A multigrid scheme for solving convection–diffusion-integral optimal control problems

Abstract

The fast multigrid solution of an optimal control problem governed by a convection–diffusion partial-integro differential equation is investigated. This optimization problem considers a cost functional of tracking type and a constrained distributed control. The optimal control sought is characterized by the solution to the corresponding optimality system, which is approximated by a finite volume and quadrature discretization schemes and solved by multigrid techniques. The proposed multigrid approach combines a multigrid method for the governing model with a fast multigrid integration method. The convergence of this solution procedure is analyzed by local Fourier analysis and validated by results of numerical experiments.

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Acknowledgements

We would like to thank Volker Schulz for many helpful comments. This work was supported in part by the DAAD, by the European Union under Grant Agreement Nr. 304617 ‘Multi-ITN STRIKE - Novel Methods in Computational Finance’, and by the Würzburg-Wroclaw Center for Stochastic Computing.

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Correspondence to Alfio Borzì.

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Communicated by Volker Schulz.

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Gathungu, D.K., Borzì, A. A multigrid scheme for solving convection–diffusion-integral optimal control problems. Comput. Visual Sci. 22, 43–55 (2019). https://doi.org/10.1007/s00791-017-0285-7

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Keywords

  • Partial-integro differential equations
  • Optimal control
  • Multigrid scheme
  • Finite differences

Mathematics Subject Classification

  • 47G20
  • 49J20
  • 49K20
  • 65N55
  • 65N06