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A certified reduced basis method for parametrized ellipticoptimal control problems∗

Published online by Cambridge University Press:  07 March 2014

Mark Kärcher  and
Martin A. Grepl
Mark Kärcher
Affiliation:
Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany. kaercher@aices.rwth-aachen.de
Martin A. Grepl
Affiliation:
Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany; grepl@igpm.rwth-aachen.de

Abstract

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

Keywords

Optimal controlreduced basis methoda posteriori error estimationmodel order reductionparameter-dependent systems

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 416 - 441
Copyright
© EDP Sciences, SMAI, 2014

References

Atwell, J.A. and King, B.B., Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33 (2001) 1–19. 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) 113–132. 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) 997–1019. Google Scholar
Dedè, L., Reduced basis method for parametrized elliptic advection-reaction problems. J. Comput. Math. 28 (2010) 122–148. Google Scholar
Dedè, L., Reduced basis method and error estimation for parametrized optimal control problems with control constraints. J. Sci. Comput. 50 (2012) 287–305. 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) 873–877. 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) 575–605. 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) 157–181. 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) 473–478. 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) 2867–2891. 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) 403–425. 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) 97–113. 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) 345–371. 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) 701–722. 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) 153–158. 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) 289–312. Google Scholar
Pierce, N.A. and Giles, M.B., Adjoint recovery of superconvergent functionals from pde approximations. SIAM Review 42 (2000) 247–264. 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) 70–80. 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) 1107–1131. 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) 229–275. 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) 355–369. Google Scholar
Tröltzsch, F. and Volkwein, S., POD a posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83–115. 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) 773–788. 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) 1007–1028. 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).