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.