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Multigrid Optimization Methods for the Optimal Control of Convection–Diffusion Problems with Bilinear Control

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Abstract

Optimal control problems, governed by convection–diffusion equations with bilinear control, are studied. For the realization of the numerical solution, the multigrid for optimization method together with finite difference discretization is utilized and investigated. In addition, the extension to constrained optimal control problems with bilinear control is considered. Results of numerical experiments show the computational performance of the proposed multigrid scheme in solving optimal control problems subject to a convection–diffusion equation with bilinear control. We obtain that the proposed multigrid strategy accelerates classical one-grid optimization schemes and inherits the order of convergence of the finite difference discretization. Moreover, the mesh independence principle is obtained, which is a typical characterization of a multigrid strategy.

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Acknowledgments

The work of E.-J. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology NRF-2012R1A2A2A01046471. M. Vallejos Lass gratefully acknowledges the support by the Office of the Vice President for Academic Affairs, of the University of the Philippines Diliman, through the Creative Work and Research Grant. M. Vallejos Lass was supported in part by the WCU program through NRF.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Würzburg, Emil-Fischer-Straße 30, 97074 , Würzburg, Germany

    A. Borzì

  2. Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Korea

    E.-J. Park

  3. Institute of Mathematics, University of the Philippines, 1101 , Quezon City, Philippines

    M. Vallejos Lass

  4. Department of Mathematics, Technische Universität Darmstadt, Dolivostraße 15, 64293 , Darmstadt, Germany

    M. Vallejos Lass

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  1. A. Borzì
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  2. E.-J. Park
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  3. M. Vallejos Lass
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Correspondence to M. Vallejos Lass.

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Communicated by Boris Vexler.

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Borzì, A., Park, EJ. & Lass, M.V. Multigrid Optimization Methods for the Optimal Control of Convection–Diffusion Problems with Bilinear Control. J Optim Theory Appl 168, 510–533 (2016). https://doi.org/10.1007/s10957-015-0791-z

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  • DOI: https://doi.org/10.1007/s10957-015-0791-z

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