Hostname: page-component-6b989bf9dc-md2j5 Total loading time: 0 Render date: 2024-04-13T11:06:59.564Z Has data issue: false hasContentIssue false

A certified reduced basis method for parametrized elliptic optimal control problems

Published online by Cambridge University Press:  07 March 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


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.

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.)


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
R. Becker and R. Rannacher, Weighted a posteriori error control in FE methods, in Proc. of ENUMATH-97. World Scientific Publishing (1998) 621–637.
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 for parametrized elliptic advection-reaction problems. J. Comput. Math. 28 (2010) 122148. 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
A.-L. Gerner and K. Veroy, Certified reduced basis methods for parametrized saddle point problems. Accepted in SIAM J. Sci. Comput. (2012).
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., Maday, Y., Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575605. 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
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
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
K. Ito and S.S. Ravindran, 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.
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 Dyn. 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).
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. System 3 (2004) 701722. Google Scholar
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971).
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D.V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Math. 331 (2000) 153158. Google Scholar
F. Negri, Reduced basis method for parametrized optimal control problems governed by PDEs, Master’s thesis. Politecnico di Milano (2011).
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).
Paraschivoiu, M., Peraire, J. and Patera, A.T., 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) 289312. Google Scholar
Pierce, N.A. and Giles, M.B., Adjoint recovery of superconvergent functionals from pde approximations. SIAM Review 42 (2000) 247264. Google Scholar
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. Fluids Engrg. 124 (2002) 7080. Google Scholar
Quarteroni, A., Lassila, T., Manzoni, A. and Rozza, G., Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM: M2AN 47 (2013) 11071131. Google Scholar
A. Quarteroni, G. Rozza, L. Dedè and A. Quaini, 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.
A.M. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, vol. 23 of Springer Series in 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. Methods Engrg. 15 (2008) 229275. 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. Modell. Dyn. Syst. 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
Veroy, K. and Patera, A.T., 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) 773788. Google Scholar
K. Veroy, C. Prud’homme, D.V. Rovas and A.T. Patera, 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.
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
G. Vossen and S. Volkwein, 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).