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Dissertation zugänglich unter
URN: urn:nbn:de:hbz:3851317
URL: http://ubt.opus.hbznrw.de/volltexte/2004/131/
Trustregion Methods for Flow Control based on Reduced Order Modelling
n.a.
Fahl, Marco
SWDSchlagwörter: 
 Kontrolltheorie , Strömungsmechanik , NavierStokesGleichung , Orthogonale Zerlegung , TrustRegionAlgorithmus 
Freie Schlagwörter (Englisch): 
 flow control , NavierStokes equations , reduced order modelling , proper orthogonal decomposition , trustregion method 
Institut: 
 Mathematik 
Fakultät: 
 Fachbereich 4 
DDCSachgruppe: 
 Mathematik 
Dokumentart: 
 Dissertation 
Hauptberichter: 
 Prof. Dr. E. Sachs 
Sprache: 
 Englisch 
Tag der mündlichen Prüfung: 
 09.02.2001 
Erstellungsjahr: 
 2001 
Publikationsdatum: 
 04.06.2004 
Kurzfassung auf Englisch: 
 The discretization of optimal control problems governed by partial differential equations typically leads to largescale optimization problems. We consider flow control involving the timedependent NavierStokes equations as state equation which is stamped by exactly this property. In order to avoid the difficulties of dealing with largescale (discretized) state equations during the optimization process, a reduction of the number of state variables can be achieved by employing a reduced order modelling technique. Using the snapshot proper orthogonal decomposition method, one obtains a lowdimensional model for the computation of an approximate solution to the state equation. In fact, often a small number of POD basis functions suffices to obtain a satisfactory level of accuracy in the reduced order solution. However, the small number of degrees of freedom in a POD based reduced order model also constitutes its main weakness for optimal control purposes. Since a single reduced order model is based on the solution of the NavierStokes equations for a specified control, it might be an inadequate model when the control (and consequently also the actual corresponding flow behaviour) is altered, implying that the range of validity of a reduced order model, in general, is limited. Thus, it is likely to meet unreliable reduced order solutions during a control problem solution based on one single reduced order model. In order to get out of this dilemma, we propose to use a trustregion proper orthogonal decomposition (TRPOD) approach. By embedding the POD based reduced order modelling technique into a trustregion framework with general model functions, we obtain a mechanism for updating the reduced order models during the optimization process, enabling the reduced order models to represent the flow dynamics as altered by the control. In fact, a rigorous convergence theory for the TRPOD method is obtained which justifies this procedure also from a theoretical point of view. Benefiting from the trustregion philosophy, the TRPOD method guarantees to save a lot of computational work during the control problem solution, since the original state equation only has to be solved if we intend to update our model function in the trustregion framework. The optimization process itself is completely based on reduced order information only.

Kurzfassung auf Deutsch: 
 n.a. 
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