In the case = 1 , we get better results as shown in Table 4. Next, Figure 5 shows the distribution of local error indicators in the two error estimators for the critical case = 0:1. Figure 6 shows the corresponding computed discrete solutions. Obviously, the weighted error estimator induces a much stronger re nement along the observation and control boundaries. N
Eh Ieff
512 8120 25544 42608 126284 2.08e-03 4.35e-05 9.26e-06 5.95e-06 8.94e-07 0.52 0.73 0.88 1.21 0.98
Table 4: Nonlinear test 2 ( = 1): E ectivity index of the weighted error estimator for computing the second stationary point.
In Figure 7, we see a faster convergence to the solution of the continuous problem with our weighted error estimator. In fact, we need in general at least a quarter less cells with the weighted error estimator.
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Figure 5: Nonlinear test 2 ( = 1): Distributions of local error indicators in the weighted error estimator (left) and the energy-error estimator (right).
Figure 6: Nonlinear test 2 ( = 1): Comparison of discrete solutions obtained by the weighted error estimator (left, N 3000 cells) and the energy-error estimator (right, N 3300 cells).
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"energy_ee" "opt3_ee"
0.0001
E_h
1e-05
1e-06
1e-07
1e-08 1000 10000 Number of elements N
100000
Figure 7: Nonlinear test 2 ( = 1): Comparison of eciency of meshes generated by the two error estimators (log = log scale).
Finally, for the third stationary point, we only indicate the eciency of the generated meshes. As seen from the e ectivity index in Table 5, the quantitative behavior of the weighted estimator is very good in this case (with interpolation constant CI = 0:1). N
1784
4544
15452
29096
77096
Eh 1.663e-05 6.02e-06 1.54e-06 7.43e-07 2.73e-07 Ieff 0.91 0.97 0.97 0.82 0.84 Table 5: Nonlinear test 2 ( = 1): E ectivity index of the weighted error estimator for computing the third stationary point.
References [1] R. Becker and H. Kapp: Optimization in PDE models with adaptive nite element discretization, Proc. ENUMATH'97, Heidelberg, Sept.29 - Oct.3, 1997, World Scienti c Publ., 1998, in press. [2] R. Becker and R. Rannacher: A feed-back approach to error control in nite element methods: Basic analysis and examples, East-West J. Numer. Math 4, 237-264 (1996). [3] S.C. Brenner and R. Scott: The Mathematical Theory of Finite Element Methods, Springer, Berlin, 1994. [4] K. Ito and K. Kunisch: Augmented Lagrangian-SQP methods for nonlinear optimal control problems of tracking type, SIAM J. Control and Optimization 34, 874-891 (1996). [5] J. L. Lions: Optimal Control of Systems Governed by Partial Di erential Equations, Springer, Berlin, 1971.
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