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