Eileen Kammann Fredi Tröltzsch Stefan Volkwein

Transcription

1 Universität Konstanz A method of a-posteriori error estimation with application to proper orthogonal decomposition Eileen Kammann Fredi Tröltzsch Stefan Volkwein Konstanzer Schriften in Mathematik Nr. 299, März 212 ISSN Fachbereich Mathematik und Statistik Universität Konstanz Fach D 197, Konstanz, Germany

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3 Mathematical Modelling and Numerical Analysis Modélisation Mathématique et Analyse Numérique Will be set by the publisher A METHOD OF A-POSTERIORI ERROR ESTIMATION WITH APPLICATION TO PROPER ORTHOGONAL DECOMPOSITION Eileen Kammann 1, Fredi Tröltzsch 2 and Stefan Volkwein 3 Abstract. We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a local solution of the reduced problem to a local solution of the original problem is estimated. Résumé Mathematics Subject Classification. 49K2, 35J61, 35K58. The dates will be set by the publisher. Introduction In this paper, we focus on the following question for a class of optimal control problems for semilinear parabolic equations: Suppose that a numerical approximation ũ for a locally optimal control is given. For instance, it might have been obtained by a numerical optimization method or as the solution to some reduced order optimization model. How far is this control from the nearest locally optimal control ū? Of course, we have to assume that such a solution ū exists in a neighborhood of ũ. Moreover, ũ should already be sufficiently close to ū. The question is to quantify the error ũ ū in an appropriate norm. In principle, this is not a paper on proper orthogonal decomposition (POD). Our estimation method is not restricted to any specific method of numerical approximation for ũ. Our primary goal is a numerical application of a perturbation method used by Arada et al. [2] in the context of FEM approximations of elliptic optimal control problems. The main idea of this method goes back to Dontchev et al. [7] and Malanowski and Maurer [18], who introduced it for the a priori error analysis of optimal control problems governed by ordinary differential equations. Keywords and phrases: Optimal control, semilinear partial differential equations, error estimation, proper orthogonal decomposition The first author was supported by SFB 91 Control of self-organizing nonlinear systems: Theoretical methods and concepts of application in Berlin. 1 Institut für Mathematik, Technische Universität Berlin, D-1623 Berlin, Germany (kammann@math.tu-berlin.de). 2 Institut für Mathematik, Technische Universität Berlin, D-1623 Berlin, Germany (troeltzsch@math.tu-berlin.de). 3 Institut für Mathematik und Statistik, Universität Konstanz, D Konstanz, Germany (Stefan.Volkwein@uni-konstanz.de). c EDP Sciences, SMAI 1999

4 2 However, we will apply our method to suboptimal controls ũ obtained by POD. Though POD is an excellent method of model reduction for many time-varying or nonlinear differential equations, it lacks an a priori error analysis that is in some sense uniform in the right-hand side of the underlying partial differential equation, say with respect to the control function. Estimates of this type are known for the method of balanced truncation (see, e.g. [4]) that is limited to linear and autonomous PDEs. There are results on a priori estimates for POD that depend on certain assumptions on the orthogonal basis generated by the selected snapshots. We refer to Kunisch and Volkwein [14], Sachs and Schu [21], or [26]. However, such estimates will in general depend on the control used for generating the snapshots. In [1] an a-priori error analysis is presented for linear-quadratic optimal control problems. If the POD basis is computed utilizing the optimal state and associated adjoint variable, a convergence rate can be shown. But in real computation we do not know the optimal solution in advance. In view of this, we are interested in a posteriori estimates for assessing the precision of optimal controls for reduced control problems set up by POD. For the reduced-basis method a-posteriori error estimates for linear-quadratic optimal control problems we refer to [8]. We extend a method suggested in [26] for linear equations to the case of semilinear equations. For this purpose, we have to assume that ū satisfies a standard second-order sufficient optimality condition. The associated coercivity constant of the reduced Hessian operator will be estimated numerically. Let us explain our idea for the following two optimal control problems in a bounded Lipschitz domain R n with boundary Γ: We consider the distributed optimal control problem min J(y, u) := 1 2 T subject to the semilinear parabolic state equation { (y(x, t) yd (x, t)) 2 + λ u(x, t) 2} dxdt (PD) y y(x, t) + d(x, t, y(x, t)) = u(x, t) in (, T ) t y(x, t) = in Γ (, T ) y(x, ) = y (x) in (.1) and to the pointwise control constraints α u(x, t) β a.e. in (, T ). Moreover, we deal with the boundary control problem min J(y, u) := 1 2 T subject to the parabolic state equation (y(x, t) y d (x, t)) 2 dxdt + λ 2 T y y(x, t) = in (, T ) t y (x, t) + b(x, t, y(x, t)) = u(x, t) in Γ (, T ) ν y(x, ) = y (x) in Γ u(x, t) 2 ds(x)dt (PB) (.2) and to the pointwise control constraints α u(x, t) β a.e. in Γ (, T ). Here, and throughout the paper, ν R n denotes the outward unit normal on Γ.

5 Our numerical analysis is based on the following assumptions: (A1) R n is a bounded domain with Lipschitz boundary Γ. (A2) The functions d : [, T ] R R and b : Γ [, T ] R R are measurable with respect to the first two variables (x, t) and twice continuously differentiable with respect to the third variable y R. The functions b and d are monotone non-decreasing with respect to the third variable y R for all fixed (x, t) Γ [, T ] and [, T ], respectively. (A3) There is a constant C > such that d(x, t, ) + d (x, t, ) y + 2 d (x, t, ) y2 C for a.a. (x, t) [, T ] and b(x, t, + b (x, t, ) y + 2 b (x, t, ) y2 C for a.a. (x, t) Γ [, T ]. Moreover, for all M > there exists a constant L M > such that 2 d y 2 (x, t, y 1) 2 d y 2 (x, t, y 2) L M y 1 y 2 holds for a.a. (x, t) [, T ] and all y 1, y 2 R satisfying y 1 y 2 < M. We require the same condition for b for a.a. (x, t) Γ [, T ]. (A4) A desired state function y d L 2 ( (, T )) and real constants λ >, α < β are given. The paper is organized as follows: After this introduction, we explain the perturbation method and its use for the simpler case of a linear equation in Section 1. In Section 2, we extend the perturbation idea to semilinear equations. To make the paper also readable for readers being not familiar with POD, we briefly introduce the main concept of POD in Section 3, and Section 4 is devoted to the numerical application of our method to different nonlinear state equations. 1. A survey on the linear-quadratic case 1.1. The perturbation method for minimizing quadratic functionals The perturbation method was introduced by Dontchev et al. [7] and Malanowski and Maurer [18] in the context of optimal control of ordinary differential equations, where it was used to derive a priori error estimates for associated numerical approximations. In a different way, it was adopted by Arada et al. [2] for elliptic control problems. Let us explain this method for the following situation: We consider the quadratic optimization problem in Hilbert space 3 min u K f(u) := 1 2 Su y H 2 H + λ 2 u 2 L 2 (D), (PQ) where H is a real Hilbert space, D R m is a measurable bounded set, K L 2 (D) is a nonempty, convex, closed and bounded set, S : L 2 (D) H a continuous linear operator, λ > a fixed number, and y H H a fixed element. In this form, optimal control problems with partial differential equations are very frequently discussed. We refer only to the monography by Lions [16] or to the more recent books by Ito and Kunisch [12], Hinze et al. [9] or to the textbook [25]. We will mainly refer to the last reference, since the discussion of optimal control problems for semilinear parabolic equations in this book is very close to our notation. Theorem 1.1. Under the assumptions formulated above, (PQ) has a unique optimal solution ū U ad. It satisfies the variational inequality {(S (Sū y H ))(x) + λ ū(x)} (u(x) ū(x)) dx u K, (1.3) D

6 4 where, S : H L 2 (D) denotes the adjoint operator to S. This is a standard result of the optimization theory, we refer e.g. to [16] or to [25, Thm. 2.22]. We introduce the adjoint state p associated with ū by p(x) := (S (Sū y H ))(x). The variational inequality can be expressed equivalently in a pointwise way, namely ( p(x) + λū(x))ū(x) = min ( p(x) + λū(x)) v (1.4) v [α,β] for almost all x D. This has important consequences. We have for almost all x D α, if p(x) + λū(x) > ū(x) = p(x)/λ, if p(x) + λū(x) = β, if p(x) + λū(x) <. While these inequalities show, how ū is determined by p + λū, the implications below concern the opposite direction; they follow directly from the discussion above. It must hold p(x) + λū(x), if ū(x) = α p(x) + λū(x) =, if α < ū(x) < β p(x) + λū(x), if ū(x) = β. The last implications are essential for understanding the perturbation method. It is helpful to answer the following question: Let ũ K be a function that need not be optimal for (PQ). Let p be the associated adjoint state. If ũ were optimal, then p(x) + λũ(x) = should be satisfied in almost all points x, where α < ũ(x) < β holds. If not, then p(x) + λũ(x) + ζ(x) =, if we define a perturbation function ζ in these points by ζ(x) = [ p(x) + λũ(x)]. In the points, where ũ(x) = α holds, the inequality p(x) + λũ(x) should be satisfied for optimality. If not, then p(x) + λũ(x) + ζ(x) is fulfilled with ζ(x) = [ p(x) + λũ(x)], where [a] is defined for a real number a by [a] = ( a a)/2. Analogously, we define ζ in the points, where ũ(x) = β. In this way, we obtain the following definition of ζ that is adopted from [2], [ p(x) + λũ(x)], if ũ(x) = α, ζ(x) := [ p(x) + λũ(x)], if α < ũ(x) < β, [ p(x) + λũ(x)] +, if ũ(x) = β. (1.5) (1.6) The main idea behind the definition of ζ is the following: Although ũ will possibly not be optimal for (PQ), it is optimal for the perturbed optimization problem min f(u) := 1 u K 2 Su y H 2 H + λ 2 u 2 L 2 (D) + ζ(x) u(x) dx. (PQ ζ ) D

7 5 This follows from our construction, because ũ satisfies the associated variational inequality D ( p(x) + λũ(x) + ζ(x))(u(x) ũ(x)) dx u K. By convexity, ũ is optimal for (PQ ζ ). Now, it is easy to estimate the distance between ũ and ū. Lemma 1.2. Suppose that ū is optimal for (PQ), ũ K is given arbitrarily, and ζ is defined by (1.6). Then there holds ũ ū L2 (D) 1 λ ζ L 2 (D). Proof. We just write down the variational inequalities satisfied by ū and ũ, and insert the other function, respectively. This yields Adding both inequalities, we arrive at (S (Sū y H ) + λū, ũ ū) L2 (D) (S (Sũ y H ) + λũ, ū ũ) L2 (D) + (ζ, ū ũ) L2 (D). S(ū ũ) 2 L 2 (D) λ ū ũ 2 L 2 (D) + (ζ, ū ũ) L 2 (D), and hence, since S(ū ũ) 2 L 2 (D), λ ū ũ 2 L 2 (D) ζ L 2 (D) ũ ū L 2 (D). Now the claimed result follows immediately. Notice that, given ũ, ζ can be computed. It is determined by the adjoint state p associated with ũ Application to a linear-quadratic control problem As a simple application, we consider the following linear-quadratic version of (PD), min J(y, u) := 1 2 T { (y(x, t) yd (x, t)) 2 + λ u(x, t) 2} dxdt (PL) subject to and to the pointwise control constraints y y = u in (, T ) t y = in Γ (, T ) y(, ) = in (1.7) α u(x, t) β a.e. in (, T ). We consider y in the space W (, T ) = {y L 2 (, T ; H 1 ()) : dy dt L2 (, T ; H 1 () )},

8 6 cf. Lions [16]. For simplicity, we have assumed y =. The case y is reduced to that with homogeneous initial condition in a standard way: We shift the solution ŷ associated with homogeneous right-hand side u but inhomogeneous initial data y to y H. Then we consider ŷ H := y H ŷ instead of ŷ. Let us introduce the notation Q := (, T ) and Σ := Γ (, T ) and define the set of admissible controls by U ad = {u L 2 (Q) : α u(x, t) β a.e. in Q}. We are going to apply Lemma 1.2 with the choice D := Q, K := U ad, y H := y d. The adjoint state p W (, T ) associated with a control u is obtained from the adjoint equation p t p = y u y d in (, T ) p = in Γ (, T ) p(, T ) = in, (1.8) where y u denotes the state associated with u. It is a standard result that (PL) has a unique optimal control ū. Let now ũ U ad be given. For instance, we might think of a control obtained as the solution to a model reduced optimal control problem. In our numerical test case, the model reduced problem is defined upon POD. To estimate the distance between ũ and the unknown exactly optimal control ū, we proceed as follows: First, we determine the exact state ỹ := yũ by solving the state equation of (PL) with control u := ũ. Next, we insert ỹ for y u in the adjoint equation (1.8) to obtain the associated adjoint state p as solution. Now we are able to determine the perturbation ζ L 2 (Q) by (1.6) (take x := (x, t) and D := Q). Finally, we arrive at the estimate ũ ū L2 (Q) 1 λ ζ L 2 (Q). In [23, 26], this method of estimation was successfully applied to different optimal control problems with quadratic objective functional and linear state equation. It is also successfully applied for other reduced-basis approximations, see [24]. For recent extension to nonlinear problems we refer to [13], where the presented error estimates are utilized in a multilevel SQP algorithm. Notice that we tacitly assume to solve the state and adjoint equation for ỹ and p exactly, i.e. we neglect associated discretization errors. The inclusion of such errors is subject of ongoing research. 2. The nonlinear case 2.1. The perturbation method for nonconvex functionals We consider now the nonconvex but smooth optimization problem min u K f(u) := 1 2 G(u) y H 2 H + λ 2 u 2 L 2 (D), (P) where G : L (D) H is a twice continuously Fréchet differentiable operator and all other quantities are defined as in (PQ). We assume that, for all u K, the first- and second-order derivatives G (u) : L (D) H and G (u) : L (D) L (D) H can be continuously extended to L 2 (D): There exists some constant c > not depending on u and v such that G (u)v H c v L2 (D) u K, v L (D) (2.9) G (u)(v 1, v 2 ) H c v 1 L2 (D) v 2 L2 (D) u K, v 1, v 2 L (D) (2.1)

9 7 with some constant c >. Then the operators G (u) and G (u) can also be applied to increments v, v 1, v 2 in L 2 (D). Therefore, we can view G (u) as continuous linear operator from L 2 (D) to H so that its adjoint operator maps continuously H to L 2 (D). The derivative f (u) is given by f (u) v = (G(u) y H, G (u) v) H + λ(u, v) L2 (D) = (G (u) (G(u) y H ) + λu, v) L 2 (D). (2.11) The L 2 (D)-function p u := G (u) (G(u) y H ) (2.12) is the adjoint state associated with u. Let us determine for convenience also the second-order derivative of f : L (D) R. Thanks to the assumptions on G, this derivative exists. To determine it, we consider the expression for f with fixed increment v := v 1 L (D) and differentiate again in the direction v 2. We find by the chain and product rule f (u)(v 1, v 2 ) = (G (u) v 2, G (u) v 1 ) H + (G(u) y H, G (u)(v 2, v 1 )) H + λ(v 2, v 1 ) L2 (D). (2.13) By our assumptions on G, also f (u) can be continuously extended to a bilinear form on L 2 (D) L 2 (D) and f (u)(v 1, v 2 ) c v 1 L 2 (D) v 2 L 2 (D) u K, v 1, v 2 L (D). Let us now assume that ū K is a locally optimal solution to (P) in the sense of L (D), i.e. there is some ρ > such that f(u) f(ū) u K with u ū L (D) ρ. We obtain the following extension of Theorem 1.1: Theorem 2.1. If ū K is a locally optimal solution of (P) in the sense of L (D), then it obeys the variational inequality {(G (ū) (G(ū) y H ))(x) + λ ū(x)} (u(x) ū(x)) dx u K. (2.14) D This result follows from the variational inequality f (ū)(u ū) u K, cf. e.g. Lemma 4.18 in [25]. Here, we use the representation (2.11) of f (ū). Let now p := pū be the adjoint state defined in (2.12). Analogously to the quadratic problem (PQ), the variational inequality can be expressed in a pointwise form, ( p(x) + λū(x))ū(x) = min ( p(x) + λū(x)) v for a.a. x D. (2.15) v [α,β] This is identical with (1.4), hence we can draw the same conclusions from (2.15) as in the linear case. Therefore, the perturbation ζ is defined exactly as in (1.6). Let now be ũ K sufficiently close to ū. We pose the same question as for (PQ): Can we quantify the distance ũ ū? Now, however, the situation is more complicated. We need some second-order information on ū. Assume that there exists some δ > such that the coercivity condition f (ū)(v, v) δ v 2 L 2 (D) v L 2 (D) (2.16)

10 8 is satisfied. Then for any < δ < δ there exists a radius r(δ ) > such that f (u)(v, v) δ v 2 L 2 (D) u with u ū L (D) r(δ ), v L 2 (D). (2.17) If ũ belongs to this neighborhood, then we are able to estimate the distance. Remark 2.2. To be on the safe side, we might select δ := δ/2 and set r := r(δ/2). This can be too pessimistic. In the application to POD we are mainly interested in the order of the error so that the factor 1/2 is not important. We use δ := δ, although this can be slightly too optimistic. Theorem 2.3. Let ū be locally optimal for (P) and assume that ū satisfies the second-order condition (2.16). If ũ K is given such that ũ ū L (D) r, where r is the radius introduced in (2.17) for δ := δ/2, then it holds where ζ is defined as in (1.6). ũ ū L2 (D) 2 δ ζ L 2 (D), Proof. As in the convex quadratic case, ũ satisfies the first-order necessary optimality conditions for the perturbed optimization problem min f(u) + (ζ, u) L u K 2 (D). (P ζ ) We insert ū in the variational inequality for ũ and ũ in the one for ū and obtain We add both inequalities and find The mean value theorem implies (f (ũ) + ζ, ū ũ) L2 (D) (f (ū), ũ ū) L2 (D). (f (ũ) f (ū), ū ũ) L 2 (D) + (ζ, ū ũ) L 2 (D). f (û)(ū ũ) 2 + (ζ, ū ũ) L2 (D) with some û {v L 2 (D) v = sū + (1 s)ũ with s (, 1)}. Now we apply (2.17) to the left-hand side and the Cauchy-Schwarz inequality to the right-hand side to deduce δ 2 ũ ū 2 L 2 (D) ζ L 2 (D) ũ ū L 2 (D). From this inequality, the assertion of the theorem follows in turn Application to the parabolic boundary control problem (PB) Our general result can be applied in various ways. Let us explain its use for the parabolic boundary control problem (PB). From the theoretical point of view, (PB) is more difficult than the distributed problem (PD). For (PB), the control-to-state mapping G is not differentiable from L 2 (Σ) to C( Q). Here, the two-norm discrepancy is a genuine issue. In (PD), this problem does not occur for the spatial dimension n = 1, while it also arises for n 2. Therefore, we selected (PB) for our analysis. We define U ad = {u L (Σ) : α u(x, t) β for a.a. (x, t) Σ}. (2.18) The following result is known on the solvability of the state equation:

11 Theorem 2.4. For all u U ad, the state equation (.2) has a unique solution y u Y := W (, T ) C( Q). The mapping G : u y u is twice continuously differentiable from L (Σ) to Y. In particular, it is twice continuously differentiable from L (Σ) to H := L 2 (Q). The first- and second-order derivatives G and G obey the extension conditions (2.9), (2.1) for the choice D := Q, K = U ad. We refer to [25], Theorems 5.9 and The extension conditions follow immediately from the representation formulas for these derivatives stated in these theorems. The same theorems include also the result for the problem (PD). Here, the reduced functional is given by f(u) := 1 T u (x, t) y d (x, t)) 2 (y 2 dxdt + λ T u(x, t) 2 ds(x)dt. 2 Γ Analogously to (2.11) and (2.12), the first derivative of the reduced functional can be expressed in the form f (u) v = T Γ (p u (x, t) + λu(x, t))v(x, t) ds(x)dt, (2.19) where p u is the adjoint state associated with u. It is the unique solution to the adjoint equation p t p = y u y d in (, T ) p ν + b y (x, t, y u) p = in Γ (, T ) p(x, T ) = in. 9 (2.2) In view of our Theorem 2.1, any locally optimal control ū of (PB) has to obey the following variational inequality: T Γ ( p(x, t) + λū(x, t))(u(x, t) ū(x, t)) ds(x)dt u U ad, (2.21) where p := pū. The general form of the second derivative of the reduced functional was determined in (2.13). In this representation, terms of the form G (u)v appeared. For given v L 2 (Σ), the function y = G (u)v is the unique solution of the linearized parabolic equation y y = in (, T ) t y ν + b y (x, t, y u) y = v in Γ (, T ) y(x, ) = in, (2.22) where y u = G(u) is the state associated with u, cf. [25, Theorem 5.9]. Therefore, the mapping v y is continuous from L 2 (Σ) to W (, T ), in particular to L 2 (Q); this yields the first condition of continuous extension (2.9). The second derivative z = G (u)(v 1, v 2 ) is the unique solution to the same equation with v substituted by v := 2 b y 2 (x, t, y u)y 1 y 2, where y i = G (u)v i, i = 1, 2, and z z = in (, T ) t z ν + b z (x, t, y u) z = 2 b y 2 (x, t, y u)y 1 y 2 in Γ (, T ) z(x, ) = in. (2.23)

12 1 This is a linear equation with control 2 b y 2 (x, t, y u)y 1 y 2. Therefore, it is not difficult to see that the second term in the representation (2.13) for f (u) can be re-written in terms of the adjoint state. Namely, it holds f (u)v 2 = J (y u, u)(y, v) 2 T = y 2 dxdt + λ T T Γ 2 b p u y 2 (x, t, y u) y 2 ds(x)dt, T v 2 ds(x)dt Γ p u 2 b y 2 (x, t, y u) y 2 ds(x)dt. (2.24) We refer, for instance to [25, Section 5.7] or to the recent paper [5], where second-order derivatives are determined for more general quasilinear equations. The second extension condition (2.1) follows immediately from (2.23). In (2.24), y is the solution to the linearized equation (2.22). The expression above is just the second-order derivative of the Lagrangian function defined upon the adjoint state p u associated with u. Therefore, the second-order sufficient optimality condition (2.16) means that T y 2 dxdt + T Γ { λ v 2 2 } b p u y 2 (x, t, y u) y 2 ds(x)dt T δ v 2 ds(x)dt v L 2 (Σ). Γ (2.25) Since y is the solution of the linearized equation that is associated to v and the left-hand side of this equation is the second derivative of the Lagrangian function, this expresses a well known formulation of the (strong) second-order condition: The second derivative of the Lagrangian function is assumed to be coercive on the subspace defined by the linearized equation Application of the perturbation method Let now ũ U ad be an arbitrary control that is close to a locally optimal control ū of (PB). We assume that ū satisfies the sufficient second-order optimality condition with some δ >. In other words, the coercivity condition (2.25) is fulfilled for u := ū. Since the second-order condition is stable with respect to L -perturbations of ū, we are justified to assume that there exists some ρ > such that f (u)v 2 δ 2 T Γ v 2 ds(x)dt v L 2 (Σ), u B ρ (ū). (2.26) We assume that ũ belongs to B ρ (ū). Now we define the perturbation function ζ as in (1.6), (x, t) [, T ] is substituted for x. Thanks to Theorem 2.3, we obtain ũ ū 2 L 2 (Σ) 2 δ ζ 2 L 2 (Σ). (2.27) We should mention already here a serious theoretical obstacle that can hardly be rigorously overcome. To apply our method of a posteriori estimation, we need the numbers ρ and δ. As ρ is concerned, we can only assume that the method of determining the (suboptimal) control ũ was sufficiently precise so that ũ ū < ρ. There exists a method by Rösch and Wachsmuth [2], [29] to verify if there exists a locally optimal control ū in a certain neighborhood of a given ũ. However, this tool is difficult to apply. It will work only for fairly special problems. In addition, we must determine the coercivity constant δ. In computations, we deal with a finite-dimensional approximation of the control problem and determine the smallest eigenvalue of the reduced Hessian. In general, this way is not reliable in estimating the coercivity constant δ for the infinite-dimensional undiscretized optimal control problem: In [29], a discouraging example is constructed, where computations with very small mesh size

13 indicate a sufficient second-order condition for a point, which is a saddle point. In special cases, the ideas of [2] and [29] might be applied that rely on deep analytical estimations. Summarizing this issue, we must confess that there is no reliable and at the same time practicable method to verify our assumptions absolutely certain. We have to trust that our problem behaves well in a neighborhood of the unknown solution ū. This is similar to problems of nonlinear programming, where optimization routines are started without evidence whether a constraint qualification is satisfied at the unknown solution or not. We assume in the sequel that our computed suboptimal control belongs to a neighborhood of a locally optimal solution ū that satisfies a second-order sufficient optimality condition. Moreover, we assume that we are able to determine the order of the coercivity constant δ by the lowest eigenvalue of the Hessian matrix associated with the suboptimal control Model reduction by POD In this section, we give a very brief survey on how to establish a reduced order model of the parabolic boundary control problem (PB) by applying standard POD. For more information and proofs we refer the reader to Kunisch and Volkwein [14] or Volkwein [28], for instance. A survey on methods of model reduction and a comparison of different reduction methods is contained in Antoulas [1] The discrete POD method Define H = L 2 () and let u L 2 (Σ) be an arbitrary control with associated state y u = G(u) Y L 2 (, T ; H). Since we cannot compute the whole trajectory y(t) for all t [, T ], we define a partition = t < t 1 <... < t n = T of the time interval [, T ] and compute shapshots y i := y(t i ) H, i = 1,..., n, of the state y at the given time instances t 1,..., t n. Our goal is to find a small Galerkin basis that well expresses the main properties of the PDE in the finite dimensional subspace H n defined by H n = span {y(t i ) i =,..., n} H with dimension d = dim H n. Let {φ 1,..., φ d } be a basis of H n that is orthonormal with respect to the inner product of L 2 (). Then every y i = y(t i ), i =,..., n, is a linear combination of φ 1,..., φ d, y i = d (y i, φ j ) H φ j. j=1 Now choose a fixed number l {1,..., d}. In the application to model reduction, l is small compared with d. We intend to find an orthonormal set of functions {ϕ j } l j=1 H such that the sum of squared errors ε i = y i l 2 (y i, ϕ j ) H ϕ j is minimized. In other words, we consider the optimization problem j=1 H min ϕ 1,...,ϕ l H n yi α i i= j=1 l 2 (y i, ϕ j ) H ϕ j =: J n (ϕ 1,..., ϕ l ), H (3.28) s.t. (ϕ i, ϕ j ) H = δ ij i, j {1,..., l},

14 12 where α,..., α n are the trapezoidal weights α = (t 1 t )/2, α i = (t i+1 t i 1 )/2, for i = 1,..., n 1, α n = (t n t n 1 )/2. The solution to (3.28) is obtained by solving a certain eigenvalue problem, described by the linear operator R n : H H n that maps z H to n R n z = α i (z, y i ) H y i. i= The following proposition characterizes the operator R n and shows that the eigenfunctions to the l largest eigenvalues of R n solve the problem (3.28). Proposition 3.1. Let H be a Hilbert space, y C([, T ], H) and H n = span {y(t i ) i =,..., n} H with dimension d = dim H n. Then the following statements hold: (i) The operator R n : H H n is bounded, self-adjoint, compact and non-negative. (ii) There exist a sequence of eigenvalues {λ n i }d with λn 1 λ n 2... λ n d and associated eigenelements H that solve the eigenvalue problem {ψ n i }d R n ψ n i = λ n i ψ n i, i = 1,..., d. (3.29) (iii) For any fixed number l {1,..., d}, the minimization problem (3.28) is solved by the eigenfunctions {ψ n 1,..., ψ n l }. (iv) The corresponding minimal value of J n is given by inf J n (ϕ 1,..., ϕ l ) = J n (ψ1 n,..., ψl n ) = ϕ 1,...,ϕ l H For a proof we refer to e.g. Volkwein [28], [27, Section 1.3] and Holmes et al. [11, Section 3]. Note that we have R n = YY, where the bounded linear operator Y : R n H n and its Hilbert space adjoint operator Y : H R n are defined by Yϕ = d i=l+1 λ n i. n α i ϕ i y i, (Y z) i = (z, y i ) H, i = 1,..., n. i= Analogously to the theory of singular value decomposition for matrices, the linear, bounded, compact, and self-adjoint operator K n = Y Y : R n R n defined by (K n ϕ) i = n α j (y j, y i ) H ϕ j, i = 1,..., n, j=1 has the same eigenvalues {λ n i }d as Rn. The corresponding eigenfunctions of K n are obtained from the ψi n transformations (Φ n i ) j = 1 (Y ψ λ n i n ) j = 1 (ψ i λ n i n, y j ) H, j {1,..., n}, i by for all i {1,..., l}. In many cases, the number n of time instances is much smaller than the number d of spatial ansatz functions. Then it is convenient to solve the eigenvalue problem instead of using the operator R n. K n Φ n i = λ n i Φ n i, i = 1,..., d, (3.3)

15 Now we apply a standard finite element approximation with respect to the spatial variable x. For this purpose, let T h be a family of regular triangulations of with mesh size h >. For each T h, let V h L 2 () be the set of piecewise linear finite element basis functions {ϕ 1,..., ϕ m }, m N. Then for every function z V h there exist z 1,..., z m R such that z(x) = m z j ϕ j (x), for all x. (3.31) j=1 Thanks to our numerical approximation of the PDE, we can assume that our snapshots y i belong to V h. Therefore, for every i {1,..., n}, m y i = y j i ϕ j. j=1 For convenience, we introduce the vectors y i = (y 1 i,..., ym i ). In this way, we have the correspondence V h y i y i R m. Let z V h be given. Then 13 (R n z)(x) = n α i (y i, z) L2 () y i(x). i= By definition, we have R n : H H n V h. Let us denote for convenience the restriction of R n to V h with range in V h by the same symbol so that R n : V h V h. Applying equation (3.31) for every node x 1,..., x m of the triangulation of, we infer for V h z z R m and V h y i y i R m (R n z)(x j ) = = = = n i= k=1 n m α i yi k ϕ k (x j ) i= l,ν=1 n i= l,ν=1 m α i y j i yl i z ν m yiz l ν l,ν=1 ϕ l (x)ϕ ν (x)dx m α i Y ji Y li (M h ) lν z ν = ϕ l (x)ϕ ν (x)dx n α i Y ji i= n α i Y ji Y M h z = ( Y D Y M h z ), j i= m l,ν=1 Y il (M h ) lν z ν where y i = (yi 1,..., ym i ), i = 1,..., n, Y = [ y 1... y n ], D = diag(α 1,..., α n ). In the first line, we used ϕ k (x j ) = δ kj. Moreover, M h with entries (M h ) k,l = ϕ k (x)ϕ l (x)dx is the mass matrix associated with the finite element basis functions. Define R n : R m R m by R n z := n α i ( y i M h z) y i. i= Since (R n z)(x j ) = = n n α i (y i, z) L2 () y i(x j ) = α i y i M h z y i (x j ) ( i= n n ) α i ( y i M h z) y j i = α i ( y i M h z) y i = ( Rn z ) j, i= i= i= j

16 14 the operator R n corresponds to the matrix R n in the numerical approximation, i.e. we have R n z = Y D Y M h z. To obtain an optimal solution of (3.28), we solve the eigenvalue problem R n ψ n i = Y DY M h ψ n i = λ n i ψ n i, i = 1,..., l. (3.32) Defining Ŷ = M 1 2 h Y D 1 2, ˆψ n i = M 1 2 h ψ n i and multiplying (3.33) by M 1 2 h from the left, we arrive at the symmetric eigenvalue problem Ŷ Ŷ ˆψn i = λ n i ˆψ n i, i = 1,..., l. Using similar arguments for the operator K, we also can solve the discretized eigenvalue problem K n Φ n i = Y M h Y D Φ n i = λ n i Φ n i, i = 1,..., l. (3.33) Defining ˆΦ n i = D 1 2 Φ n i and multiplying (3.33) by D 1 2 from the left, we infer Ŷ Ŷ ˆΦ n i = λ n ˆΦ i n i, i = 1,..., l. Summarizing, we compute the POD basis of rank l by the method of snapshots as follows: Algorithm 3.2 (Method of Snapshots). Let m n. (1) Solve the symmetric n n eigenvalue problem Ŷ Ŷ ˆΦ n i = λ n ˆΦ i n i, i = 1,..., l. (2) Transform the eigenvectors ˆΦ n 1,..., ˆΦ n l by singular value decomposition to the POD basis vectors ψ n 1,..., ψ n l ψ n i = M 1 2 h ˆψ i n = 1 M 1 2 λ n i h Ŷ ˆΦ n i = 1 λ n i Y D 1 2 ˆΦn i, i {1,..., l}. Then, the POD basis functions V h ψi n ψ i n R m, i = 1,..., l, are given by m ψi n = ( ψ i n ) j ϕ j (x). j=1 Remark 3.3. (1) In the method of snapshots, we do not have to set up the matrix M 1 2 h, since Ŷ Ŷ = D 1 2 Y M h Y D 1 2. (2) Note that (3.32) is an m m eigenvalue problem, whereas (3.33) has dimension n n. In many applications we have m n. Hence, the operator K n is preferred for generating the POD basis. In the case, m n, one should solve the problem (3.32) instead. On the other hand, the singular value decomposition is much more stable in computing the small eigenvalues, see e.g. [15, 23] for this aspect in the context of POD.

17 3.2. POD Galerkin projection Let us consider exemplarily the state equation (.2) of our boundary control problem (PB). As above, we denote by V h the space of piecewise linear functions spanned by the FE basis functions {ϕ 1,..., ϕ m } defined on a regular triangulation T h of. For convenience, we use the short notation (, ) := (, ) L 2 () and (, ) Γ := (, ) L2 (Γ) for inner products. Then the variational formulation of the equations (.2) is given by 15 d dt (y(t), ϕ) + ( y(t), ϕ) + (b(, t, y(t)), ϕ) Γ = (u(t), ϕ) Γ (y(), ϕ) = (y, ϕ), (3.34) for all ϕ H 1 () and for almost all t [, T ]. Under our assumptions on the given data, to each u L r (Γ (, T )) with r > 2, there exists a unique solution y W (, T ) C( Q) of (.2). Let now {t i } n i= be an equidistant partition of [, T ]. To generate the snapshots we select a fixed reference control u s L r (Γ) and solve equation (3.34) at the given time instances t i. We denote the snapshots by ys i = y s (t i ), i =,..., n and, as before, H n = span {y s (t i ) i {,..., n}} with dimension d N. Next, we fix a small natural number l d and compute the POD basis functions {ψ1 n,..., ψl n} as described in Section 3.1. Notice that all ψn i, i = 1,..., l, are linear combinations of the FE basis functions ϕ 1,..., ϕ m. Applying a Galerkin scheme based on the (small) POD basis, we obtain a nonlinear initial value problem of finding a function y l with l y l (t) = η i (t)ψi n, such that d dt (yl (t), ψi n ) + ( y l (t), ψi n ) + (b(, t, y l (t)), ψi n ) Γ = (u(t), ψi n) Γ (3.35) (y l (), ψi n) = (y, ψi n) is satisfied for almost all t [, T ] and all i {1,..., l}. This is the state equation of a low size optimal control problem with state space of dimension l, see problem (PB.1 l ) below. This small optimal control problem is then solved to obtain a suboptimal control ū l. Remark 3.4. It is not obvious that the reduced equation (3.35) has a unique solution, because the reduced Galerkin basis might destroy the monotonicity of the nonlinearity. To avoid this theoretical difficulty, we can truncate b as follows: b(x, t, c) if y c b c (x, t, y) = b(x, t, y) if y c/2 b(x, t, c) if y c, where b c is defined in ( c, c/2) (c/2, c) such that b c is C 2 with respect to y. Then b c is uniformly Lipschitz with respect to y so that the theorem by Picard and Lindelöf permits to show existence and uniqueness of a solution of (3.35) for all T >. In our computations, the function y l was uniformly bounded so that we did not follow this idea. 4. A posteriori error estimation for POD solutions To apply our a posteriori error estimation technique described in Section 2, we select a solution of some POD reduced optimal control problem. This is convenient, since model reduction by POD does not provide a rigorous a priori error analysis. Nevertheless, it often gives excellent results. This will be illustrated in the following two numerical examples.

18 Example 1 We first discuss a one-dimensional boundary control problem that is governed by a semilinear parabolic equation. The boundary control u is to drive a spatio-temporal temperature distribution to a predefined, desired temperature distribution at the final time T. Let = (, 1), T = 1.58 and Q := (, T ). The optimal control problem is given by min ( y(x, T ) 1 ) 2 2 (1 1 T x2 ) dx + u(t) 2 dt =: J (y, u) 2 (PB.1) subject to the heat equation with Stefan-Boltzmann type boundary condition y t (x, t) y xx (x, t) = in Q y x (, t) = in (, T ) y x (1, t) + y(1, t) + y 4 (1, t) = u(t) in (, T ) y(x, ) = in (4.36) and to the bilateral control constraints 1 u(t) 1, t (, T ). This is a well-known test example that goes back to Schittkowski [22]. Remark 4.1. Formally, the function y y 4 does not fulfill our assumption of monotonicity. Therefore, often y y y 3 is considered instead. In our numerical tests, all occuring state functions y were non-negative. Therefore, we are justified to write y 4. The corresponding adjoint equation is defined by p t (x, t) p(x, t) = p x (, t) = p x (1, t) + p(1, t) + 4 y 3 (1, t)p(1, t) =, p(x, T ) = y(x, T ) y d (x). (4.37) To reduce the state equation by POD, we selected an equidistant partition of the time interval, = t < t 1 <... < t n = T, and a piecewise linear finite element discretization on the interval with standard FE basis ( hat ) functions {ϕ 1,..., ϕ m }. A semi-implicit Euler scheme in time was then used to solve the resulting semidiscrete equation (3.35). Associated with the fixed control u.5, the shapshots { y j = y(t j ) = m y i jϕ i}n j= V h L 2 () were generated. Now we collect the snapshot vectors y j = (y 1 j,..., ym j ), j =,..., n, in the matrix Y = [ y... y n ] R m n. As above, M denotes the mass matrix associated with {ϕ 1,..., ϕ m } with entries M ij = (ϕ i, ϕ j ) L2 (), i, j {1,..., m}. For convenience, we ignore the dependence on h. Furthermore, let t = T/n and α,..., α n be the trapezoidal weights α = α n = t/2, α i = t, for i = 1,..., n 1.

19 Define D = diag(α,..., α n ). Then the small orthonormal system {ψ 1,..., ψ l } is obtained from the eigenvalue decomposition of Y MY D, see Section 3.1, Algorithm 3.2. We use the POD Galerkin ansatz for both the state y from (4.36) and the corresponding adjoint state p that solves the adjoint equation (4.37), y l (x, t) = l y i (t)ψ i (x), p l (x, t) = The associated POD reduced optimal control problem is subject to min l p i (t)ψ i (x). ( y l (x, T ) 1 ) 2 2 (1 1 T x2 ) dx + u(t) 2 dt (PB.1 l ) 2 17 and M l t y(t) + (Kl + F l ) y(t) + G l ( y(t)) = B l u(t) 1 u(t) 1, y() =, (4.38) for almost all t (, T ). Here, y : [, T ] R l denotes the vector function y(t) = (y 1 (t),..., y l (t)). The matrices M l, K l, F l R l l, B l R l are given by M l ij = (ψ i, ψ j ) L 2 (), K l ij = (ψ i, ψ j) L 2 (), F l ij = ψ i (1)ψ j (1), B l i = ψ i (1), and G l : R l R l is defined by y (y B l ) 4 B l. The adjoint state p : [, T ] R l associated with y is the unique solution of the equation M l t p(t) + (Kl + F l ) p(t) + 4H l ( y(t))f l p(t) = p(t ) = y(t ) y d. (4.39) with H l : R l R l given by y (y B l ) 3 B l. The state equation of the associated discretized optimal control problem is solved again by a semi-implicit Euler scheme and the control functions u are chosen as step function according to the given partition {t,..., t n } by u(t) = u i, t i 1 t < t i, i = 1,..., n. Therefore, we have the equivalence u( ) (u 1,..., u n ) = u R n. This discretized optimal control problem was solved by an SQP method, where the associated sequence of linear-quadratic control problems systems was treated by a primal-dual active set strategy. Let us denote by ū l = (ū l 1,..., ū l n) the obtained locally suboptimal control vector, and by ū l τ ( ) ū l the corresponding step function. We are interested in estimating the distance between ū l τ and the next locally optimal control ū. For this purpose, we apply our perturbation method that was formulated in Section 2.3. This needs two full PDE solves. First, we have to determine the state ȳ l = G(ū l τ ), i.e. the solution of the parabolic boundary value problem (4.36) associated with ū l τ. Second, we must compute the solution p l of the full adjoint equation (4.37). Then the time-dependent perturbation ζ l is also a step function on [, T ]. It is equivalent to the vector (ζ 1,..., ζ n )

20 18 defined by for all i {1,..., n} and [ ] 1 ti p l (1, t)dt + λ ū l i, if ū l i τ = 1, t [ i 1 ] 1 ti ζ i = p l (1, t)dt + λ ū l i, if 1 < ū l i τ < 1, t [ i 1 ] 1 ti p l (1, t)dt + λ ū l i, if ū l i τ = 1. t i 1 + ζ l (t) = ζ l i, t i 1 t < t i. (4.4) We assume that the suboptimal control ū l τ belongs to an L -ball around a locally optimal control ū, where the coercivity condition (2.26) is satisfied. In other words, we assume that ū l τ is sufficiently precise and the optimal control problem behaves well around ū. Then we need an approximation of the coercivity constant δ. This will be accomplished by that of the reduced Hessian matrix Ψ l, associated with the suboptimal control ū l. We have to assume that Ψ l is positive definite. Let be σmin l the smallest eigenvalue of Ψl. Then there holds for all vectors u R l and their associated step functions u τ that u Ψ l u σmin u l 2 2 = σl min u τ 2 L τ 2 (,T ). Therefore, if the control problem behaves well around ū, the coercivity constant δ can be approximated by /τ. Assuming that σ σ l min l min δ τ holds, we can deduce that the distance of ū l to the unknown locally optimal control ū can be estimated by ū l ū L2 (,T ) τ σ l min ζ l L2 (,T ). In our numerical test, we computed a sufficiently precise approximation of ū, since this control was not exactly known. For this purpose, the interval = (, 1) was split in 4 intervals [x j, x j+1 ), j = 1,..., m 1, (step size h = 1 4 ), and we selected a time step size τ = T n = T 2. The solution of the accordingly discretized optimal control problem (PB.1) was the function ū τ shown in Figure 1. Table 1 compares the a posteriori error estimate for ū ū l τ L 2 (,T ) determined by our method with the exact numerical error ū τ ū l τ L 2 (,T ). l ū τ ū l τ τ L 2 (,T ) ζ l σmin l L 2 (,T ) σmin l e e e e e-2 8.6e e e e e e e e e e e e e-4 Table 1. Example 1: Numerical errors ū τ ū l τ L2 (,T ) and error estimates for ū ū l τ L2 (,T ) with corresponding eigenvalues σ l min, l = 1..., 6.

21 19 Figure 1. Example 1: Optimal FE control u τ. Remark 4.2. So far, we have tacitly assumed in our error estimates that, to determine a perturbation ζ, we are able to solve the state equation and the adjoint equation exactly. This is not possible in numerical applications. In solving these equations, numerical errors are unavoidable. There are two sources of errors, which are related to the approximation in time and the approximation in space. It is known from an early paper by Malanowski [17] and a recent publication by Neitzel and Vexler [19] that the error between a locally optimal control ū and the corresponding discretized locally optimal control ū τ,h satisfies the a priori error estimate ū ū τ,h L 2 (Q) C (τ + h 2 ) with some constant C. In our method, we want to estimate the error ū ū l τ L 2 (,T ), but actually we estimate the norm of ū τ,h ū l τ. By the triangle inequality and the a priori error estimate mentioned above, we obtain ū ū l τ L 2 (,T ) ū ū τ,h L 2 (,T ) + ū τ,h ū l τ L 2 (,T ) C (τ + h 2 ) + ū τ,h ū l τ L 2 (,T ) C τ + ū τ,h ū l τ L 2 (,T ) (4.41) since in our computations we have τ h 2. To make this estimate work, we need an estimation for C that is described below. Let us tacitly assume that the discretization in space can be neglected and denote by ū τ the solution of the full system discretized only in time. To determine C in (4.41), we solve the (full size) discretized optimal control problem with mesh sizes τ and τ/2, respectively. Then it holds so that we can expect that ū τ ū τ/2 L2 (,T ) ū τ ū L 2 (,T ) + ū ū τ/2 L2 (,T ) C τ + C τ 2 = 3τ 2 C C 2 3τ ū τ ū τ/2 L2 (,T ). In our example above, we applied this rough estimation technique by solving the problem with mesh sizes τ = T/2 and τ/2 = T/4, respectively. We obtained C.2928 and therefore the discretization error can be estimated by ū ū τ L2 (,T )

22 2 According to Table 1, this is of higher order than the error ū τ ū l τ L 2 (,T ). Nevertheless, the computations show that a more detailed analysis on the influence of the discretization errors is desirable. Here, we do not consider this issue. By the same SQP type optimization algorithm as for the full discretized problem, the POD-reduced optimal control problem (PB.1 l ) with l = 4 was solved within 1.2 seconds, whereas the optimization of the full problem (PB.1) needed 143 seconds, see Table 2. It turns out that the difference between the minimal values of the objective function J for the full FE solution and the POD solutions with l = 4 only amounts to the order FEM POD (l = 4) CPU time for optimization 143 s 1.2 s Minimal value of J Table 2. Example 1: CPU times for POD optimization and corresponding minimal values of the objective function in comparison with FE method Example 2: A distributed control problem In our second example, we consider a distributed 2D optimal control problem, which is essentially more demanding, since distributed controls have more direct influence on the solution of the parabolic equation than boundary controls. Moreover, the dimension two of increases the computational effort. The system dynamics of the problem is given by a semilinear parabolic partial differential equation in = (, π) 2. We constructed the example in such a way that we know the exact optimal solution. Here, a desired spatio-temporal heat distribution y Q is to be pursued as close as possible. Let be T = 1 and Q := (, T ). Four time dependent controls u 1,..., u 4 L 2 (, T ) are given that are distributed to by some weight functions α i L 2 (), i = 1,..., 4. Thus, we consider a distributed control u of the form u(x, t) = 4 α i (x)u i (t). (4.42) Furthermore, we define some academic functions a y L 2 () and d L 2 (Q), which help us constructing an explicitely known optimal solution. Our objective functional to be minimized is the function J (y, u 1,..., u 4 ) = 1 2 y y Q 2 L 2 (Q) + (a y, y(, T )) L 2 () u i 2 L 2 (,T ). (PD.2) The state equation is the semilinear heat equation y t (x, t) y(x, t) + y3 (x, t) + d(x, t) = 4 α i (x)u i (t) in Q y (x, t) ν = in Σ y(x, ) = y (x) in (4.43) with given initial function y L 2 () and homogeneous Neumann boundary condition. Moreover, we impose pointwise box constraints on the four controls, u i (t) 1, a.e. in (, T ), i = 1,..., 4.

23 Remark 4.3. The problem (4.43) does not exactly fit in problem (.1). First, we have a Neumann boundary condition instead of a Dirichlet one. Second, the control function has the form (4.42). However, the theory is similar to that explained at the beginning of the paper. Moreover, we detail the necessary changes. The treatment of homogeneous Neumann boundary conditions is completely analogous to the Dirichlet case. All given parameters are chosen such that locally optimal solutions are known in advance. For this purpose, we define with x = (x 1, x 2 ) y (x) = cos(x 1 ) cos(x 2 ), y Q (x, t) = cos(x 1 ) cos(x 2 )(1 +.2(t t 2 ).3 t 2 cos 2 (x 1 ) cos 2 (x 2 )), a y (x) =.1 cos(x 1 ) cos(x 2 ), α 1 (x) = max{, 1 5(x y)(x + y π 2 )}, α 2 (x) = max{, 1 5(x y π 2 )(x + y π)}, α 3 (x) = max{, 1 5(x y)(x + y 3π 2 )}, α 4 (x) = max{, 1 5(x y + π 2 )(x + y π)}, { } β i (t) = IP [ 1,1] t 2 α i (x) cos(x 1 ) cos(x 2 )dx, i = 1,..., 4, d(x, t) = 4 α i (x)β i (t) 2 cos(x 1 ) cos(x 2 ) cos 3 (x 1 ) cos 3 (x 2 ). The graphs of the four functions α 1,..., α 4 are shown in Figure 2. Consequently, locally optimal controls 21 can be specified exactly as Figure 2. Example 2: Distributing functions α 1,..., α 4. ū i (t) = IP [ 1,1] { t 2 } a i (x) cos(x 1 ) cos(x 2 )dx, i = 1,..., 4.

24 22 They are shown in Figure 3. The associated (locally) optimal state ȳ and the corresponding adjoint state p are given by ȳ(x, t) = cos(x 1 ) cos(x 2 ), p(x, t) = 1 1 t2 cos(x 1 ) cos(x 2 ). Figure 3. Example 2: Optimal controls ū 1, ū 2. For symmetry reasons it holds ū 1 = ū 3, ū 2 = ū 4. The domain is discretized by a regular triangulation so that an FE space of m = 687 piecewise linear ansatz functions is obtained. For the snapshot generation we took an equidistant partition of the time interval [, T ] with step size τ = 1/119, so that n = 12 snapshots were computed. Then a number l was chosen, the POD basis {ψ i } l was computed as explained in Section 3.1 and we applied the standard POD Galerkin ansatz for both the state y and the adjoint state p, y l (x, t) = l y i (t)ψ i (x), p l (x, t) = as well as for the functions α 1,..., α 4 L 2 () and d L 2 (Q), d l (x, t) = l d i (t)ψ i (x), αk(x) l = to obtain the reduced order model (PD.2 l ) subject to min 1 2 l p i (t)ψ i (x), l ( α k ) i ψ i (x), k = 1,..., 4, y l y Q 2 L 2 (Q) + ( a y, y l (, T ) ) L 2 () M l 4 t y(t) + Kl y(t) + G l ( y(t)) + M l d(t) = M l α i u i (t) 4 u i 2 (PD.2 l ) L 2 (,T ) (4.44) y() =, and u i (t) 1, i = 1,..., 4, for almost all t (, T ). As above, y : [, T ] R l denotes the vector function y(t) = (y 1 (t),..., y l (t)), and the matrices M l, K l R l l are given by M l ij = (ψ i, ψ j ) L2 (), K l ij = ( ψ i, ψ j ) L2 (), i, j = 1,..., l,

25 23 and G l : R l R l is defined by ( G l (y) ) i = l j,k,l=1 y j y k y l ψ j (x)ψ k (x)ψ l (x)ψ i (x), i = 1,..., l. After discretizing the semidiscrete problem (PD.2 l ) by the semi-implicit Euler scheme with an equidistant partition {t i } n i= of [, T ], we applied an SQP method for solving the discretized reduced problem. In this context, the integration of linearized terms such as (y k (x, t)) 2 y(x, t)ψ j (x)dx = = ( l ) 2 l yi k (t)ψ i (x) y l (t)ψ l (x)ψ j (x)dx, l i,µ,l=1 yi k (t)yµ(t)y k l (t) l=1 ψ i (x)ψ µ (x)ψ l (x)ψ j (x)dx We applied a prismoidal formula, i.d. we interpolated the node data of the POD basis functions ψ 1,..., ψ l linearly to the triangle midpoints x τν of the finite elements τ ν such that the cubature is given by ψ i (x)ψ µ (x)ψ l (x)ψ j (x)dx m ν=1 A τν ψ i (x τν )ψ µ (x τν )ψ l (x τν )ψ j (x τν ), where the corresponding areas of the finite elements are denoted by A τν and m is the number of triangles of the approximation of. The linear-quadratic subproblems in each level of the SQP method were treated by the primal-dual active set strategy. Remark 4.4. We should mention in this context that this ad-hoc method can certainly be improved by the application of the discrete empirical interpolation method (DEIM) as it was suggested by Patera [3] or Sorensen [6]. Let us also refer to the recent paper [15]. However, we did not concentrate on this technique, because our main concern is the problem of a posteriori error estimation. Now denote the obtained POD-optimal control set by (ū l 1,..., ū l 4) and by ȳ l the associated state. Define the control-to-state mapping G : L (, T ; R 4 ) L 2 (Q) that maps the control terms u = (u 1, u 2, u 3, u 4 ) L (, T ; R 4 ) to the associated state y. We obtain the following corollary of Theorem 2.3 on error estimation: Corollary 4.5. Suppose there are a radius r > and some δ > such that there holds f (u)(v, v) δ v 2 L 2 (,T ;R 4 ) for all u with u ū L (,T ;R 4 ) r and for all v L 2 (, T ; R 4 ). L (, T ; R 4 ). If ū l ū L (,T ;R 4 ) r, then we have Let l N be a fixed number and ū l ū l ū 2 L 2 (,T ;R 4 ) 1 δ ζ L 2 (,T ;R 4 ). As before, the coercivity parameter δ is approximated by means of the smallest eigenvalue of the reduced Hessian matrix associated with the POD solution ū l, as described in Section 4.2. Under the assumptions of colorred Corollary 4.5, ū l ū 2 L 2 (,T ;R 4 ) τ ζ L2 (,T ;R 4 ) σ l min