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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| Veröffentlicht in: | Computing and visualization in science Jg. 13; H. 6; S. 249 - 264 |
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01.08.2010
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| 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. |
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| AbstractList | 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. |
| Author | Sorensen, Danny C. Hoppe, Ronald H. W. Heinkenschloss, Matthias Antil, Harbir |
| Author_xml | – sequence: 1 givenname: Harbir surname: Antil fullname: Antil, Harbir organization: Department of Computational and Applied Mathematics, MS-134, Rice University – sequence: 2 givenname: Matthias surname: Heinkenschloss fullname: Heinkenschloss, Matthias email: heinken@rice.edu organization: Department of Computational and Applied Mathematics, MS-134, Rice University – sequence: 3 givenname: Ronald H. W. surname: Hoppe fullname: Hoppe, Ronald H. W. organization: Department of Mathematics, University of Houston, Institute of Mathematics, University of Augsburg – sequence: 4 givenname: Danny C. surname: Sorensen fullname: Sorensen, Danny C. organization: Department of Computational and Applied Mathematics, MS-134, Rice University |
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| Cites_doi | 10.1007/PL00000950 10.1007/s007910050039 10.1109/TAC.1981.1102568 10.2514/2.1576 10.1007/978-3-540-69777-0_46 10.1016/j.paerosci.2003.12.001 10.1051/m2an:2005044 10.1016/S0168-874X(03)00054-4 10.1002/rnc.657 10.1051/m2an:2001126 10.1016/S0045-7825(01)00302-4 10.1007/3-540-27909-1_10 10.1080/00207178408933239 10.1023/A:1022205420182 10.1137/1.9780898718713 10.1007/978-1-4757-3290-0 10.1137/1.9781611970920 10.1137/1.9780898717761 10.1007/3-540-27909-1 10.1007/978-94-017-1247-7_7 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables |
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