A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Control Problems with Control Constraints★

A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Control Problems with Control Constraints★

8th Vienna International Conference on Mathematical Modelling February 18 - 20, 2015. Vienna University of Technology, Vienna, February - 20, 2015. Vi...

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8th Vienna International Conference on Mathematical Modelling February 18 - 20, 2015. Vienna University of Technology, Vienna, February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna18International Conference on Mathematical Modelling 8th Conference on Mathematical Modelling Austria 8th Vienna18International International Conference on Mathematical Modelling Available online at www.sciencedirect.com Austria February 20, 2015. Vienna University of Technology, Vienna, February February 18 18 -- 20, 20, 2015. 2015. Vienna Vienna University University of of Technology, Technology, Vienna, Vienna, Austria Austria Austria

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IFAC-PapersOnLine 48-1 (2015) 719–720 A A Certified Certified Reduced Reduced Basis Basis Approach Approach for for A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Parametrized Linear-Quadratic Optimal Control Problems with Parametrized Linear-Quadratic Optimal Control Problems with Control Control Constraints Constraints  Control Problems with Control Constraints  Control Problems with  Control Constraints ∗ ∗∗ ∗∗∗

Eduard a Martin Eduard Bader Bader ∗ Mark Mark K¨ K¨ archer rcher ∗∗ ∗∗∗∗ Martin Grepl Grepl ∗∗∗ ∗ ∗∗ ∗∗∗ Karen Veroy-Grepl ∗ Mark ∗∗ ∗∗∗∗ ∗∗∗ Eduard Bader K¨ a rcher Martin Karen Veroy-Grepl ∗ Mark K¨ ∗∗ Martin Grepl ∗∗∗ Eduard a rcher Grepl Eduard Bader BaderKaren Mark K¨ a rcher Martin Grepl ∗∗∗∗ ∗∗∗∗ Veroy-Grepl ∗∗∗∗ Karen Veroy-Grepl ∗ Veroy-Grepl ∗ Aachen Institute forKaren Aachen Institute for Advanced Advanced Study Study in in Computational Computational Engineering Engineering ∗ Science (AICES), RWTH Aachen University, Schinkelstraße 2, ∗ Aachen(AICES), Institute RWTH for Advanced Advanced Study in Computational Computational Engineering Science AachenStudy University, Schinkelstraße 2, 52062 52062 ∗ Aachen Institute for in Engineering Aachen Institute for Advanced Study in Computational Engineering Aachen, Germany (e-mail: [email protected]). Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany (e-mail: [email protected]). Science (AICES), RWTH Aachen University, Schinkelstraße 2, ∗∗ Science (AICES), RWTH AachenStudy University, Schinkelstraße 2, 52062 52062 Aachen Institute for Advanced in Computational Engineering ∗∗ Aachen, Germany (e-mail: [email protected]). Aachen Institute for Advanced Study in Computational Engineering Aachen, Germany (e-mail: [email protected]). Aachen, Germany (e-mail: [email protected]). ∗∗ Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 ∗∗ Aachen Institute for in Engineering Science (AICES), RWTH AachenStudy University, Schinkelstraße 2, 52062 ∗∗ Aachen Institute for Advanced Advanced Study in Computational Computational Engineering Aachen Institute for Advanced Study in Computational Engineering Aachen, Germany (e-mail: [email protected]) Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 52062 Aachen, Germany (e-mail: [email protected]) Science (AICES), RWTH Aachen University, Schinkelstraße ∗∗∗ Science (AICES), RWTH [email protected]) University, Schinkelstraße 2, 2, 52062 Numerical Mathematics, RWTH ∗∗∗ Aachen, Germany (e-mail: Numerical Mathematics, RWTH Aachen Aachen University, University, Aachen, Germany (e-mail: [email protected]) Aachen, Germany (e-mail: [email protected]) ∗∗∗ Templergraben 55, Germany (e-mail: ∗∗∗ Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056 52056 Aachen, Aachen, Germany (e-mail: ∗∗∗ Numerical Mathematics, RWTH Aachen University, Numerical Mathematics, RWTH Aachen University, [email protected]) Templergraben 55, 52056 Aachen, Germany (e-mail: [email protected]) Templergraben 55, 55,for 52056 Aachen, Aachen, Germany Germany (e-mail: ∗∗∗∗Templergraben 52056 (e-mail: ∗∗∗∗ Aachen Institute [email protected]) Aachen Institute for Advanced Advanced Study Study in in Computational Computational [email protected]) [email protected]) ∗∗∗∗ Engineering Science RWTH Aachen University, ∗∗∗∗ Aachen Institute for Advanced Study in Computational Engineering Science (AICES), (AICES), RWTH Aachen University, ∗∗∗∗ Aachen for Advanced Study in Aachen Institute Institute for Advanced Study in Computational Computational Schinkelstraße 2, 52062 Aachen, Germany (e-mail: Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany (e-mail: Engineering Science (AICES), RWTH Aachen University, Engineering Science (AICES), RWTH Aachen (e-mail: University, [email protected]) Schinkelstraße 2, 52062 Aachen, Germany [email protected]) Schinkelstraße 2, 2, 52062 52062 Aachen, Aachen, Germany Germany (e-mail: (e-mail: Schinkelstraße [email protected]) [email protected]) [email protected])

Abstract: Abstract: In Abstract: In this this talk, talk, we we consider consider the the efficient efficient and and reliable reliable solution solution of of distributed distributed optimal optimal control control Abstract: Abstract: problems governed by parametrized elliptic partial differential equations involving constraints In this talk, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations involving constraints In this talk, we consider the efficient and reliable solution of distributed optimal control In this talk, weThe consider thebasis efficient andpartial reliable solution of distributed optimal control on the control. reduced method is used as aa low-dimensional surrogate model to problems governed by parametrized elliptic differential equations involving constraints on the control. The reduced basis method is used as low-dimensional surrogate model to problems governed by parametrized elliptic partial differential equations involving constraints problems governed by parametrized elliptic partial differential equations involving constraints solve the optimal control problem. To this end, we introduce reduced basis spaces not only on the control. The reduced basis method used aa low-dimensional surrogate to solve the optimal control problem. To this is end, we as introduce reduced basis spaces model not only on the control. The reduced basis method is used as low-dimensional surrogate model to on the control. The reduced basis method isforused as a low-dimensional surrogate model to for the state and adjoint variable also distributed control variable and propose solve optimal To this we introduce reduced spaces only for thethe state and control adjoint problem. variable but but also end, for the the distributed control basis variable and not propose solve the optimal control problem. To this end, we introduce reduced basis spaces not only solve the optimal control problem. To this end, we introduce reduced basis spaces not only rigorous error bounds for the error in the optimal control. The reduced basis optimal control for the state adjoint variable also for thecontrol. distributed control variable and propose rigorous error and bounds for the error but in the optimal The reduced basis optimal control for the and adjoint variable but also for distributed control variable and for the state state and adjoint variable but also for the the distributed control variable and propose propose problem and associated aa posteriori error bounds can be efficiently evaluated in an offline-online rigorous error bounds for the error in the optimal control. The reduced basis optimal control problem and associated posteriori error bounds can be efficiently evaluated in an offline-online rigorous error bounds for the error in optimal control. The basis optimal control rigorous errorassociated bounds for the error error in the the optimal Theinreduced reduced basis optimal control computational procedure, thus our approach many-query or problem and and a posteriori posteriori bounds cancontrol. berelevant efficiently evaluated in an an offline-online computational procedure, thus making making our approach relevant in the the many-query or real-time real-time problem associated a error bounds can be efficiently evaluated in offline-online problem and associated a posteriori error bounds can be efficiently evaluated in an offline-online context. We present numerical results for a model problem to show the validity of our approach. computational procedure, thus making our approach relevant in the many-query or real-time context. We present numerical results forour a model problem to show the many-query validity of our approach. computational procedure, thus making approach relevant in or real-time computational procedure, thusresults making our approach relevant in the the many-query or approach. real-time context. We present numerical for aa model problem to show the validity of our context. We present numerical results for model problem to show the validity of our approach. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. context. We present numerical results for a model problem to show the validity of our approach. Many trol problem problem is is generally generally sub-optimal sub-optimal and and reliable reliable error error Many problems problems in in science science and and engineering engineering can can be be modmod- trol eled in terms of optimal control problems governed by estimation is thus crucial. Besides serving as aa certificate Many problems in science and engineering can be modtrol problem is generally sub-optimal and error is thus crucial. sub-optimal Besides serving asreliable certificate eled inproblems terms ofinoptimal control problems can governed by estimation trol problem is generally and reliable error Many science and engineering be modMany problems inoptimal science and engineering can be While modtrol problem isthe generally sub-optimal and reliable error parametrized partial differential equations (PDEs). of fidelity for sub-optimal solution, our a certificate posteriori eled in terms of control problems governed by estimation is thus crucial. Besides serving as a of fidelity for the sub-optimal solution, our posteriori parametrized partial differential equations (PDEs). While estimation is thus crucial. Besides serving as a certificate eled in terms of optimal control problems governed by eled in terms of optimal control problems governed by estimation is thus crucial. Besides serving as a certificate the PDE describes the underlying system or component error bounds are also a crucial ingredient in generating parametrized partial differential equations (PDEs). While of fidelity for the sub-optimal our posteriori error bounds are also a crucialsolution, ingredient ina generating the PDE describes underlying system (PDEs). or component fidelity for the sub-optimal solution, our posteriori parametrized partialthe differential equations While parametrized partial differential equations (PDEs). While of of fidelity forbasis the with sub-optimal solution, ourina a generating posteriori behavior, the often to aa particreduced algorithms. the PDE describes the underlying or component error bounds are also agreedy crucial ingredient the reduced basis with greedy algorithms. behavior, the parameters parameters often serve servesystem to identify identify partic- the error bounds are also a crucial ingredient in generating the PDE describes the underlying system or component the PDE describes the underlying system or component error bounds are also a crucial ingredient in generating ular configuration of the component — such as boundary behavior, the parameters often serve to identify a particthe reduced basis with greedy algorithms. ular configuration of the component — as boundary approach efficient the reduced basis with greedy algorithms. behavior, the parameters often serve serve to such identify partic- A behavior, the parameters often to identify aa particthe reduced basisfor with greedycomputation algorithms. of A new new approach for efficient computation of error error bounds bounds and initial conditions, material properties, and geometry. ular configuration of the component — such as boundary and initial conditions, material properties, and geometry. for unconstrained distributed control problems was proular configuration of the component — such as boundary A new approach for efficient computation of error bounds ular configuration of the component — such as boundary for unconstrained distributed control problems was proIn such cases — in addition to solving the optimal conA new approach for efficient computation of error bounds and initial conditions, material geometry. In such cases — in addition to properties, solving theand optimal con- A new approach for efficient computation of errorwas bounds posed in K¨ a rcher et al. (2014). This approach, however, and initial conditions, material properties, and geometry. for unconstrained distributed control problems proand initial conditions, material properties, and geometry. posed in K¨ a rcher et al. (2014). This approach, however, trol problem itself — one is often interested in explorfor unconstrained distributed control problems was proIn such cases — in addition solving the optimal control problem itself — one is to often interested in explorfor unconstrained distributed control problems was proand all other existing approaches in the literature, see In such cases — in addition to solving the optimal conposed in K¨ a rcher et al. (2014). This approach, however, In such cases — in addition to solving the optimal conand all other existing approaches in the literature, see ing many different parameter configurations and thus in posed in K¨ a rcher et al. (2014). This approach, however, trol problem itself — one is often interested in exploring many different parameter configurations and thus in posed in K¨ a rcher et al. (2014). This approach, however, e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), trol problem itself — one is often interested in explorand all other existing approaches in the literature, see trol problem itself — one is often interested in explore.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), speeding up the solution of the optimal control problem. and all other other existing approaches in the literature, literature, see ing different parameter and thus in speeding the solution of theconfigurations optimal control all existing approaches in the see are not directly applicable to the important case with ing many manyup different parameter configurations andproblem. thus as in and e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), ing many different parameter configurations and thus in are not directly applicable to the important case with However, using classical discretization techniques such e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), speeding up the solution of the optimal control problem. However, using classical discretization techniques such as e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), additional constraints on control. speeding up the theorsolution solution of the the optimal optimal control problem. problem. are not directly applicable the important case with speeding up of control additional constraints on the the to control. finite elements finite even single is not applicable to the However, using classical discretization such as finite elements or finite volumes volumes even aatechniques single solution solution is are are not directly directly applicable to the important important case case with with However, using classical discretization techniques such as additional constraints on the control. However, using classical discretization techniques such as often computationally expensive and aa In this work we extend the methodology presented in additional constraints on the control. finite elements or finite volumes even single solution is often computationally expensive and aatime-consuming, time-consuming, additional constraints on the control. In this work we extend the methodology presented in finite elements or finite volumes even single solution is finite elements orexploration finiteexpensive volumes even atime-consuming, singleOne solution is parameter-space thus prohibitive. way to Zhang et al. (2014) to consider PDE-constrained optimal often computationally and a parameter-space exploration thus prohibitive. One way to In this work we extend the methodology presented in Zhang et al. (2014) to consider PDE-constrained optimal often computationally expensive and time-consuming, a In this work we extend the methodology presented in often computationally expensive and time-consuming, a In thisetproblems. work we extend the methodology presented in decrease the burden is model control The authors in Zhang et al. (2014) parameter-space exploration thus prohibitive. One way to decrease the computational computational burden is the the surrogate surrogate model Zhang al. (2014) to consider PDE-constrained optimal control problems. The authors in Zhang et al. (2014) parameter-space exploration thus prohibitive. One way to Zhang et al. (2014) to consider PDE-constrained optimal parameter-space exploration thus prohibitive. One way to Zhang et al. (2014) to consider PDE-constrained optimal approach, where the original high-dimensional model is redevelop a certified Reduced Basis (RB) method that decrease computational is surrogate approach,the where the originalburden high-dimensional modelmodel is re- develop control problems. problems. The authorsBasis in Zhang Zhang et al. (2014) (2014) a certifiedThe Reduced (RB) et method that decrease the computational burden is the the These surrogate model control authors in al. decrease the computational burden is the surrogate model control problems. The authorsBasis inposteriori Zhang al. bounds (2014) placed reduced order approximation. ideas have provides and inexpensive a error approach, the original high-dimensional replaced by by aawhere reduced order approximation. Thesemodel ideas is have develop asharp certified Reduced (RB) et method that provides sharp and inexpensive a posteriori error bounds approach, where the original high-dimensional model is redevelop a certified Reduced Basis (RB) method that approach, theorder original high-dimensional model is re- develop asharp certified ReducedInBasis (RB) the method that received aaawhere lot of in various model for variational inequalities. particular, approach placed by reduced approximation. These ideas have received lot of attention attention in the the past past and and various model provides and inexpensive a posteriori error bounds for variational inequalities. In particular, the approach placed by a reduced order approximation. These ideas have provides sharp and inexpensive a posteriori error bounds placed by a reduced order approximation. These ideas have provides sharp and inexpensive a posteriori error bounds order reduction techniques have been used in this context. has advantages compared to prior work on variational received aa lot attention in past and model order reduction have been this context. for variational particular, the approach has advantages inequalities. compared toIn prior work on variational received lot of oftechniques attention in the the pastused andinvarious various model for variational inequalities. In particular, the approach received athe lot of attention in the past and various model variational inequalities. Inprior particular, the approach However, of reduced order optimal coninequalities with the RB method Haasdonk et al. (2012). order reduction techniques have been used in this context. However, the solution solution of the the reduced order optimal con- for has advantages compared to work on variational inequalities with the RB method Haasdonk et al. (2012). order reduction techniques have been used in this context. has advantages compared to prior work on variational order reduction techniques have been used in this context. has advantages compared to prior work on variational The methodology in Zhang et al. (2014) not only (i However, the solution of the reduced order optimal coninequalities with the RB method Haasdonk et al. (2012).  The methodology in Zhang et al. (2014) not only (i )) However, the solution of the reduced order optimal coninequalities with the RB method Haasdonk et al. (2012). This work was supported by the Excellence Initiative of the  However, thewassolution of the reduced orderInitiative optimalofconinequalities with error the RB method Haasdonk et al.only (2012). provides sharper bounds that mimic the convergence This work supported by the Excellence the The methodology in Zhang et al. (2014) not (i )) provides sharper error bounds that mimic the convergence German federal and state governments and the German Research The methodology in Zhang al. (2014) not only (i  The of methodology in bounds Zhang et et al.mimic (2014) not only (i ) German federal and state governments and the German Research work was supported by the Excellence Initiative of the  rate the RB approximation, but also (ii) does so at an This work was supported by the Excellence Initiative of the  This provides sharper error that the convergence rate of the RB approximation, but also (ii) does so at an Foundation through Grant GSC 111. This work was supported by111. the Excellence Initiative of the provides sharper error bounds that mimic the convergence Foundation through Grant GSC German federal and state governments and the German Research provides sharper error bounds that mimic the convergence German federal and state governments and the German Research rate of of the RB RB approximation, but but also (ii) (ii) does so so at an an German federal and Grant state governments and the German Research rate Foundation rate of the the RB approximation, approximation, but also also (ii) does does so at at an Foundation through through Grant Grant GSC GSC 111. 111. Foundation through GSC 111.

Copyright © 2015, IFAC 719 Copyright © 2015, 2015,IFAC IFAC (International Federation of Automatic Control) 719Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © Copyright © 2015, IFAC 719 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2015, IFAC 719 Copyright © 2015, IFAC 719 10.1016/j.ifacol.2015.05.167

MATHMOD 2015 720 February 18 - 20, 2015. Vienna, Austria

Eduard Bader et al. / IFAC-PapersOnLine 48-1 (2015) 719–720

online cost that is independent of the high dimension of the original problem. In particular we use the approach presented in Zhang et al. (2014) (i ) to construct a feasible RB approximation of the control and (ii) to derive efficiently computable a posteriori error bounds. The main idea is to generate two RB–systems. The first one is “standard” and is used to construct low dimensional approximations for the state and the Lagrange multiplier. In the second one we construct nonnegative slack variables for the control and so can generate feasible low dimensional surrogates for the control. Finally, we extend the proof of the a posteriori error bounds from K¨archer et al. (2014) to derive efficient a posteriori bounds for the control error in the constraint case. REFERENCES Haasdonk, B., Salomon, J., and Wohlmuth, B. (2012). A reduced basis method for parametrized variational inequalities. SIAM J. Numer. Anal., 50(5), 2656–2676. K¨archer, M., Grepl, M., and Veroy, K. (2014). Certified reduced basis methods for parametrized distributed optimal control problems. submitted to SIAM Journal on Control and Optimization. Negri, F., Rozza, G., Manzoni, A., and Quarteroni, A. (2013). Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. Accepted for publication. Negri, F. (2011). Reduced basis method for parametrized optimal control problems governed by PDEs. Master’s thesis, Politecnico di Milano. Rozza, G., Manzoni, A., and Negri, F. (2012). Reduction strategies for pde-constrained optimization problems in haemodynamics. In Proceedings ECCOMAS Congress, Vienna Austria. Zhang, Z., Bader, E., and Veroy, K. (2014). A duality approach to error estimation for variational inequalities. submitted to SIAM Journal on Scientific Computing. URL http://arxiv.org/abs/1410.2095.

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