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A path-following inexact Newton method for PDE-constrained optimal control in BV

Author

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  • D. Hafemeyer

    (TU München)

  • F. Mannel

    (University of Graz)

Abstract

We study a PDE-constrained optimal control problem that involves functions of bounded variation as controls and includes the TV seminorm of the control in the objective. We apply a path-following inexact Newton method to the problems that arise from smoothing the TV seminorm and adding an $$H^1$$ H 1 regularization. We prove in an infinite-dimensional setting that, first, the solutions of these auxiliary problems converge to the solution of the original problem and, second, that an inexact Newton method enjoys fast local convergence when applied to a reformulation of the auxiliary optimality systems in which the control appears as implicit function of the adjoint state. We show convergence of a Finite Element approximation, provide a globalized preconditioned inexact Newton method as solver for the discretized auxiliary problems, and embed it into an inexact path-following scheme. We construct a two-dimensional test problem with fully explicit solution and present numerical results to illustrate the accuracy and robustness of the approach.

Suggested Citation

  • D. Hafemeyer & F. Mannel, 2022. "A path-following inexact Newton method for PDE-constrained optimal control in BV," Computational Optimization and Applications, Springer, vol. 82(3), pages 753-794, July.
    • Handle: RePEc:spr:coopap:v:82:y:2022:i:3:d:10.1007_s10589-022-00370-2
    DOI: 10.1007/s10589-022-00370-2
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    References listed on IDEAS

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    1. Ole Løseth Elvetun & Bjørn Fredrik Nielsen, 2016. "The split Bregman algorithm applied to PDE-constrained optimization problems with total variation regularization," Computational Optimization and Applications, Springer, vol. 64(3), pages 699-724, July.
    2. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
    3. Li, Hongyi & Wang, Chaojie & Zhao, Di, 2020. "Preconditioning for PDE-constrained optimization with total variation regularization," Applied Mathematics and Computation, Elsevier, vol. 386(C).
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