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Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. (English) Zbl 1220.65074

Summary: We introduce a technique for the dimension reduction of a class of partial differential equation (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.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M27 Decomposition methods
49Q10 Optimization of shapes other than minimal surfaces

Software:

INTLAB; KELLEY
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Full Text: DOI Link

References:

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