6.5 Application: Multiobjective optimal control
6.5.3 Multiobjective optimization by the weighted sum method
The multiobjective optimal control problem (6.9) is now solved by the weighted sum method in reduced order for a discrete parameter set MWSM ={µi}Ni=1 ⊂Mad with a number ofN = 500 randomly chosen admissible parameters. If we assume for each chosen parameter µ ∈ MWSM the average optimization times according to the reduced basis generation as given in Table 6.8, we gain a MOR speed-up of factor >5 with respect to the high-dimensional (FE) optimization.
The resulting set of efficient points Z = ˆJ(U) ⊂ R3 for the Pareto optimal control set U = {¯u(µ)}µ∈MWSM, representing the Pareto front, is shown in Figure 6.9. A more detailed view is presented in Figure 6.10 by taking a closer look on an extraction of the Pareto front. For illustration we also added some non-efficient points, which are obtained without optimization by just evaluating J(y, u) for arbitrarily chosen controlsu∈Uad and their corresponding state solution y=y(u).
6.5.4 Conclusion and outlook
Once a reduced order model is derived, the multiobjective optimization can be done by the weighted sum method in reduced dimension at (much) lower computational costs. During the reduced basis generation by the POD-greedy algorithm, the weak a-posteriori error estimaton WAPEE (121.1 sec), that utilizes the grid approximation approach (GRID) for the computation of an approximative
6.5 Application: Multiobjective optimal control
1 2 3 4 5 6 7 8 9 10
ℓ 10-4
10-2 100 102
Error estimates
Max FULLHS Max GRID Maxkuh−uℓk Min FULLHS Min GRID Minkuh−uℓk Avg FULLHS Avg GRID Avgkuh−uℓk
Figure 6.7: Error estimates for increasing number ` of basis functions. For each` we indicate by
“avg” the error estimates averaged over the total number of parameters inStrain.
lower bound of the smallest eigenvalue, is more than twice as fast as the a-posteriori error estimate APEE (272.5 sec), where the smallest eigenvalue of the Hessian is computed by FULLHS. If we compare only the computation times for the eigenvalue approximation (0.001 sec by GRID) and eigenvalue computation (>1.1sec by FULLHS), we even gain a tremendous computational speed-up of factor > 1000. Unfortunately, this is almost fully compensated by the evaluation of the perturbation function ζ (around 1.0 sec) in the error estimator, for which high-dimensional (FE) solves of the (nonlinear) state and adjoint equations are needed. Moreover, one has also to take the additional numerical costs caused by the pre-offline phase into account, which took here a total computation time of370sec. Of course, this makes the GRID approach for this constellation, where only one POD-greedy run for a single training setStrainwas performed, in sum more expensive. But for those cases, where reduced bases have to be generated for a multitude N of large and varying training setsStrain(1)6=Strain(2) 6=...6=Strain(N), it will represent a cost-efficient implementation, since the pre-computation of the eigenvalues on the control grid Ξgrid is parameter-independent.
Concrete, it would be here the case for a numberN >3. In this regard let us recall the important property, that the pre-offline phase is ideally suited for parallel computation as shown in Section 6.4.3, Table 6.4, where a computational speed-up in time of factor around 3 is gained. Another promising possibility of saving computational time (and hence a suggestion for further research) could be a numerical efficient estimate for the perturbation function ζ that does not rely on high dimensional (FE) solves of the (nonlinear) state and adjoint equation, as it is successfully applied for linear elliptic and parabolic problems by Kärcher and Grepl in [KG14a, KG14b]. Since both (FE) solutions are essential for the computation of the Hessian, they have always to be computed for the FULLHS approach.
Remark 6.5.4For a (very) small parameter training setStrainit might be also numerical efficient to apply the “strong” POD-greedy algorithm, i.e., we solve for allµ∈Straina high-dimensional optimal
0 5 10 15 20
6.5 Application: Multiobjective optimal control
0.185
6.5 Application: Multiobjective optimal control control problem and are therefore able to compute during basis generation always the actual error kuh(µ)−u`(µ)kU instead of the a-posteriori estimate. In this case we would need in total (for a number of20parameters and the corresponding high-dimensional optimization runs) only108sec, based on the average FE optimization time given in Table 6.8. Here, the crucial thing is the RO optimization speed-up factor of “only” around5, compared to a factor of up to14 in the previous Section 5.1. This can be attributed primarily to the underlying coarser FE discretization, so that the accompanying reduced order approach can also develop only less potential. But already for an average RO optimization speed-up factor of around7, which is only half the factor as presented in Section 5.1 for a higher FE discretization, the weak a-posteriori error estimation again turns out to be the computational faster (online) approach. Let us also refer here to the already mentioned fact, that this behaviour is especially consolidated for a multitudeN of large and varying parameter training sets {S(i)train}Ni=1. Nevertheless, let us conclude, that for this constellation it would be the method of choice, and – as far as possible – an (a-priori) investigation in the numerical effort for additional a-posteriori error estimation techniques might be quite reasonable.
Summary
Finally we present a brief recap of the observations and conclusions made in this work as they can be found in detail in Sections 5.4, and 6.5.4 as well as at the end of Sections 6.4.3 and 6.4.4.
In this thesis an a-posteriori error estimation for POD reduced-order (parametrized) optimal control problems, governed by nonlinear (parameter-independent) partial differential equations, was investigated with regard to numerical reliability as well as efficiency. For this purpose, special focus was put on suitable approaches for the computation and approximation of the smallest eigenvalue of the Hessian at the computed optimal control, being an integral part of the underlying error estimation formula. For all presented approaches in this work, the quality of the provided second-order derivative information was sufficient to establish reliable a-posteriori error estimates for any numerical test and application considered to this end. However, this applied not in general for the purpose of computational efficiency, as our observations and results have demonstrated.
For the case of non-parametrized problems, only the introduced approach using a Quasi-Newton BFGS approximation of the Hessian turned out to be a competitive candidate, that does not consume the computational benefit strived by reduced order optimization under given conditions.
But since its general application is subject to specified limitations, we suggest this approach for further investigation. Further on, for the extended case of parametrized problems, we presented an online-efficient (weak) a-posteriori error estimate in the framework of a POD-greedy algorithm, that ensured a computational speed-up during the reduced basis generation process. Nevertheless, for a reasonable statement in terms of numerical efficiency as a whole, also the computational costs for the offline phase had to be taken into account. Hence, for the treated application in context of multiobjective control, we stated a recommendation for a minimum number of parameter training sets, that should have been required to justify the offline effort against the online benefit. For comprehensive and specified suggestions concerning further research as well as possibilities of (computational and implementational) enhancement, we refer to relevant sections mentioned at the beginning.
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