A posteriori snapshot location for POD in optimal control of linear parabolic equations

Alla, Alessandro; Grässle, Carmen; Hinze, Michael

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1847-1873.

In this paper we study the approximation of an optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. We propose to determine the time instances (snapshot locations) by an a posteriori error control concept. The proposed method is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and show the effectiveness of the method in comparison to existing approaches.

Classification : 49J20, 65N12, 78M34
Mots clés : Optimal control, model order reduction, proper orthogonal decomposition, optimal snapshot location

[1] K. Afanasiev and M. Hinze, Adaptive Control of a Wake Flow Using Proper Orthogonal Decomposition. Vol. 216 of Lecture Notes in Pure and Applied Mathematics. Shape Optimization & Optimal Design, Marcel Dekker (2001) 317–332. | MR | Zbl

[2] A. Alla and M. Falcone, An Adaptive POD Approximation Method for the Control of Advection-Diffusion Equations. Vol. 164 of Control and Optimization with PDE Constraints, edited by K. Kunisch, K. Breides, C. Clason, G. Von Winckel. International Series of Numerical Mathematics, Birkhäuser, Basel (2013) 1–17. | MR | Zbl

[3] A. Alla, C. Gräßle and M. Hinze, A Residual Based Snapshot Location Strategy for POD in Distributed Optimal Control of Linear Parabolic Equations. Vol. 49 of 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2016 (2016) 13–18. | MR

[4] E. Arian, M. Fahl and E. Sachs, Trust-region proper orthogonal decomposition models by optimization methods. In Proc. of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada (2002) 3300–3305.

[5] L.C. Evans, Partial Differential Equations. Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2010). | MR | Zbl

[6] J. Ghiglieri and S. Ulbrich, Optimal flow control based on POD and MPC and an application to the cancellation of tollmien-schlichting waves. Optim. Methods Softw. 29 (2014) 1042–1074. | DOI | MR | Zbl

[7] W. Gong, M. Hinze and Z.J.Zhou, Space-time finite element approximation of parabolic optimal control problems. J. Numer. Math. 20 (2012) 111–145. | DOI | MR | Zbl

[8] C. Gräßle, M. Gubisch, S. Maier, S. Rogg and S. Volkwein, POD basis updates for nonlinear PDE control. Automatisierungstechnik 65 (2017) 298–307. | DOI

[9] M. Gubisch and S. Volkwein, Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control. Model Reduction and Approximation: Theory and Algorithms, edited by P. Benner, A. Cohen, M. Ohlberger and K. Willcox. SIAM, Philadelphia, PA (2017). | DOI | MR

[10] M. Gubisch, I. Neitzel and S. Volkwein, Model reduction and approximation: theory and algorithms. Computational Science and Engineering. SIAM (2016).

[11] M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. | DOI | MR | Zbl

[12] M. Hinze and M. Vierling, Variational discretization and semi-smooth Newton methods; implementation, convergence and globalization in PDE constrained optimization with control constraints. Optim. Methods Softw. 27 (2012) 933–950. | DOI | MR | Zbl

[13] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. S. Comput. Optim. Appl. 39 (2008) 319–345. | DOI | MR | Zbl

[14] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints. Vol. 23 of Mathematical Modelling: Theory and Applications. Springer Verlag (2009). | MR | Zbl

[15] R.H.W. Hoppe and Z. Liu, Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolicpartial differential equations. J. Numer. Math. 22 (2014) 1–32. | DOI | MR | Zbl

[16] E. Kammann, F. Tröltzsch and S. Volkwein, A method of a posteriori error estimation with application to proper orthogonal decomposition. ESAIM: M2AN 47 (2013) 555–581. | Numdam

[17] Z. Kanar Seymen, H. Yücel and B. Karasözen, Distributed optimal control of time-dependent diffusion-convection-reaction equations using space-time discretization. J. Comput. Appl. Math. 261 (2014) 146–157. | DOI | MR | Zbl

[18] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. | DOI | MR | Zbl

[19] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM, J.Numer. Anal. 40 (2002) 492–515. | DOI | MR | Zbl

[20] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems. ESAIM: M2AN 42 (2008) 1–23. | DOI | Numdam | MR | Zbl

[21] K. Kunisch and S. Volkwein, Optimal Snapshot Location for computing POD basis functions. ESAIM: M2AN 44 (2010) 509–529. | DOI | Numdam | MR | Zbl

[22] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer (1971). | MR | Zbl

[23] I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numerische Mathematik 120 (2012) 345–386. | DOI | MR | Zbl

[24] I. Neitzel, U. Prüfert and T. Slawig, A smooth regularization of the projection formula for constrained parabolic optimal control problems. Numer. Funct. Anal. Optim. 32 (2011) 1283–1315. | DOI | MR | Zbl

[25] J. Nocedal and S.J. Wright, Numerical Optimization, second edition. Springer Series in Operation Research (2006). | MR | Zbl

[26] N.C. Nguyen, G. Rozza and A.T. Patera, Reduced basis approximation and a posteriori error estimation for time dependent viscous Burgers equation. Calcolo 46 (2009) 157–185. | DOI | MR | Zbl

[27] G.M. Oxberry, T. Kostova-Vassilevska, B. Arrighi and K. Chand, Limited-memory adaptive snapshot selection for proper orthogonal decomposition. Int. J. Numer. Methods Eng. 109 (2017) 198–217. | DOI | MR

[28] A.T. Patera and G. Rozza, Reduced Basis Approximation and A posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering. Oxford University (2006).

[29] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods. Eng. 15 (2008) 229–275. | DOI | MR | Zbl

[30] L. Sirovich, Turbulence and the dynamics of coherent structures. Parts I–II. Q. Appl. Math. XVL (1987) 561–590. | DOI | Zbl

[31] F. Tröltzsch, Optimal control of partial differential equations: theory, methods and application. Am. Math. Soc. (2010). | MR | Zbl

[32] F. Tröltzsch and S. Volkwein, POD a posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83–115. | DOI | MR | Zbl

[33] S. Volkwein, Optimality system POD and a posteriori error analysis for linear-quadratic problems. Control Cybern. 40 (2011) 1109–1125. | MR | Zbl

[34] S. Volkwein and A. Studinger, Numerical Analysis of POD a posteriori error estimation for optimal control. Control and Optimization with PDE Constraints, edited by K. Kunisch, K. Breides, C. Clason and G. Von Winckel. In Vol. 164 of International Series of Numerical Mathematics. Birkhäuser, Basel (2013) 137–158. | MR | Zbl