A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1615-1638.

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach.

DOI: 10.1051/m2an/2014012
Classification: 49K20,  49M29,  35K15,  65M15,  93C20
Keywords: optimal control, reduced basis method, a posteriori error estimation, model order reduction, parameter-dependent systems, partial differential equations, parabolic problems
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     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {1615--1638},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
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     doi = {10.1051/m2an/2014012},
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     url = {http://archive.numdam.org/articles/10.1051/m2an/2014012/}
}
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Kärcher, Mark; Grepl, Martin A. A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1615-1638. doi : 10.1051/m2an/2014012. http://archive.numdam.org/articles/10.1051/m2an/2014012/

[1] N. Altmüller and L. Grüne, Distributed and boundary model predictive control for the heat equation. GAMM Mitteilungen 35 (2012) 131-145. | MR | Zbl

[2] A.C. Antoulas, Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM (2005). | MR | Zbl

[3] J.A. Atwell and B.B. King, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33 (2001) 1-19. | MR | Zbl

[4] R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39 (2000) 113-132. | MR | Zbl

[5] P. Benner, V. Mehrmann and D. Sorensen, Dimension reduction of large-scale systems, vol. 45 of Lect. Notes Computational Science and Engineering. Berlin, Springer (2005). | MR | Zbl

[6] L. Dedè, Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32 (2010) 997-1019. | MR | Zbl

[7] L. Dedè, Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287-305. | MR | Zbl

[8] J. Eftang, D. Huynh, D. Knezevic and A. Patera, A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 28-58. | MR | Zbl

[9] A.-L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34 (2012) A2812-A2836. | MR | Zbl

[10] M.A. Grepl and M. Kärcher, Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C.R. Math. 349 (2011) 873-877. | MR | Zbl

[11] M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | MR | Zbl

[12] M. Gunzburger and A. Kunoth, Space-time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations. SIAM J. Control Optim. (2011) 1150-1170. | MR | Zbl

[13] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277-302. | Numdam | MR

[14] W. Hager, Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27 (1990) 1061-1080. | MR | Zbl

[15] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Math. Model. Theor. Appl. Springer (2009). | MR | Zbl

[16] D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C.R. Math. 345 (2007) 473-478. | MR | Zbl

[17] L. Iapichino, S. Ulbrich and S. Volkwein. Multiobjective PDE-constrained optimization using the reduced-basis method. Technical report, Universität Konstanz (2013).

[18] K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems. ESAIM: COCV 8 (2002) 741-760. | Numdam | MR | Zbl

[19] K. Ito and K. Kunisch, Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems. SIAM J. Numer. Anal. 46 (2008) 2867-2891. | MR | Zbl

[20] K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403-425. | MR | Zbl

[21] K. Ito and S.S. Ravindran, A reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid D. 15 (2001) 97-113. | MR | Zbl

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

[23] M. Kärcher and M.A. Grepl. A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM: COCV 20 (2014) 416-441. | Numdam | Zbl

[24] K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999) 345-371. | MR | Zbl

[25] K. Kunisch, S. Volkwein and L. Xie, HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3 (2004) 701-722. | MR | Zbl

[26] G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich and S. Ulbrich, Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics. Birkhäuser Basel (2012). | MR | Zbl

[27] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971). | MR | Zbl

[28] K. Malanowski, C. Buskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems, vol. 195. CRC Press (1997) 253-284. | MR | Zbl

[29] F. Negri, G. Rozza, A. Manzoni and A. Quarteroni, Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316-A2340. | MR | Zbl

[30] I. B. Oliveira, A “HUM” Conjugate Gradient Algorithm for Constrained Nonlinear Optimal Control: Terminal and Regulator Problems. Ph.D. thesis, Massachusetts Institute of Technology (2002).

[31] C. Prud'Homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluid. Eng. 124 (2002) 70-80.

[32] A.M. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Ser. Comput. Math. Springer (2008). | Zbl

[33] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Method. E. 15 (2008) 229-275. | MR

[34] G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1244-1260. | MR | Zbl

[35] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and pod a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Model. Dyn. 17 (2011) 355-369. | MR

[36] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83-115. | MR | Zbl

[37] K. Urban and A.T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations. C. R. Math. 350 (2012) 203-207. | MR | Zbl

[38] K. Veroy, D.V. Rovas and A.T. Patera, A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations: “convex inverse” bound conditioners. Special Volume: A tribute to J.L. Lions. ESAIM: COCV 8 (2002) 1007-1028. | Numdam | MR | Zbl

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