POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems

Kahlbacher, Martin; Volkwein, Stefan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 491-511

An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD a-posteriori error estimator developed by Tröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83-115] the difference of the suboptimal to the (unknown) optimal solution of the linear-quadratic subproblem is estimated. Hence, the inexactness of the discrete solution is controlled in such a way that locally superlinear or even quadratic rate of convergence of the SQP is ensured. Numerical examples illustrate the efficiency for the proposed approach.

Classification : 35J47, 49K20, 49M15, 90C20
Keywords: optimal control, inexact SQP method, proper orthogonal decomposition, a-posteriori error estimates, bilinear elliptic equation

[1] W. Alt, The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201-224. | Zbl | MR

[2] A.C. Antoulas, Approximation of Large-Scale Dynamical Systems. Advances in Design and Control, SIAM, Philadelphia (2005). | Zbl | MR

[3] N. Arada, E. Casas and F. Tröltzsch. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201-229. | Zbl | MR

[4] E. Arian, M. Fahl and E.W. Sachs, Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000).

[5] J.A. Atwell, J.T. Borggaard and B.B. King, Reduced order controllers for Burgers' equation with a nonlinear observer. Int. J. Appl. Math. Comput. Sci. 11 (2001) 1311-1330. | Zbl | MR

[6] P. Benner and E.S. Quintana-Ortí, Model reduction based on spectral projection methods, in Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, edited by P. Benner, V. Mehrmann and D.C. Sorensen (2005) 5-48. | Zbl | MR

[7] P. Deuflhard, Newton Methods for Nonlinear Problems : Affine Invariance and Adaptive Algorithms, Springer Series in Comput. Math. 35 (2004). | Zbl | MR

[8] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island 19 (2002). | Zbl | MR

[9] R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28 (1974) 963-971. | Zbl | MR

[10] T. Gänzler, S. Volkwein and M. Weiser, SQP methods for parameter identification problems arising in hyperthermia. Optim. Methods Softw. 21 (2006) 869-887. | Zbl | MR

[11] M. Hintermüller, On a globalized augmented Lagrangian SQP-algorithm for nonlinear optimal control problems with box constraints, in Fast solution methods for discretized optimization problems, International Series of Numerical Mathematics. edited by K.-H. Hoffmann, R.H.W. Hoppe and V. Schulz, Birkhäuser publishers, Basel 138 (2001) 139-153. | Zbl | MR

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

[13] A. Kröner and B. Vexler, A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Math. 230 (2009) 781-802. | Zbl | MR

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

[15] H.V. Ly and H.T. Tran, Modeling and control of physical processes using proper orthogonal decomposition. Math. Comput. Model. 33 (2001) 223-236. | Zbl

[16] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in Mathematical Programming with Data Perturbation, edited by A.V. Fiacco and M. Dekker. Inc., New York (1997) 253-284. | Zbl

[17] 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 (2006).

[18] S.S. Ravindran, Adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 28 (2002) 1924-1942. | Zbl | MR

[19] M. Read and B. Simon, Methods of Modern Mathematical Physics I : Functional Analysis. Academic Press, Boston (1980). | Zbl | MR

[20] E.W. Sachs and S. Volkwein, Augmented Lagrange-SQP methods with Lipschitz-continuous Lagrange multiplier updates. SIAM J. Numer. Anal. 40 (2002) 233-253. | Zbl | MR

[21] L. Sirovich, Turbulence and the dynamics of coherent structures, parts I-III. Quart. Appl. Math. XLV (1987) 561-590. | Zbl | MR

[22] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a-posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Modelling of Dynam. Systems 17 (2011) 355-369. | MR

[23] F. Tröltzsch, Optimal Control of Partial Differential Equations : Theory, Methods and Applications, Graduate Studies in Mathematics. American Mathematical Society 112 (2010). | Zbl

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

[25] M. Vallejos and A. Borzì, Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing 82 (2008) 31-52. | Zbl | MR

[26] S. Volkwein, Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces. SIAM J. Control Optimization 38 (2000) 767-785. | Zbl | MR