Hamburger Beiträge zur Angewandten Mathematik

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1 Hamurger Beiträge zur Angewandten Mathematik POD asis updates for nonlinear PDE control Carmen Gräßle, Martin Guisch, Simone Metzdorf, Sarina Rogg, Stefan Volkwein Nr August 16

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3 C. Gräßle, M. Guisch, S. Metzdorf, S. Rogg, and S. Volkwein POD asis updates for nonlinear PDE control Astract: In the present paper a semilinear oundary control prolem is considered. For its numerical solution proper orthogonal decomposition (POD) is applied. POD is ased on a Galerkin type discretization with asis elements created from the evolution prolem itself. In the context of optimal control this approach may suffer from the fact that the asis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. Therefore, different POD asis update strategies which avoids this prolem of unmodelled dynamics are compared numerically. 1 Introduction In this paper we consider an optimal control prolem governed y a semilinear paraolic equation together with control constraints. For the numerical solution we apply a Galerkin approximation, which is ased on proper orthogonal decomposition (POD), a method for deriving reduced-order models of dynamical systems; cf. Holmes et al. (1). However, to otain the state data underlying the POD model, it is necessary to solve once the full state system using a reference control. Consequently, the POD approximations depend on the chosen reference control, so that the choice of a reference control turned out to e essential for the computation of a POD asis for the optimal control prolem. To overcome this prolem we investigate two different asis update strategies for improving the POD asis: optimality-system POD (cf. Kunisch, Volkwein (8)) and trust-region POD (cf. Arian et al. ()). In oth update strategies the POD asis is changed in the optimization method in order to ensure convergence and a certain accuracy for the otained controls. Let us also refer to the papers Yue, Meerergen (13) and Qian et al. (16), where the trust-region optimization is efficiently comined with the reduced-asis method. The paper is organized as follows: In Section we introduce our semilinear control prolem and review optimality conditions. Our applied numerical methods are explained in Section 3. Section 4 is devoted to explain the application of the POD method to our optimal control prolem and in Section 5 we present our numerical experiments. Optimal control prolem Let R n, n {1,, 3}, e a ounded, open domain with Lipschitz-continuous oundary Γ =. The oundary is divided into m disjunct segments Γ i with Γ = m Γ i. The corresponding characteristic functions are denoted y χ Γi, i = 1,..., m. We consider the time interval [, T ] with T > and define the sets Q = (, T ), Σ = (, T ) Γ. The quadratic ojective has the form J(y, u) = 1 + σ u y(t, x) yd (x) dx T Γ u i (t)χ Γi (x) dxdt. (1) In (1) the desired state y d : R is assumed to e ounded and σ u > holds. The admissile controls elong to the closed, convex set U ad = { u : [, T ] R m ua u u in [, T ] }, () where u a, u : [, T ] R m are ounded functions and is understood componentwise in R m. For a given control u U ad the state y = y(t, x) satisfies the semilinear paraolic differential equation cy t (t, x) y(t, x) + y(t, x) 3 =, (t, x) Q, y m n (t, x) + qy(t, x) = u i (t)χ Γi (x), (t, x) Σ, (3) y(, x) = y (x), x. C. Gräßle: Universität Hamurg, Fachereich Mathematik, Bundesstraße 55, 146 Hamurg, Germany, carmen.graessle@unihamurg.de M. Guisch, S. Metzdorf, S. Rogg, S. Volkwein: Universität Konstanz, Fachereich Mathematik und Statistik, Universitätsstraße 1, Konstanz, Germany, {martin.guisch,simone.metzdorf,sarina.rogg,stefan.volkwein}@unikonstanz.de We suppose that the initial condition y : R is ounded and c, q are positive constants. Throughout the paper we focus on the specific nonlinearity y 3, which is utilized in our numerical tests. Of course, other nonlinearities satisfying certain oundedness and monotonicity conditions are also possile; see (Tröltzsch, 1, Chapter 5.1). In (3) the vector n = n(x) stands for the normal

4 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control vector defined on Γ and y n is the normal derivative. Recall that H = L () and V = H 1 () are defined as { ( ) } 1/ H = ϕ : R ϕ H = ϕ(x) dx <, { V = ϕ H ϕ V = ( ϕ H + n ) } 1/ ϕxi H <, respectively, where ϕ xi denotes the (weak) partial derivative of ϕ with respect to i-th component x i of x. A solution to (3) is understood in the following weak sense: y(t) = y(t, ) V satisfies y() = y in and cy t (t)ϕ + y(t) ϕ + y(t) 3 ϕ dx + Γ qy(t)ϕ dx = u i (t) Γ i ϕ dx for all ϕ V, where we set Γ i = Γ i 1dx. Let us define ( ) e(y, u), (p, p ) = y() y p dx + Γ cy t (t)p(t) + y(t) p(t) + y(t) 3 p(t) dx + qy(t)p(t) dx u i (t) Γ i p(t) dx (4) for all (test) functions p : Q R with T p(t) V dt < and p H. Then, the operator equation e(y, u) = is equivalent to the fact that y = y(u) is a weak solution to (3) for given u U ad. Now the optimal control prolem is expressed as min J(y, u) s.t. e(y, u) = and u U ad, (P) where s.t. stands for suject to. Under appropriate assumptions on y we can ensure that (3) has a unique weak solution y(u) for any admissile control u U ad ; i.e., we have e(y(u), u) =. This motivates the introduction of the reduced ojective Ĵ(u) = J(y(u), u). We consider instead of (P) the reduced prolem min Ĵ(u) s.t. u U ad. (ˆP) The notion reduced expresses the fact that Ĵ depends on the control only, whereas J is dependent on the state and control. It follows from (Tröltzsch, 1, Chapter 5.3) that (ˆP) admits an optimal solution ū U ad. If ȳ = y(ū) is the unique weak solution to (3), then x = (ȳ, ū) is an optimal solution to (P). To compute a locally optimal solution we make use of the following first-order necessary optimality condition for a locally optimal solution (Tröltzsch, 1, Chapter.8): Ĵ (ū)(u ū) for all u U ad, (5) where Ĵ (ū) denotes the (Gateaux) derivative of Ĵ at ū. Following the general approach in Chapter 5.5 in Tröltzsch (1) or the one for our present optimal control prolem in Gräßle (14); Rogg (14) we derive that (5) is equivalent to the variational inequality T ( σ u ū i (t) Γ i p(t) dx ) (u i (t) ū i (t)) dt (6) for all u U ad, where p(t) = p(t, ) V is the (unique) weak solution to the dual or adjoint equation c p t (t, x) p(t, x) = 3ȳ(t, x) p(t, x), (t, x) Q, p (t, x) + q p(t, x) =, (t, x) Σ, n p(t, x) = y d (x) ȳ(t, x), x (7) and ȳ = y(ū) is solves (4) for the optimal control u = ū. The associated dual variale p is determined y p = c p(, x) for all x. Note that (ˆP) and also (P) are nonlinear (and therefore non convex) prolems. To ensure that the solution ū to (5) is a locally optimal solution to ( ˆP) we have to investigate second-order sufficient optimality conditions; see, e.g., Chapter 5.7 in Tröltzsch (1) and Section.3 in Gräßle (14). 3 Optimization methods In this section we riefly review the numerical algorithms utilized to numerically solve (P) and ( ˆP), respectively. For more details we refer to Gräßle (14); Metzdorf (15); Rogg (14). 3.1 Sequential quadratic programming We introduce the Lagrangian associated with (P) as L(y, u, p, p ) = J(y, u) + e(y, u), (p, p ), where the mapping e(y, u) has already een defined in Section. It can e shown that the Lagrangian is twice continuously (Fréchet) differentiale. The principal idea of the sequential quadratic programming (SQP) method is to solve (P) in an iterative procedure, wherey in

5 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control 3 each SQP iteration k a linear-quadratic SQP suprolem has to e solved. This SQP suprolem is otained y a quadratic approximation of the Lagrangian L and a linearization of the equality constraint at the current iterates x k = (y k, u k ) and z k = (p k, p k ). For revity, we set L k = L(x k, z k ) and e k = e(x k ). The linear-quadratic SQP suprolem is given as min L k + L k xx k x k =(yk,uk ) + 1 Lk xx(x k, xk ) s.t. e k + e k xx k = and uk + u k U ad. By x we denote the partial (Fréchet) derivative with respect to the argument x = (y, u). Prolem (8) is welldefined if second-order sufficient optimality conditions hold at the point (x k, z k ). If x is determined, the new SQP iterate is x k+1 = x k + x k. For the Lagrange multiplier z k the update is given in Algorithm 1. It can e shown that the SQP method is locally equivalent to the Newton method, which consists in finding a stationary point of the Lagrangian. Hence, the rate of convergence is locally quadratic, provided that the second-order sufficient optimality condition holds; cf. Hinze et al. (9). 3. Primal-dual active set strategy (8) Since (8) involves inequality constraints, we utilize a primal-dual active set strategy (PDASS); cf. Bergounioux et al. (1997). Let k and j denote the current SQP and PDASS iteration level, respectively. The PDASS method works on the primal variale u k, which has to satisfy the inequality constraint u k +u k U ad, and on the dual variale µ k, which corresponds to the control constraint. For a given iterate (u k,j, µ k,j ) of the PDASS algorithm we define the active and inactive sets A k,j a A k,j = { t [, T ] : u k,j (t) + u k,j (t) + µ k,j (t) < u a (t) }, = { t [, T ] : u k,j (t) + u k,j (t) + µ k,j (t) > u (t) }, A k,j = A k,j a L k uu e k, u e k y e k u A k,j, I k,j = [, T ] \ A k,j, where the inequalities are understood componentwise. Then, the solution of (8) is computed y solving its associated KKT system: L k yy e k, y y L k y + = u A k,j = χ A k,j a where we set x k,j+1 and e k, e k, y u z µ k,j+1 A k,j L k u e k (u a u k ) + χ A k,j (u u k ), (9) = (y, u ) and z k,j+1 = z. By u we denote the dual operators of e k y and e k u, Algorithm 1: SQP method with PDASS Require: Initial value (x, z ), set k = ; 1: repeat : Initialize (u k,, µ k, ) and set j = ; 3: repeat 4: Set j = j + 1; 5: Determine A k,j a, A k,j, A k,j and I k,j ; 6: Solve (9) for (y, u, z ) and µ k,j A k,j ; 7: Set j = j + 1, x k,j = (y, u ), z k,j = z, and µ k,j I k,j 1 = ; 8: until j 1, A k,j a = A k,j 1 a, A k,j = A k,j 1 9: Set x k+1 = x k + x k,j and z k+1 = z k + z k,j 1: until SQP stopping criterium is fulfilled; ; ; respectively. The SQP method with PDASS is summarized in Algorithm 1. In general, Algorithm 1 is only locally convergent. In order to ensure convergence for any starting point (x, z ), a gloalization strategy is utilized, which is made from two ingredients: a modification of the second-order term L k xx to ensure coercivity and the inclusion of an Armijo acktracking line search for an l 1 -merit function. For more details we refer to Hintermüller (1) for the general theory and to Gräßle (14) for our specific optimal control prolem. 3.3 Trust region method In this paper we compare Algorithm 1 with the trustregion (TR) method for the solution of (ˆP); cf. Conn et al. (). The key idea of TR methods is as follows: At a current iterate the ojective Ĵ is replaced y a simpler (often quadratic) model function which is then approximately minimized in a TR to find the next iterate. Assume we are in iteration k with current iterate u k. Then, we uild a model function m k to approximate the ojective Ĵ at uk. In Algorithm, we restrict ourselves to the quadratic model m k (d) = Ĵ(uk ) + g k d + 1 Hk (d, d), (1) where d is a mapping from [, T ] to R m, g k = Ĵ (u k ) holds and H k denotes the Hessian Ĵ (u k ) or a self-adjoint approximation to it. In line a trial step d k is computed y approximately solving the TR suprolem. In lines 4 to 13 it is then decided whether or not to accept the trial point u k + d k as next iterate and the TR radius k gets adjusted. This decision is ased upon the quotient ρ k in line 3 which compares the reduction pred k of the model function to the reduction ared k of the cost functional and

6 4 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control Algorithm : (Trust region method) Require: Initial TR radius >, initial point u, constants η 1, η, γ 1, γ, γ 3 satisfying < η 1 η < 1, < γ 1 γ < 1 γ 3 ; set k = ; 1: Build up the model function m k ; : Determine an approximate solution d k to min d m k (d) s.t. u k + d U ad and d k ; 3: Compute the quotient ρ k = ared k pred k = Ĵ(uk ) Ĵ(uk +d k ) m k () m k (d k ) ; 4: if ρ k η then 5: Set u k+1 = u k + d k and k+1 [ k, γ 3 k ]; 6: Set k = k + 1 and go to line 1; 7: else if η 1 ρ k < η then 8: Set u k+1 = u k + d k and k+1 [γ (k), k ]; 9: Set k = k + 1 and go to line 1; 1: else if ρ k < η 1 then 11: Set u k+1 = u k and k+1 [γ 1 k, γ k ]; 1: Set k = k + 1 and go to line ; 13: end if which provides a measure for the approximation quality of the model function. In case ρ k η 1 (e.g. η 1 =.), the trial point is accepted. If even ρ k η (e.g. η =.8), the approximation quality of the model function is as good that the TR radius can e increased for the next iteration. Otherwise the radius should e kept or decreased. In case ρ k < η 1, the quadratic model is a poor approximation to the cost functional on the TR, the trial point is rejected and the TR suprolem gets again solved for a decreased TR radius. Disregarding standard assumptions, Algorithm is gloally convergent in the sense lim k g k = if the trial steps d k lead to a model decrease pred k at least as good as that of the so-called Cauchy step. If the model function possesses inexact gradient information g k Ĵ (u k ), gloal convergence still holds if the Carter condition Ĵ (u k ) g k ζ g k, ζ (, 1 η ) (11) 4 POD reduced-order modelling In this section we recall the asics of the POD method for optimal control prolems. For more details we refer, e.g., to Benner et al. (14); Sachs, Volkwein (1). 4.1 POD method Suppose that we are given trajectories z ν (t) = z ν (t, ) H, t [, T ], 1 ν and N. We introduce the snapshot suspace as V = span { z ν (t) t [, T ] and ν = 1,..., } with d = dim V. The inner product in H is given as ϕ, φ H = ϕ(x)φ(x) dx for all ϕ, φ H. For every l with 1 l d, a POD asis of rank l is defined as a solution to the minimization prolem min T ν=1 z ν (t) l z ν (t), ψ i H ψ i dt s.t {ψ i } l H, ψ i, ψ j H = ij, 1 i, j l. A solution to (1) is given y the eigenvalue prolem H (1) Rψ i = λ i ψ i for λ 1 λ... λ l λ d > (13) with the linear, ounded integral operator R : H V H, Rψ = T ν=1 z ν (t), ψ H z ν (t) dt. In our numerical experiments we consider =. We include y z 1 and z information of the (linearized) state and the dual equation, respectively. 4. Reduced-order modelling for the control prolem is satisfied in every iteration; see Carter (1991). We compute truncated Newton steps using the Steihaug-CG algorithm, see Nocedal, Wright (6), together with active set projection according to Kelley (1999). Hence, we fulfill this condition. Note that the computation of Ĵ (u k )d requires the solutions to a linearized state and linearized adjoint equation; cf. Hinze et al. (9). Suppose that a POD asis {ψ i } l is computed. We introduce the suspace V l = span {ψ 1,..., ψ l } and define the orthogonal projection P l ϕ = l ϕ, ψ i H ψ i for ϕ H. The POD scheme for (4) is as follows: y l (t) V l satisfies y l () = P l y in H and cyt(t)ψ l + y l (t) ψ + y(t) 3 ψ dx + Γ qy l (t)ψ dx = u i (t) Γ i ψ dx (14)

7 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control 5 for all ψ V l and t (, T ]. Then, we replace (P) y the following POD approximation min J(y l, u) s.t. (y l, u) satisfies (14) and u U ad. (P l ) We assume that (14) has a unique solution y l (u) for any u U ad. Then, setting Ĵl(u) = J(y l (u), u) for any u U ad we otain the POD approximation of (ˆP) min Ĵl(u) s.t. u U ad. (ˆP l ) Suppose that ū l is a locally solution to (ˆP l ). We interpret ū l as a suoptimal solution to (ˆP). First-order necessary optimality conditions for (ˆP l ) are given y (compare (6)) T ( σ u ū l i(t) Γ i ) p l (t) dx (u i (t) ū l i(t)) dt for all u U ad, where p l (t) = p(t, ) V l is the weak solution to a POD Galerkin scheme for (7) utilizing the optimal POD state ȳ l = y l (ū). More details are presented in Gräßle (14); Metzdorf (15); Rogg (14). 4.3 POD asis update strategies The choice of a reference control u ref U ad turned out to e essential for the computation of a POD asis of rank l. When using an aritrary control, the otained accuracy was not at all satisfying even when using a huge numer of asis functions. On the other hand, an optimal POD asis (computed from the FE optimally controlled state) led to far etter results; Hinze, Volkwein (8). To overcome this prolem different techniques for improving the POD asis have een proposed Optimality system POD Let us just roughly recall optimality system POD (OS- POD) introduced in Kunisch, Volkwein (8). The idea of OS-POD is to include the equations determining the POD asis in the optimization process min Ĵl(u) s.t. u U ad and y(u) is the solution to (4), {ψ i } l solves (13) for = 1, z 1 = y(u), y l (u) satisfies (14). (P l os) A therey otained asis is optimal for the considered prolem. Note that (P l os) is more complex than the original prolem (P). Therefore, we do not solve (P l os). Starting with an initial control u U ad, we compute a few projected gradient steps for (P l os) in order to find a POD asis, which is appropriate for the underlying optimal control prolem. Then, this POD asis is kept fix and the POD approximation (P l ) of (P) is derived. We discuss two possiilities to use OS-POD: 1) First, starting with a reference control u ref U ad we apply a few projected gradient steps to solve (P l os) numerically. Then, the otained POD asis is kept fix and the POD approximation (P l ) is solved numerically y the SQP method. This approach is used in Grimm et al. (14); Metzdorf (15) for linear paraolic equations. ) In the second approach we utilize OS-POD within the SQP framework. In each SQP iteration a linearquadratic optimal control prolem has to e solved up to a certain accuracy. For each linear-quadratic SQP suprolem we formulate the OS-POD optimization prolem as in (P l os). Again, we apply a few (projected) gradient steps, ut now to the linearquadratic prolem in order to find a POD asis which is appropriate for the current SQP suprolem. Then, we apply the PDASS algorithm to solve the linear-quadratic SQP suprolem using the POD asis found y the OS-POD gradient steps Trust region POD Within the trust-region POD (TR-POD) algorithm we guarantee that the reduced-order models are sufficiently accurate y ensuring gradient accuracy according to the Carter condition (11); see Schu (1). This is realized y successively adapting the reduced-order models in the course of optimization just y increasing the numer l of POD asis functions or, if this does not lead to a sufficient accuracy, y computing a new POD asis from the current control iterate (and then adapting the numer l). Recall the approximative model gradient g k and the approximative Hessian H k from (1). In the TR-POD approach g k and H k are computed from the respective POD approximations of Ĵ: m k l (d) = Ĵ(uk ) + gl k d + 1 Hk l (d, d). The TR-POD method is summarized in Algorithm 3. The lower part is completely identical to Algorithm. But here, at the eginning of each iteration, the approximation quality of the current POD asis is tested and if necessary recomputed.

8 6 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control Algorithm 3: TR-POD method Require: Initial TR radius >, initial point u, a maximal numer l max of POD asis functions and constants η 1, η, γ 1, γ, γ 3 satisfying < η 1 η < 1, < γ 1 γ < 1 γ 3 ; set k = ; 1: Compute a reduced-order model for some l l max ; : Guarantee (11) y computing new POD asis and/or expanding l l max ; 3: Build up the model function m k l (d); 4: Determine an approximate solution d k to min d m k l (d) s.t. uk + d U ad and d k ; 5: Compute the quotient ρ k = ared k pred k = Ĵ(uk ) Ĵ(uk +d k ) m k l () mk l (dk ) ; 6: Lines 4 to 13 of Algorithm. method ε u as ε u rel SQP-POD, no asis update OS-POD approach OS-POD approach SQP-POD, optimal asis TR-POD Tale 1. Run 1: errors etween POD and FE optimal controls t 1 SQP TR Numer i POD ased inexact SQP method For large-scale systems it is costly to solve each SQP suprolem (9) exactly. Thus, an inexact version of the SQP method is used, where the inexactness is caused y replacing (9) y its POD surrogate model; cf. Kahlacher, Volkwein (11). An a-posteriori error ound for linearquadratic optimal control prolems (see Tröltzsch, Volkwein (9)) is used to control the accuracy of the POD model and hence guarantee convergence. For more details we refer to Grimm et al. (14); Sachs, Volkwein (1). 5 Numerical Results Prolem setting. We choose = (, 1) (, 1) R and T =. The parameters are c = 1 and q =.1. We divide the oundary Γ into m = 4 parts so that the characteristic functions {χ Γi } 4 represent each of the four oundary parts. The cost parameter is σ u = For the spatial discretization we use piecewise linear finite elements (FE) with diameter h max =.6 leading to 498 degrees of freedom. As time integration scheme we utilize the implicit Euler method with N t = 5 equidistant time steps. The reference control for snapshot generation in the context of Section 4.1 is u ref =. The snapshot set for POD asis computation differs depending on the chosen optimization strategy. In case of TR-POD, state and adjoint state snapshots are taken, whereas for SQP linearized state and linearized adjoint state snapshots are utilized. Moreover, we define the asolute and relative Fig. 1. Run 1, left: SQP-POD controls {u i (t)} 4 for a fixed POD asis with l = 1 (lack line) and SQP-FE solution (lack dotted). Right: Decay of the eigenvalues for SQP-POD and TR- POD. errors ε u as = ūh u and ε u rel = εu as / ūh with u := ( () (u i ) t N t + (u (j) i j=1 i ) ) + (u(n t+1) to measure the difference etween any computed control u and the optimal FE control ū h. Run 1 (unconstrained case). At first we consider (P) without control constraints. We choose the desired state y d (x) = sin(π(x 1.5)) + 4x 1 x and the initial condition y (x) = sin(π(x 1.5)) for x = (x 1, x ). In the left plot of Fig. 1 we present the SQP-FE optimal controls (dotted lines) compared to the SQP-POD optimal controls otained with a fixed POD asis of rank l = 1. As can e seen, the accuracy is not satisfying; see also Tale 1, row 1. This confirms the necessity of a POD asis update strategy. The POD eigenvalues computed for SQP-POD and TR-POD are shown in the right plot of Figure 1. As first POD asis update strategy, we use the OS-POD approach 1) of Section One gradient step is applied to (P l os). Then, the POD asis of rank l = 1 is fixed and (P l ) is solved y the SQP method. This update strategy leads to an improvement of the error in the control variale of one order (see Tale 1, row ). The next POD asis update strategy consists in a comination of the POD ased inexact SQP method with OS-POD approach ) of Section Two SQP iterations are needed for convergence. The first SQP iteration is cheap, since l = 5 POD ases suffice. In the second SQP iteration, it is detected that a higher accuracy of the POD surro- )

9 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control t t Fig.. Run 1: POD controls {u i (t)} 4 (lack line) and FE solution (lack dotted), OS-POD asis update (left) and TR-POD (right). gate model for the linear-quadratic SQP suprolem (8) is needed. Thus, we apply one gradient step to solve the OS-POD optimization prolem for the linear-quadratic SQP suprolem and the numer of utilized POD asis functions is increased to l = 1. The POD solution ū l for the time dependent control intensities coincides convincingly with the FE solution (see Fig., left and Tale 1, row 3). Since the POD asis is adapted to the desired accuracy within the solution process, this strategy leads to etter results than the first POD asis update strategy. For comparison, the error etween the FE solution ū h and the SQP-POD solution ū l with optimal POD asis of length l = 1 is listed in Tale 1, row 4. Now, we compare these results to those otained y TR-POD. Recall that the TR-POD algorithm guarantees convergence to a stationary point. This is why the deviation of the TR-POD optimal control from the FE optimal control is within discretization accuracy; see the last row in Tale 1 and Fig., right plot. The iterations of the TR-POD method are very similar to those of the POD ased inexact SQP method with OS-POD approach. In the first iteration l = 4 POD asis functions suffice to fulfill the Carter condition (11). In the second iteration a new POD asis gets computed and l = 1 POD modes are required. We visualize the adaptation of the POD asis during optimization y means of the shape of the first three asis functions. In Fig. 3 the adaptation corresponding the POD ased inexact SQP method with OS-POD approach is shown. For comparison, in Fig. 4 we present the POD asis adaptation otained from the TR-POD algorithm. Run (constrained case). Now we impose the ox constraints u a = and u = 7. We choose y d (x) = + x 1 x and y (x) = 3 4(x.5) for x = (x 1, x ). The approximation results for the POD solution ū l (t) with snapshots computed from the reference controls u ref = and u ref = ū h (t) are listed in Tale, row 1 and 4, respectively. The application of the POD asis update strategy y OS-POD approach 1) leads to a reduction of the error in the control y a factor of two in comparison to the case Fig. 3. Run 1, OS-POD approach ): POD ases ψ 1, ψ, ψ 3 computed in the first (top) and second iteration (middle) of SQP- POD method; optimal asis functions computed y reference control u ref = ū h (ottom). Fig. 4. Run 1, TR-POD: POD ases ψ 1, ψ, ψ 3 computed in the first (top) and second iteration (middle); optimal asis functions computed y the reference control u ref = ū h (ottom). method ε u as ε u rel SQP-POD, no asis update OS-POD approach OS-POD approach SQP-POD, optimal asis TR-POD Tale. Run : errors etween POD and FE optimal controls.

10 8 Gräßle, Guisch, Metzdorf, Rogg & Volkwein, POD asis updates for nonlinear PDE control t t Fig. 5. Run : controls {u i (t)} 4, lack dotted: FE solution, lack line: POD solution. Right: TR-POD. with no POD asis update. Finally, we utilize OS-POD approach ) in comination with the inexact SQP-POD method. In this case, four SQP iterations are needed for convergence. In the first two iterations, l = 3 POD modes suffice. In the third iteration, one gradient step is performed to compute approximatively a solution for the OS-POD optimization prolem for the linear-quadratic SQP suprolem. Moreover, the numer of utilized POD asis functions is increased to l = 7. In the last iteration, the numer of POD modes is increased to the maximum numer l = 1. The resulting POD controls are compared to the FE controls in the left plot of Fig. 5 as well as in row 3 of Tale. From the computational point of view, OS-POD approach 1) is the fastest. The computational time for snapshot generation and OS-POD asis computation is.6 sec. The SQP-POD run then takes 7.3 sec. In comparison, the SQP-FE run takes 85.8 sec. The TR- POD Algorithm needs five iterations for optimization. In the first three iterations the initial POD asis suffices to guarantee the Carter condition with only 3, 3 and 5 POD asis functions. In the fourth iteration a new POD asis is computed and l = 7 POD modes are needed. This asis of rank seven can then e kept for the last step. Acknowledgment: C. Gräßle and S. Metzdorf have een supported partially y the project A-posteriori error estimation for TR-POD methods in Design and Control financed y the University of Konstanz. References Arian, E., Fahl M., Sachs, E.W. Trust-region proper orthogonal decomposition for flow control. Technical Report -5, ICASE,. Bergounioux, M., Ito, K., Kunisch, K. Primal-dual strategy for constrained optimal control prolems. SIAM J. Control Optim., (1997) Benner, P., Sachs, E.W., Volkwein, S. Model order reduction for PDE constrained optimization. Internat. Ser. Numer. Math., , (14) Carter, R.G. On the gloal convergence of trust region algorithms using inexact gradient information. SIAM J. Numer. Anal., (1991) Conn, A.R., Gould, N.I.M., Toint, P.L. Trust-region methods. SIAM () Gräßle, C. POD ased inexact SQP methods for optimal control prolems governed y a semilinear heat equation. Diploma Thesis, University of Konstanz (14), Grimm, E., Guisch, M., Volkwein, S. A-Posteriori Error Analysis and Optimality-System POD for Constrained Optimal Control. Comp. Sci. Eng., 15, (14) Holmes, P., Lumley, J.L., Berkooz, G., Romley, C.W. Turulence, Coherent Structures, Dynamical Systems and Symmetry. Camridge Monographs on Mechanics, Camridge University Press (1) Hintermüller, M. On a gloalized augmented Lagrangian-SQP algorithm for nonlinear optimal control prolems with ox constraints. Internat. Ser. Numer. Math., (1) Hinze, M., Pinnau, R., Ulrich, M., Ulrich, R. Optimization with ODE constraints. Math. Mod., 3, Springer Series (9) Hinze, M., Volkwein, S. Error estimates for astract linearquadratic optimal control prolems using proper orthogonal decomposition. Comput. Optim. Appl., (8) Kahlacher, M., Volkwein, S. POD a-posteriori error ased inexact SQP method for ilinear elliptic optimal control prolems. ESAIM: MAN, (11) Kelley, C.T. Iterative Methods for Optimization. Frontiers in Applied Mathematics, SIAM, Philadelphia, PA (1999) Kunisch, K., Volkwein, S. Proper orthogonal decomposition for optimality systems. ESAIM: MAN 4, 1-3 (8) Metzdorf, S. Optimality system POD for timevariant, linear-quadratic control prolems. Diploma Thesis, University of Konstanz (15), Nocedal, J., Wright, S.J. Numerical Optimization. nd ed., Springer, New York (6) Qian, E., Grepl, M., Veroy, K., Willcox, K. A certified trust region reduced asis approach to PDE-constrained optimization. ACDL Technical Report TR16-3, 16 Rogg, S. Trust region POD for optimal oundary control of a semilinear heat equation. Diploma Thesis, University of Konstanz (14), Sachs, E.W., Volkwein, S. POD Galerkin approximations in PDE-constrained optimization. GAMM-Mitt. 33, (1) Schu, M. Adaptive trust-region POD methods and their applications in finance. Ph.D thesis, University of Trier (1) Tröltzsch, F. Optimal Control of Partial Differential Equations: Theory, Methods and Applications. AMS American Mathematical Society, nd ed. (1) Tröltzsch, F., Volkwein, S. POD a-posteriori error estimates for linear-quadratic optimal control prolems. Comput. Optim. Appl., (9) Yue, Y., Meerergen, K. Accelerating PDE-constrained optimization y model order reduction with error control. SIAM J. Optim. 3, (13)