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

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H. Antil

Department of Mathematics

University of Houston

M. Heinkenschloss

Department of Computational and Applied Mathematics

Rice University

R. H. W. Hoppe

Department of Mathematics

University of Houston

Department of Mathematics

University of Augsburg

D. C. Sorensen

Department of Computational and Applied Mathematics

Rice University

Computing and Visualization in Science, Vol. 13, No. 6 (2010) pp. 249-264

### Abstract

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.
#### Keywords

Optimal control, shape optimization, domain decomposition, model reduction.