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Multiobjective PDE-constrained optimization using the reduced-basis method

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Abstract

In this paper the reduced basis (RB) method is applied to solve quadratic multiobjective optimal control problems governed by linear parametrized variational equations. These problems often arise in applications, where the quality of the system behavior has to be measured by more than one criterium. The weighted sum method is exploited for defining scalar-valued linear-quadratic optimal control problems built by introducing additional optimization parameters. The optimal controls corresponding to specific choices of the optimization parameters are efficiently computed by the RB method. The accuracy is guaranteed by an a-posteriori error estimate. An effective sensitivity analysis allows to further reduce the computational times for identifying a suitable and representative set of optimal controls.

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References

  1. Babuška, I.: The finite element method with Lagrange multipliers. Numer. Math. 20, 179–192 (1973)

    Article  MATH  Google Scholar 

  2. Banholzer, S., Beermann, D., Volkwein, S.: POD-based bicriterial optimal control by the reference point method. IFAC-PapersOnLine 49(8), 210–215 (2016)

    Article  MathSciNet  Google Scholar 

  3. Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339(9), 667–672 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. Springer, New York (2011)

    MATH  Google Scholar 

  5. Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. ESAIM: Math. Modell. Numer. Anal. 8, 129–151 (1974)

    MATH  Google Scholar 

  6. Brezzi, F., Fortin, M.: Mixed and hybrid finite elements methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  7. Das, I.: newblock nonlinear multicriteria optimization and robust optimality. Ph.D thesis. Rice University, Houston, Texas (1997)

    Google Scholar 

  8. Dede, L.: Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32, 997–1019 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ehrgott, M.: Multicriteria optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  10. Evans, L.C.: Partial differential equations. American Mathematics of Society Providence, Rhode, Island (2008)

    MATH  Google Scholar 

  11. Göpfert, A., Nehse, R.: Vektoroptimierung. BSB Teubner Verlagsgesellschaft, Leibzig (1990)

    MATH  Google Scholar 

  12. Grepl, M., Kärcher, M.: A posteriori error estimation for reduced order solutions of parametrized parabolic optimal control problems. To appear in ESAIM: Mathematical Modelling and Numerical Analysis (2014)

  13. Griesse, R., Vexler, B.: Numerical sensitivity analysis for the quantity of interest in PDE-constrained optimization. SIAM J. Sci. Comput. 29, 22–48 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hesthaven, J., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations (2016)

  15. Hillermeier, C.: Nonlinear multiobjective optimization. A generalized homotopy approach. Basel, Birkhäuser Verlag (2001)

  16. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE constraints (2009)

  17. Huynh, D.B.P., Knezevic, D.J., Chen, Y., Hesthaven, J.S., Patera, A.T.: A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199, 1963–1975 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Iapichino, L., Volkwein, S.: Optimization strategy for parameter sampling in the reduced basis method. IFAC-PapersOnLine 48, 707–712 (2015)

  19. Iapichino, L. Reduced basis methods for the solution of parametrized PDEs in repetitive and complex networks with application to CFD. Ph.D thesis, École Polytechnique Fédérale de Lausanne, 2012. N. 5529, http://infoscience.epfl.ch

  20. Iapichino, L., Volkwein, S. Greedy sampling of distributed parameters in the reduced-basis method by numerical optimization Konstanzer Schriften in Mathematik No. 308 University of Konstanz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:352-228388

  21. Iapichino, L., Trenz, S., Volkwein, S.: Reduced-order multiobjective optimal control of semilinear parabolic problems. In: Numerical Mathematics and Advanced Applications (ENUMATH 2015), Lecture Notes in Computational Science and Engineering, vol. 112, pp. 389–397 (2016)

  22. Kammann, E., Tröltzsch, F., Volkwein, S.: A method of a-posteriori error estimation with application to proper orthogonal decomposition. Math. Modell. Numer. Anal. 47(933–934), 218–225 (2013)

    MATH  Google Scholar 

  23. Kuhn, H., Tucker, A.: Nonlinear programming. In: Newman, J. (ed.) Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp 481–492. University of California Press, Berkeley (1951)

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

  25. Nocedal, J., Wright, S.J.: Numerical optimization, 2nd edn. Springer Series in Operation Research (2006)

  26. Patera, A.T., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2006)

  27. Peitz, S., Oder-Blöbaum, S., Dellnitz, M.: Multiobjective optimal control methods for fluid flow using reduced order modeling. Submitted (2015)

  28. Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equatiuons an introduction (2016)

  29. 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, 229–275 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  30. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Meth. Appl. Mech. Engr. 7, 1244–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Stadler, W.: Multicriteria optimization in engineering and in the sciences. Plenum Press, New York (1988)

  32. Tröltzsch, F.: Optimal control of partial differential equations. Theory, Methods and applications, vol. 112. American Math. Society, Providence (2010)

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Urban, K., Volkwein, S., Zeeb, O.: Greedy sampling using nonlinear optimization. In: Quarteroni, A., Rozza, R. (eds.) Reduced Order Methods for Modeling and Computational Reduction, pp 137—157. Springer (2014)

  35. Zadeh, L.: OptiMality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control, 8 (1963)

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Authors and Affiliations

  1. Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

    L. Iapichino

  2. Department of Mathematics, Darmstadt University of Technology, Dolivostraße 15, Darmstadt, Germany

    S. Ulbrich

  3. Department of Mathematics and Statistics, University of Konstanz, Universitätsstraße 10, D-78457, Konstanz, Germany

    S. Volkwein

Authors
  1. L. Iapichino
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  2. S. Ulbrich
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  3. S. Volkwein
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Correspondence to S. Volkwein.

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Communicated by: Karsten Urban

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Iapichino, L., Ulbrich, S. & Volkwein, S. Multiobjective PDE-constrained optimization using the reduced-basis method. Adv Comput Math 43, 945–972 (2017). https://doi.org/10.1007/s10444-016-9512-x

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