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A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-Sided Control Constraints

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Model Reduction of Parametrized Systems

Part of the book series: MS&A ((MS&A,volume 17))

Abstract

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and two-sided control constraints, which play an important role in many industrial and economical applications. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using two slack variables, we then develop a reduced basis approximation for both slack problems by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for a model problem.

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Notes

  1. 1.

    The subscript “e” denotes the “exact” infinite-dimensional continuous problem setting.

  2. 2.

    The framework of this work directly extends to Neumann boundary controls U e = L 2(∂Ω) or finite dimensional controls \(U_{\mathrm{e}} = \mathbb{R}^{m}\). Also distributed controls on a subdomain Ω U ⊂ Ω or Neumann boundary controls on a boundary segment Γ U ⊂ ∂Ω are possible.

  3. 3.

    Alternative methods to deal with the non-negativity can be found in [2].

References

  1. Bader, E., Kärcher, M., Grepl, M.A., Veroy, K.: Certified reduced basis methods for parametrized distributed optimal control problems with control constraints. SIAM J. Sci. Comput. 38(6), A3921–A3946 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balajewicz, M., David, A., Farhat, C.: Projection-based model reduction for contact problems. Int. J. Numer. Methods Eng. 106(8), 644–663 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Haasdonk, B., Salomon, J., Wohlmuth, B.: A reduced basis method for parametrized variational inequalities. SIAM J. Numer. Anal. 50(5), 2656–2676 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications, vol. 23. Springer, Berlin (2009)

    Google Scholar 

  5. Kärcher, M.: Certified reduced basis methods for parametrized PDE-constrained optimization problems. PhD thesis, RWTH Aachen University (2016)

    Google Scholar 

  6. Kärcher, M., Grepl, M.A.: A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM Control Optim. Calculus Var. 20(2), 416–441 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kärcher, M., Grepl, M.A., Veroy, K.: Certified reduced basis methods for parameterized distributed optimal control problems. Technical report (2014)

    Google Scholar 

  8. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, New York (1971)

    Book  MATH  Google Scholar 

  9. Maday, Y., Patera, A.T., Rovas, D.V.: A blackbox reduced-basis output bound method for noncoercive linear problems. In: Cioranescu, D., Lions, J.L. (eds.) Studies in Mathematics and Its Applications, vol. 31, pp. 533–569. Elsevier Science B.V., Amsterdam (2002)

    Google Scholar 

  10. Negri, F., Rozza, G., Manzoni, A., Quarteroni, A.: Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35(5), A2316–A2340 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Negri, F., Manzoni, A., Rozza, G.: Reduced basis approximation of parametrized optimal flow control problems for the stokes equations. Comput. Math. Appl. 69(4), 319–336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44, 83–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Z., Bader, E., Veroy, K.: A slack approach to reduced-basis approximation and error estimation for variational inequalities. C.R. Math. 354(3), 283–289 (2016)

    Google Scholar 

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Authors and Affiliations

  1. Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062, Aachen, Germany

    Eduard Bader & Karen Veroy

  2. Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Martin A. Grepl

Authors
  1. Eduard Bader
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  2. Martin A. Grepl
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  3. Karen Veroy
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Corresponding author

Correspondence to Eduard Bader .

Editor information

Editors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

  2. Institute for Computational and Applied Mathematics, University of Münster , Münster, Germany

    Mario Ohlberger

  3. Massachusetts Institute of Technology, MIT , Cambridge, Massachusetts, USA

    Anthony Patera

  4. SISSA MathLab, International School for Adv. Studies , Trieste, Italy

    Gianluigi Rozza

  5. Institute of Numerical Mathematics, Ulm University , Ulm, Germany

    Karsten Urban

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Bader, E., Grepl, M.A., Veroy, K. (2017). A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-Sided Control Constraints. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_3

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