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...

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Bibliographic Details
Published in:Computing and visualization in science Vol. 13; no. 6; pp. 249 - 264
Main Authors: Antil, Harbir, Heinkenschloss, Matthias, Hoppe, Ronald H. W., Sorensen, Danny C.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.08.2010
Springer
Subjects:
Algorithms
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Computational Mathematics and Numerical Analysis
Computer Applications in Chemistry
Computer science; control theory; systems
Exact sciences and technology
Fluid dynamics
Fundamental areas of phenomenology (including applications)
Mathematics
Mathematics and Statistics
Numerical Analysis
Physics
Regular Article
Software
Software engineering
Solid mechanics
Structural and continuum mechanics
Turbulence simulation and modeling
Turbulent flows, convection, and heat transfer
Visualization
Model reduction
Domain decomposition
Optimal control
Shape optimization
Geometrical shape
Software development
Error estimation
Linear time
Advection diffusion equation
Substructure
Modeling
Constrained optimization
Truncation
Time dependence
Dimension reduction
Reduced order systems
Coupling
Optimal control (mathematics)
Mathematical programming
ISSN:1432-9360, 1433-0369
Online Access:Get full text
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Summary: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.
ISSN:1432-9360
1433-0369
DOI:10.1007/s00791-010-0142-4