Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-20T14:51:33.650Z Has data issue: false hasContentIssue false

A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

Published online by Cambridge University Press:  09 September 2014

Mark Kärcher
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany..
Martin A. Grepl
Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany. ;
Get access


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.

Research Article
© EDP Sciences, SMAI 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Altmüller, N. and Grüne, L., Distributed and boundary model predictive control for the heat equation. GAMM Mitteilungen 35 (2012) 131145. Google Scholar
A.C. Antoulas, Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM (2005).
Atwell, J.A. and King, B.B., Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33 (2001) 119. Google Scholar
Becker, R., Kapp, H. and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39 (2000) 113132. Google Scholar
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).
Dedè, L., Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems. SIAM J. Sci. Comput. 32 (2010) 9971019. Google Scholar
Dedè, L., Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287305. Google Scholar
Eftang, J., Huynh, D., Knezevic, D. and Patera, A., A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 2858. Google Scholar
Gerner, A.-L. and Veroy, K., Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34 (2012) A2812A2836. Google Scholar
Grepl, M.A. and Kärcher, M., Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C.R. Math. 349 (2011) 873877. Google Scholar
Grepl, M.A. and Patera, A.T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157181. Google Scholar
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.
Haasdonk, B. and Ohlberger, M., Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277302. Google Scholar
Hager, W., Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27 (1990) 10611080. Google Scholar
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Math. Model. Theor. Appl. Springer (2009).
Huynh, D.B.P., Rozza, G., Sen, S. and Patera, A.T., A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C.R. Math. 345 (2007) 473478. Google Scholar
L. Iapichino, S. Ulbrich and S. Volkwein. Multiobjective PDE-constrained optimization using the reduced-basis method. Technical report, Universität Konstanz (2013).
Ito, K. and Kunisch, K., Receding horizon optimal control for infinite dimensional systems. ESAIM: COCV 8 (2002) 741760. Google Scholar
Ito, K. and Kunisch, K., Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems. SIAM J. Numer. Anal. 46 (2008) 28672891. Google Scholar
Ito, K. and Ravindran, S.S., A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403425. Google Scholar
Ito, K. and Ravindran, S.S., A reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid D. 15 (2001) 97113. Google Scholar
M. Kärcher, The reduced-basis method for parametrized linear-quadratic elliptic optimal control problems. Master’s thesis, Technische Universität München (2011).
Kärcher, M. and Grepl, M.A.. A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM: COCV 20 (2014) 416441. Google Scholar
Kunisch, K. and Volkwein, S., Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999) 345371. Google Scholar
Kunisch, K., Volkwein, S. and Xie, L., HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3 (2004) 701722. Google Scholar
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).
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971).
K. Malanowski, C. Buskens and H. Maurer, Convergence of approximations to nonlinear optimal control problems, vol. 195. CRC Press (1997) 253–284.
Negri, F., Rozza, G., Manzoni, A. and Quarteroni, A., Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316A2340. Google Scholar
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).
Prud’homme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T. and Turinici, G.. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluid. Eng. 124 (2002) 7080. Google Scholar
A.M. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Ser. Comput. Math. Springer (2008).
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Method. E. 15 (2008) 229275. Google Scholar
Rozza, G. and Veroy, K., On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196 (2007) 12441260. Google Scholar
Tonn, T., Urban, K. and Volkwein, S., 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) 355369. Google Scholar
Tröltzsch, F. and Volkwein, S., POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83115. Google Scholar
Urban, K. and Patera, A.T., A new error bound for reduced basis approximation of parabolic partial differential equations. C. R. Math. 350 (2012) 203207. Google Scholar
Veroy, K., Rovas, D.V. and Patera, A.T., 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) 10071028. Google Scholar