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
Mots clés : optimal control, inexact SQP method, proper orthogonal decomposition, a-posteriori error estimates, bilinear elliptic equation

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