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


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

Mathematics Subject Classification

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