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.
Mots clés : optimal control, reduced basis method, a posteriori error estimation, model order reduction, parameter-dependent systems, partial differential equations, parabolic problems
@article{M2AN_2014__48_6_1615_0,
author = {K\"archer, Mark and Grepl, Martin A.},
title = {\protect\emph{A {Posteriori} {}Error} {Estimation} for {Reduced} {Order} {Solutions} of {Parametrized} {Parabolic} {Optimal} {Control} {Problems}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1615--1638},
publisher = {EDP-Sciences},
volume = {48},
number = {6},
year = {2014},
doi = {10.1051/m2an/2014012},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2014012/}
}
TY - JOUR AU - Kärcher, Mark AU - Grepl, Martin A. TI - A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 1615 EP - 1638 VL - 48 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2014012/ DO - 10.1051/m2an/2014012 LA - en ID - M2AN_2014__48_6_1615_0 ER -
%0 Journal Article %A Kärcher, Mark %A Grepl, Martin A. %T A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 1615-1638 %V 48 %N 6 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2014012/ %R 10.1051/m2an/2014012 %G en %F M2AN_2014__48_6_1615_0
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 , Tome 48 (2014) no. 6, pp. 1615-1638. doi : 10.1051/m2an/2014012. http://www.numdam.org/articles/10.1051/m2an/2014012/
[1] and , Distributed and boundary model predictive control for the heat equation. GAMM Mitteilungen 35 (2012) 131-145. | MR | Zbl
[2] , Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM (2005). | MR | Zbl
[3] and , Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33 (2001) 1-19. | MR | Zbl
[4] , and , Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39 (2000) 113-132. | MR | Zbl
[5] , and , Dimension reduction of large-scale systems, vol. 45 of Lect. Notes Computational Science and Engineering. Berlin, Springer (2005). | MR | Zbl
[6] , 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] , Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287-305. | MR | Zbl
[8] , , and , A two-step certified reduced basis method. J. Sci. Comput. 51 (2012) 28-58. | MR | Zbl
[9] and , Certified reduced basis methods for parametrized saddle point problems. SIAM J. Sci. Comput. 34 (2012) A2812-A2836. | MR | Zbl
[10] and , Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C.R. Math. 349 (2011) 873-877. | MR | Zbl
[11] and , A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | MR | Zbl
[12] and , Space-time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations. SIAM J. Control Optim. (2011) 1150-1170. | MR | Zbl
[13] and , Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277-302. | Numdam | MR
[14] , Multiplier methods for nonlinear optimal control. SIAM J. Numer. Anal. 27 (1990) 1061-1080. | MR | Zbl
[15] , , and , Optimization with PDE Constraints, vol. 23 of Math. Model. Theor. Appl. Springer (2009). | MR | Zbl
[16] , , and , 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] , and . Multiobjective PDE-constrained optimization using the reduced-basis method. Technical report, Universität Konstanz (2013).
[18] and , Receding horizon optimal control for infinite dimensional systems. ESAIM: COCV 8 (2002) 741-760. | Numdam | MR | Zbl
[19] and , Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems. SIAM J. Numer. Anal. 46 (2008) 2867-2891. | MR | Zbl
[20] and , A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403-425. | MR | Zbl
[21] and , A reduced basis method for optimal control of unsteady viscous flows. Int. J. Comput. Fluid D. 15 (2001) 97-113. | MR | Zbl
[22] , The reduced-basis method for parametrized linear-quadratic elliptic optimal control problems. Master's thesis, Technische Universität München (2011).
[23] and . A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM: COCV 20 (2014) 416-441. | Numdam | Zbl
[24] and , 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] , and , HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 3 (2004) 701-722. | MR | Zbl
[26] , , , , , , and , Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics. Birkhäuser Basel (2012). | MR | Zbl
[27] , Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971). | MR | Zbl
[28] , and , Convergence of approximations to nonlinear optimal control problems, vol. 195. CRC Press (1997) 253-284. | MR | Zbl
[29] , , and , Reduced basis method for parametrized elliptic optimal control problems. SIAM J. Sci. Comput. 35 (2013) A2316-A2340. | MR | Zbl
[30] , A “HUM” Conjugate Gradient Algorithm for Constrained Nonlinear Optimal Control: Terminal and Regulator Problems. Ph.D. thesis, Massachusetts Institute of Technology (2002).
[31] , , , , , and . Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluid. Eng. 124 (2002) 70-80.
[32] and , Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Ser. Comput. Math. Springer (2008). | Zbl
[33] , and , 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] and , 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] , and , 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] and , POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83-115. | MR | Zbl
[37] and , A new error bound for reduced basis approximation of parabolic partial differential equations. C. R. Math. 350 (2012) 203-207. | MR | Zbl
[38] , and , 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
Cité par Sources :
