Methodology

snapshot locations low. In particular, the POD reduced-order modeling of the optimal control problem including offline costs for snapshot location and POD basis computation shall give a notable speed up in comparison to solving the full-order model. For this reason, a coarse spatial resolution is used for solving the elliptic systems (3.17) and (3.21). This will be heuristically jus-tified in the numerical tests where it turns out that spatial and temporal discretization decouple for the considered problem setting and data.

Besides the computation of suitable time instances for snapshot generation, the proposed method produces an approximation of the optimal control at the same time which serves as input control for snapshot generation.

Moreover, it turns out that it is beneficial to use these snapshot locations also as time grid points in the POD reduced optimization process, since they are related to the optimal solution.

We investigate three possible approaches:

(i) Snapshot location tailored for the state. We solve the elliptic system (3.17) adaptively with respect to time using the error estimator (3.25) according to Algorithm 2. This computation is done using a coarse spatial resolution in order to keep the computational costs low. As a result, we get time points which are related to the optimal state variable.

At the same time an approximation of the optimal state is computed. The solution of the adjoint equation (3.10) on a coarse spatial resolution with the approximate state as right-hand side delivers an approximate adjoint state. From the optimality condition (3.11) we get an approximation of the optimal control with respect to a coarse spatial resolution which is utilized as input control for snapshot generation. A summary of this approach is given by Algorithm 4.

(ii) Snapshot location tailored for the adjoint state / control. We solve the elliptic system (3.21) adaptively with respect to time using the error estimator (3.27) on a coarse spatial grid.

As a result, we obtain time grid points which are related to the optimal adjoint state. At the same time, an approximation of the optimal adjoint state is computed. According to the optimality condition (3.11) we compute an approximation of the optimal control with respect to a coarse spatial resolution. This control is then utilized as input control for snapshot generation. A summary of this approach is given by Algorithm 5.

(iii) Snapshot location tailored for both state and adjoint state / control. We run both strategies (i) and (ii) and then use the union of the time grid points as snapshot locations. At the same time, we compute two approximations of the optimal control on a coarse spatial mesh at different time points. From these, we compute a joint approximation as input control for snapshot generation. This approach is summarized in Algorithm 6.

Note that by Algorithm 2 and Algorithm 3 we obtain the state and adjoint solution to the biharmonic problem, respectively, which live in a space-time finite element space, i.e. yh ∈ Yh

and p_h ∈ Y˜_h,0. In order to compute an approximation for the optimal control which lives for each time instance in the finite element space V_h, we need to utilize an interpolation. This is possible due to the specific construction of the triangulation into space-time slabs (tj−1, tj)×Ω.

In particular, we use the Lagrange interpolationπ in order to interpolate the space-time discrete functions y_h ∈ Y_h and p_h ∈ Y˜_h,0 for each time instance t_j into the finite element space V_h, i.e.

π(yh(tj)), π(ph(tj))∈Vh. Discussion

We discuss the approaches (i), (ii), (iii) with regard to the suitability for the optimization inter-est, the computational efforts and the heuristic flavor.

Suitability for the optimization interest. If one expects significant variations of the dynamics

3.6 Snapshot location in optimal control for POD model order reduction 43

Algorithm 4Adaptive snapshot location selection for linear-quadratic optimal control problems (tailored for the state).

Input: Coarse uniform spatial grid with resolution h⁺.

Output: Snapshot locations{t_j}ⁿ_j=0, approximation of the optimal controlu_h⁺.

1: Call Algorithm 2 in order to solve (3.17) adaptively w.r.t. time using the error estimator (3.25) with spatial resolution h⁺. Obtain time instances{t_j}ⁿ_j=0 =I and the solution y_h⁺ ∈Yh.

2: Solve (3.10) onIwith uniform spatial resolution h⁺and right-hand sideπy_h⁺ to obtainp_h⁺.

3: Setu_h⁺ =−_α¹p_h⁺.

Algorithm 5Adaptive snapshot location selection for linear-quadratic optimal control problems (tailored for the adjoint state / control).

Input: Coarse uniform spatial grid with resolution h⁺.

Output: Snapshot locations{t_j}ⁿ_j=0, approximation of the optimal controlu_h⁺.

1: Call Algorithm 3 in order to solve (3.21) adaptively w.r.t. time using the error estimator (3.27) with spatial resolutionh⁺. Obtain time instances {t_j}ⁿ_j=0 =I and the solution p_h⁺.

2: Setu_h⁺ =−_α¹πp_h⁺.

in time in either of the state or control variable while the other variable is expected to behave mild, then an adaptive snapshot location strategy which is tailored for either the state or control might be advantageous. However, if the dynamical behavior of both state and adjoint state differs significantly over time and strong variations in the temporal domain are expected, the resulting time grids from Algorithms 4 and 5 might also differ strongly. Thus, the adaptive time grid tailored for the state variable constructed according to Algorithm 4 might not be a suitable time grid for the adjoint variable and vice versa. Only Algorithm 6 ensures to capture both dynamical evolutions of the state and adjoint state over time. However, if one expects a similar temporal behavior of the state and adjoint variables, it might suffice to only consider one of the Algorithms 4 and 5 depending on the quantity of interest and the application.

Computational efforts. Concerning the snapshot location computation, Algorithms 4 and 5 have the same computational effort. However, the computation of the approximate control is cheaper in Algorithm 5 since it only requires the solution of (3.11), while in Algorithm 4 it requires additionally one solve of (3.10). The approach (iii) is the most expensive approach. The compu-tational costs are the sum of the compucompu-tational costs of Algorithms 4 and 5 plus the additional costs for the construction of the approximate control u_h+ according to step 3 in Algorithm 6.

Heuristic flavor. The efficiency of the proposed approaches is based on the assumption that Algorithm 6Adaptive snapshot location selection for linear-quadratic optimal control problems (tailored for both state and adjoint state / control).

Input: Coarse uniform spatial grid with resolution h⁺.

Output: Snapshot locations{t_j}ⁿ_j=0, approximation of the optimal controlu_h⁺.

1: Run Algorithm 4 and obtain snapshot locations{t¹_j}ⁿ_j=0¹ and an approximation of the optimal controlu¹_h+.

2: Run Algorithm 5 and obtain snapshot locations{t²_j}ⁿ_j=0² and an approximation of the optimal controlu²_h+.

3: Set{t_j}ⁿ_j=0={t¹_j}ⁿ_j=0¹ ∪ {t²_j}ⁿ_j=0² and set forj= 0, . . . , n u_h⁺(tj) =







u¹_h+(t_j) ift_j ∈ {t¹_j}ⁿ_j=0¹ and t_j ∈ {t/ ²_j}ⁿ_j=0² , u²_h+(tj) iftj ∈ {t²_j}ⁿ_j=0² and tj ∈ {t/ ¹_j}ⁿ_j=0¹ ,

1

2(u¹_h+(tj) +u²_h+(tj)) iftj ∈ {t¹_j}ⁿ_j=0¹ and tj ∈ {t²_j}ⁿ_j=0² .

temporal and spatial discretization decouple for the considered problem setting and data. This behavior is discovered in numerical experiments. In this way, the method has a certain heuristic flavor. Based on this observation we use a coarse spatial resolution in the offline phase in or-der to generate suitable snapshot locations. In general, the issue of when spatial and temporal discretization decouple is not easy to answer. Related to this issue, we refer to [178, Chapter 3], where the relationship of temporal and spatial discretization is analyzed in the context of unsteady internal flow. In [30] a combination of time adaptivity and spatial adaptivity is con-sidered including a-priori and a-posteriori error analysis.

POD reduced-order modeling for the optimal control problem

After the computation of the snapshot locations and an approximate controlu_h⁺ using either of the Algorithms 4 to 6, the usual POD offline phase begins. For this, we sample state and adjoint snapshots at the computed snapshot locations with respect to a fine spatial resolution by solving (3.15) with control inputIu_h⁺ and (3.16), whereIdenotes a suitable interpolation of the control u_h⁺ from a coarse spatial grid onto the fine finite element grid.

In order to derive a POD reduction of the optimal control problem (3.9), we follow the first-reduce-then-optimize approach, see Section 2.4.2. We utilize an integrated snapshot ensemble, i.e. we take into account both state and adjoint snapshots, compare Remark 2.47. In this way, the POD projection error for both state and adjoint variables is determined by the sum of the neglected eigenvalues. Assume we have constructed a POD reduced spaceX_{`</sub>⊂H<sub>0</sub><sup>1</sup>(Ω) according to (2.7). Let us denote by ˜y<sub>0</sub> a suitable approximation of the initial valuey<sub>0</sub> in the reduced space X<sub>`}. For example, it can be computed as the solution to

(˜y₀, ψ)_L²_(Ω) = (y₀, ψ)_L²_(Ω) ∀ψ∈X_`.

Note that it is also possible to utilize a homogenization with respect to the initial condition (according to e.g. [89, Remark 3.3]) which then allows to construct a reduced POD model for which the POD reduced state fulfills the initial condition exactly, see [89, Section 3.3]. The POD reduced-order optimal control problem reads as follows:

min

(y`,u`)∈W(0,T;X_{`</sub>)×L<sup>2</sup>(0,T;Ω)J`(y`, u`) s.t. y`=S<sub>`}(u`,y˜0), (3.34) where the linear operator

S_{`</sub>:L<sup>2</sup>(0, T;H<sup>−1</sup>(Ω))×X<sub>`}→W(0, T;X_{`</sub>), (ˆu,y˜<sub>0</sub>)7→y<sub>`} :=S_`(ˆu,y˜₀)

is defined analogously to the solution operator S defined in (3.5). In particular, we replace the spaceH₀¹(Ω) in (3.2) byX_{`</sub>. This leads to the following reduced-order state equation: for a given initial conditiony<sub>`}(0) := ˜y₀ determine a functiony_{`</sub> ∈W(0, T;X<sub>`}) such that

d

dt(y`(t), ψ)<sub>L</sub><sup>2</sup><sub>(Ω)</sub>+ (∇y<sub>`</sub>(t),∇ψ)_L2(Ω)= (u`(t), ψ)<sub>L</sub><sup>2</sup><sub>(Ω)</sub> fort∈[0, T], ψ∈X`. (3.35) Note that we use the notationu_{`</sub> for the control in order to indicate the affiliation of the control to the reduced-order optimization problem. This helps to distinguish notationally between the solutionuto (3.9) and the solutionu<sub>`}to (3.34). However, note carefully that we do not impose a POD reduction for the control space. In order to achieve a conservative reduction of the control variable, the reduction of the control is induced by the reduction of the adjoint variable due to the optimality condition

αu_{`</sub>+p<sub>`} = 0 in [0, T]×Ω, (3.36)

where p_{`</sub> denotes the solution to the following reduced-order adjoint equation: determine a functionp<sub>`} ∈W(0, T;X_{`</sub>) such thatp<sub>`}(T) = 0 and

−d

dt(p_{`</sub>(t), ψ)<sub>L</sub><sup>2</sup><sub>(Ω)</sub>+ (∇p<sub>`}(t),∇ψ)_L2(Ω)= ((y_{`</sub>−y<sub>d</sub>)(t), ψ)<sub>L</sub><sup>2</sup><sub>(Ω)</sub> fort∈[0, T], ψ∈X<sub>`}, (3.37a)

3.6 Snapshot location in optimal control for POD model order reduction 45

Existence of a unique solution y_{`</sub> and p<sub>`} to (3.35) and (3.37), respectively, as well as stability results follow analogously to the infinite-dimensional case according to [69] and X`⊂H₀¹(Ω).

Now, the POD online phase begins in which (3.34) is solved. Since we already have a time grid at hand which is related to the optimal control problem, it makes sense to use this time grid for solving the problem (3.34). The POD reduced-order modeling for the linear-quadratic optimal control problems using adaptive snapshot computation is summarized in Algorithm 7.

Algorithm 7 POD model order reduction for linear-quadratic optimal control problems using adaptive snapshot locations.

Input: Snapshot locations {t_j}ⁿ_j=0 = I and input control u_h⁺ from either of the Algorithms 4 to 6, fine spatial resolutionh^∗, number of POD basis functions `

Output: y_{`</sub>, u<sub>`}

1: Sample state and adjoint snapshots in a simulation of (3.15) and (3.16) with spatial resolution h^∗ and input controlIu_h⁺ on I.

2: Compute a POD basis of rank`according to (2.7) using an integrated snapshot ensemble.

3: Set up and solve the POD-ROM (3.34) onI to obtain y`, u`.

Remark 3.9. In principle, it is conceivable to derive a POD reduced-order formulation for the biharmonic systems (3.17)and (3.21), respectively. However, this approach is not followed here, since our motivation is to use the usual optimization algorithm, for example of Remark 2.45, and utilize existing implementations. The snapshot location strategy is considered as an add-on with the goal to deliver time grid points and an input control which are associated with the optimal solution. For a space-time POD Galerkin approach we refer to [26, 187].