Domain Decomposition and Model Reduction for the Numerical Solution of PDE Constrained Optimization Problems with Localized Optimization Variables

  • We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.

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Author:Harbir AntilGND, Matthias Heinkenschloss, Ronald H. W. HoppeORCiDGND, Danny C. Sorensen
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Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2009-11)
Publishing Institution:Universität Augsburg
Contributing Corporation:University of Houston, Rice University
Release Date:2009/04/30
Tag:optimal control; shape optimization; domain decomposition; balanced truncation model reduction
GND-Keyword:Optimale Kontrolle; Mathematisches Modell; Ordnungsreduktion; Partielle Differentialgleichung; Gebietszerlegungsmethode
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand