Numerical solution of a nonlinear parabolic control problem by a reduced SQP method
Identifieur interne : 002D53 ( Main/Exploration ); précédent : 002D52; suivant : 002D54Numerical solution of a nonlinear parabolic control problem by a reduced SQP method
Auteurs : F. S. Kupfer [Allemagne] ; E. W. Sachs [Allemagne]Source :
- Computational Optimization and Applications [ 0926-6003 ] ; 1992-10-01.
Abstract
Abstract: We consider a control problem for a nonlinear diffusion equation with boundary input that occurs when heating ceramic products in a kiln. We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.
Url:
DOI: 10.1007/BF00247656
Affiliations:
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S." last="Kupfer">F. S. Kupfer</name><affiliation wicri:level="1"><country xml:lang="fr">Allemagne</country><wicri:regionArea>Universität Trier FB IV-Mathematik, Postfach 3825, W-5500, Trier</wicri:regionArea><wicri:noRegion>Trier</wicri:noRegion><wicri:noRegion>Trier</wicri:noRegion></affiliation></author><author><name sortKey="Sachs, E W" sort="Sachs, E W" uniqKey="Sachs E" first="E. W." last="Sachs">E. W. 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We interpret this control problem as a constrained optimization problem, and we develop a reduced SQP method that presents for this problem a new and efficient approach of its numerical solution. As opposed to Newton's method for the unconstrained problem, where at each iteration the state must be computed from a set of nonlinear equations,in the proposed algorithm only the linearized state equations need to be solved. Furthermore, by use of a secant update formula, the calculation of exact second derivatives is avoided. In this way the algorithm achieves a substantial decrease in the total cost compared to the implementation of Newton's method in [2]. Our method is practicable with regard to storage requirements, and by choosing an appropriate representation for the null space of the Jacobian of the constraints we are able to exploit the sparsity pattern of the Jacobian in the course of the iteration. We conclude with a presentation of numerical examples that demonstrate the fast two-step superlinear convergence behavior of the method.</div></front></TEI><affiliations><list><country><li>Allemagne</li></country></list><tree><country name="Allemagne"><noRegion><name sortKey="Kupfer, F S" sort="Kupfer, F S" uniqKey="Kupfer F" first="F. S." last="Kupfer">F. S. Kupfer</name></noRegion><name sortKey="Sachs, E W" sort="Sachs, E W" uniqKey="Sachs E" first="E. W." last="Sachs">E. W. Sachs</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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