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Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

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Computing and Visualization in Science

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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Authors and Affiliations

  1. Department of Computational and Applied Mathematics, MS-134, Rice University, 6100 Main Street, Houston, TX, 77005-1892, USA

    Harbir Antil, Matthias Heinkenschloss & Danny C. Sorensen

  2. Department of Mathematics, University of Houston, 4800 Calhoun, Houston, TX, 77204-3008, USA

    Ronald H. W. Hoppe

  3. Institute of Mathematics, University of Augsburg, 86159, Augsburg, Germany

    Ronald H. W. Hoppe

Authors
  1. Harbir Antil
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  2. Matthias Heinkenschloss
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  3. Ronald H. W. Hoppe
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  4. Danny C. Sorensen
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Corresponding author

Correspondence to Matthias Heinkenschloss.

Additional information

Communicated by Gabriel Wittum.

The research of RH was supported in part by NSF grants DMS-0511624, DMS-0707602, DMS-0810176, DMS-0811153 and by the German National Science Foundation (DFG) within the Priority Program SPP 1253. The research of MH was supported in part by NSF grant DMS-0915238 and AFOSR grant FA9550-09-1-0225.

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Antil, H., Heinkenschloss, M., Hoppe, R.H.W. et al. Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables. Comput. Visual Sci. 13, 249–264 (2010). https://doi.org/10.1007/s00791-010-0142-4

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  • DOI: https://doi.org/10.1007/s00791-010-0142-4

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