In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption of affine parameter dependence, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. Numerical results are presented to confirm the validity of our approach.

Keywords: optimal control, reduced basis method, a posteriori error estimation, model order reduction, parameter-dependent systems

@article{COCV_2014__20_2_416_0, author = {K\"archer, Mark and Grepl, Martin A.}, title = {A certified reduced basis method for parametrized elliptic optimal control problems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {416--441}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, doi = {10.1051/cocv/2013069}, mrnumber = {3264210}, zbl = {1287.49032}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2013069/} }

TY - JOUR AU - Kärcher, Mark AU - Grepl, Martin A. TI - A certified reduced basis method for parametrized elliptic optimal control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 416 EP - 441 VL - 20 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013069/ DO - 10.1051/cocv/2013069 LA - en ID - COCV_2014__20_2_416_0 ER -

%0 Journal Article %A Kärcher, Mark %A Grepl, Martin A. %T A certified reduced basis method for parametrized elliptic optimal control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 416-441 %V 20 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013069/ %R 10.1051/cocv/2013069 %G en %F COCV_2014__20_2_416_0

Kärcher, Mark; Grepl, Martin A. A certified reduced basis method for parametrized elliptic optimal control problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 416-441. doi : 10.1051/cocv/2013069. http://www.numdam.org/articles/10.1051/cocv/2013069/

[1] Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33 (2001) 1-19. | MR | Zbl

and ,[2] Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optim. 39 (2000) 113-132. | MR | Zbl

, and ,[3] Weighted a posteriori error control in FE methods, in Proc. of ENUMATH-97. World Scientific Publishing (1998) 621-637. | Zbl

and ,[4] 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

,[5] Reduced basis method for parametrized elliptic advection-reaction problems. J. Comput. Math. 28 (2010) 122-148. | MR | Zbl

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

,[7] Certified reduced basis methods for parametrized saddle point problems. Accepted in SIAM J. Sci. Comput. (2012). | MR | Zbl

and ,[8] Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Math. 349 (2011) 873-877. | MR | Zbl

and ,[9] Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575-605. | Numdam | MR | Zbl

, , and ,[10] A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157-181. | Numdam | MR | Zbl

and ,[11] 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

, , and ,[12] Reduced-order optimal control based on approximate inertial manifolds for nonlinear dynamical systems. SIAM J. Numer. Anal. 46 (2008) 2867-2891. | MR | Zbl

and ,[13] A reduced basis method for control problems governed by pdes, in Control and Estimation of Distributed Parameter Systems, vol. 126 of Internat. Series Numer. Math., edited by W. Desch, F. Kappel and K. Kunisch. Birkhäuser Basel (1998) 153-168. | MR | Zbl

and ,[14] A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403-425. | MR | Zbl

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

and ,[16] The reduced-basis method for parametrized linear-quadratic elliptic optimal control problems, Master's thesis. Technische Universität München (2011).

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

and ,[18] HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. System 3 (2004) 701-722. | MR | Zbl

, and ,[19] Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971). | MR | Zbl

,[20] Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Math. 331 (2000) 153-158. | MR | Zbl

, , , and ,[21] Reduced basis method for parametrized optimal control problems governed by PDEs, Master's thesis. Politecnico di Milano (2011).

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

,[23] A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Symposium on Advances in Computational Mechanics. Comput. Methods Appl. Mechanics Engrg. 150 (1997) 289-312. | MR | Zbl

, and ,[24] Adjoint recovery of superconvergent functionals from pde approximations. SIAM Review 42 (2000) 247-264. | MR | Zbl

and ,[25] Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002) 70-80.

, , , , , and ,[26] Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM: M2AN 47 (2013) 1107-1131. | Numdam | MR

, , and ,[27] Numerical approximation of a control problem for advection-diffusion processes, System Modeling and Optimization, in vol. 199 of IFIP International Federation for Information Processing. Edited by F. Ceragioli, A. Dontchev, H. Futura, K. Marti and L. Pandolfi. Springer (2006) 261-273. | MR | Zbl

, , and ,[28] Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Series in Comput. Math. Springer (2008). | Zbl

and ,[29] Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Engrg. 15 (2008) 229-275. | MR

, and ,[30] Comparison of the reduced-basis and pod a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Modell. Dyn. Syst. 17 (2011) 355-369. | MR

, and ,[31] POD a posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83-115. | MR | Zbl

and ,[32] Certifed real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Intern. J. Numer. Methods Fluids 47 (2005) 773-788. | MR | Zbl

and ,[33] A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proc. of the 16th AIAA Computational Fluid Dynamics Conference. AIAA Paper (2003) 2003-3847.

, , and ,[34] 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

, and ,[35] Model reduction techniques with a posteriori error analysis for linear-quadratic optimal control problems, in vol. 298 of Konstanzer Schriften in Mathematik. Universität Konstanz (2012). | MR | Zbl

and ,*Cited by Sources: *