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Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

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Abstract

In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach–Nečas–Babuška theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.

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Notes

  1. The subscripts “\(\mathrm {e}\)” denote “exact”.

  2. Our framework covers spatially distributed controls \(U_\mathrm {e}= L^2({\varOmega }_U)\), \({\varOmega }_U\subset {\varOmega }\), and Neumann boundary controls \(U_\mathrm {e}= L^2({\varGamma }_U)\), \({\varGamma }_U\subset {\varGamma }\). It also applies to finite-dimensional control spaces \(U_\mathrm {e}= \mathbb {R}^m\).

  3. Here and in the following we often omit the dependence on \(\mu \) to simplify notation.

  4. We again note that we omit the dependence on \(\mu \) to simplify notation, i.e., we write \(y = y(\mu )\), \(p = p(\mu )\), and \(u = u(\mu )\).

  5. The bilinear form \(b(\cdot ,\cdot ;\mu ): U\times Y \rightarrow \mathbb {R}\) defines an associated mapping \(\mathscr {B}(\mu ): U \rightarrow Y'\) given by \(\langle \mathscr {B}(\mu ) \psi , \phi \rangle _{Y',Y} = b(\psi ,\phi ;\mu )\) for all \(\psi \in U\), \(\phi \in Y\), \(\mu \in \mathscr {D}\).

  6. The error for a POD basis of size \(N_\text {POD}\) is given by \((\sum _{i=N_\text {POD}+1}^{n_\mathrm {train}} \sigma _i^2)^{1/2}\), where \(\sigma _i\), \(1 \le i \le n_\mathrm {train}\), are the singular values (in decreasing order) of \(\frac{1}{\sqrt{n_\mathrm {train}}} \mathbb {U}^{1/2} S\). Here, \(\mathbb {U}\) is the finite element matrix associated with the reference inner product \((\cdot ,\cdot )_{U} = (\cdot ,\cdot )_{U(\mu ^\mathrm {ref})}\), and \(S \in \mathbb {R}^{\mathscr {N}_U\times n_\mathrm {train}}\) is the snapshot matrix of optimal controls \(u^*(\mu )\) for all \(\mu \in {\varXi }_\mathrm {train}\). See also [36].

  7. The superscript “o” indicates quantities related to the original parameter-dependent domain \({\varOmega }^o(\mu )\), whereas no superscript refers to the parameter-independent reference domain \({\varOmega }= {\varOmega }^o\big (\mu ^\mathrm {ref}\big )\).

References

  1. Atwell, J.A., King, B.B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33(1–3), 1–19 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39(1), 113–132 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benner, P., Sachs, E., Volkwein, S.: Model Order Reduction for PDE Constrained Optimization. Springer International Publishing, New York (2014). doi:10.1007/978-3-319-05083-619

    Book  MATH  Google Scholar 

  4. Chen, P., Quarteroni, A.: Weighted reduced basis method for stochastic optimal control problems with elliptic PDE constraint. SIAM/ASA J. Uncertain. Quantif. 2(1), 364–396 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, P., Quarteroni, A., Rozza, G.: Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59(1), 187–216 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Y., Hesthaven, J.S., Maday, Y., Rodríguez, J.: Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell’s problem. ESAIM Math. Model. Numer. Anal. 43(6), 1099–1116 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Das, S.K., Clegg, S.T., Samulski, T.V.: Computational techniques for fast hyperthermia temperature optimization. Med. Phys. 26(2), 319–328 (1999)

    Article  Google Scholar 

  8. Davidson, S.R.H., Sherar, M.D.: Theoretical modelling, experimental studies and clinical simulations of urethral cooling catheters for use during prostate thermal therapy. Phys. Med. Biol. 48, 729–744 (2003)

    Article  Google Scholar 

  9. Dedè, L.: Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32(2), 997–1019 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Elman, H.C., Qifeng, L.: Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA J. Uncertain. Quantif. 1(1), 192–217 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, Berlin (2010)

    Google Scholar 

  12. Gerner, A.L., Veroy, K.: Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34(5), A2812–A2836 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grepl, M.A., Kärcher, M.: Reduced basis a posteriori error bounds for parametrized linear–quadratic elliptic optimal control problems. C. R. Math. 349(15–16), 873–877 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gubisch, M., Neitzel, I., Volkwein, S.: A-posteriori error estimation of discrete POD models for PDE-constrained optimal control. Tech. rep., Konstanzer Schriften in Mathematik (2016)

  15. Hager, W.W.: Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27(4), 1061–1080 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Problems. Springer Briefs in Mathematics. Springer, Berlin (2015)

    Google Scholar 

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

    MATH  Google Scholar 

  18. Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345(8), 473–478 (2007)

  19. Ito, K., Kunisch, K.: Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems. SIAM J. Numer. Anal. 46(6), 2867–2891 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ito, K., Ravindran, S.S.: A reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid D. 15, 97–113 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kammann, E., Tröltzsch, F., Volkwein, S.: A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM Math. Model. Numer. Anal. 47(2), 555–581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kärcher, M.: The reduced-basis method for parametrized linear-quadratic elliptic optimal control problems. Master’s thesis, Technische Universität München (2011)

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

  24. Kärcher, M., Grepl, M.A.: A posteriori error estimation for reduced order solutions of parametrized parabolic optimal control problems. ESAIM Math. Model. Numer. Anal. 48(6), 1615–1638 (2014)

  25. Kärcher, M., Grepl, M.A.: A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM Control Optim. Calc. Var 20(2), 416–441 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kunisch, K., Volkwein, S.: Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory App. 102, 345–371 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kunisch, K., Volkwein, S., Xie, L.: HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3, 701–722 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S. (eds.): Constrained Optimization and Optimal Control for Partial Differential Equations. (International Series of Numerical Mathematics). Birkhäuser Basel (2012)

  29. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  30. Malanowski, K., Büskens, C., Maurer, H.: Convergence of approximations to nonlinear optimal control problems. In: Fiacco, A.V. (ed.) Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Applied Mathematics, pp. 253–284. Marcel Dekker, Inc., New York City (1997)

    Google Scholar 

  31. Negri, F.: Reduced basis method for parametrized optimal control problems governed by PDEs. Master’s thesis, Politecnico di Milano (2011)

  32. 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 

  33. 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 

  34. Pennes, H.H.: Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1(2), 93–122 (1948)

    Article  Google Scholar 

  35. Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluid. Eng. 124(1), 70–80 (2002)

    Article  Google Scholar 

  36. Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. Springer, Berlin (2015)

    MATH  Google Scholar 

  37. 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 

  38. Rozza, G., Veroy, K.: On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196(7), 1244–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Tokoutsi, Z., Kärcher, M., Grepl, M., Veroy, K., Baragona, M., Maessen, R.: Real time optimization of thermal ablation cancer treatments (2017). In preparation

  40. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  41. Tröltzsch, F., Volkwein, S.: POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44(1), 83–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference. AIAA Paper 2003-3847 (2003)

  43. Veroy, K., Rovas, D.V., Patera, A.T.: A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “convex inverse” bound conditioners. ESAIM Control Optim. Calc. Var 8, 1007–1028 (2002)

    Article  MathSciNet  MATH  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

    Mark Kärcher, Zoi Tokoutsi & Karen Veroy

  2. Philips Research Eindhoven, Eindhoven, The Netherlands

    Zoi Tokoutsi

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

    Martin A. Grepl

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  1. Mark Kärcher
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  2. Zoi Tokoutsi
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  3. Martin A. Grepl
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  4. Karen Veroy
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Correspondence to Martin A. Grepl.

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This work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111 and by the European Commission through the Marie Skłodowska-Curie Actions (Innovative Training Program—European Industrial Doctorate, Project Number 642445).

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Kärcher, M., Tokoutsi, Z., Grepl, M.A. et al. Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls. J Sci Comput 75, 276–307 (2018). https://doi.org/10.1007/s10915-017-0539-z

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