Abstract
An accurate and fast solution scheme for parabolic bilinear optimization problems is presented. Parabolic models where the control plays the role of a reaction coefficient and the objective is to track a desired trajectory are formulated and investigated. Existence and uniqueness of optimal solution are proved. A space-time discretization is proposed and second-order accuracy for the optimal solution is discussed. The resulting optimality system is solved with a nonlinear multigrid strategy that uses a local semismooth Newton step as smoothing scheme. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the ability of the multigrid scheme to solve the given optimization problems with mesh-independent efficiency.
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Communicated by: M. Stynes
Supported in part by the European Union under Grant Agreement Nr. 304617 Marie Curie Research Training Network ‘Multi-ITN STRIKE - Novel Methods in Computational Finance’, and by BMBF Verbundproject 05M2013 ‘ROENOBIO: Robust energy optimization of fermentation processes for the production of biogas and wine’, and by EFP Research Fellowship awarded by the Program Committee of the IMU Berlin Einstein Foundation Program 2011.
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Borzì, A., González Andrade, S. Second-order approximation and fast multigrid solution of parabolic bilinear optimization problems. Adv Comput Math 41, 457–488 (2015). https://doi.org/10.1007/s10444-014-9369-9
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DOI: https://doi.org/10.1007/s10444-014-9369-9
Keywords
- Multigrid methods
- Newton methods
- Finite differences
- Parabolic partial differential equations
- Bilinear control
- ptimal control theory