IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v70y2018i1d10.1007_s10589-018-9979-0.html
   My bibliography  Save this article

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

Author

Listed:
  • Eduardo Casas

    (Universidad de Cantabria)

  • Mariano Mateos

    (Universidad de Oviedo)

  • Arnd Rösch

    (Universtät Duisburg-Essen)

Abstract

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.

Suggested Citation

  • Eduardo Casas & Mariano Mateos & Arnd Rösch, 2018. "Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity," Computational Optimization and Applications, Springer, vol. 70(1), pages 239-266, May.
  • Handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-018-9979-0
    DOI: 10.1007/s10589-018-9979-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-018-9979-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-018-9979-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Christian Clason & Karl Kunisch, 2012. "A measure space approach to optimal source placement," Computational Optimization and Applications, Springer, vol. 53(1), pages 155-171, September.
    2. Eduardo Casas & Fredi Tröltzsch, 2012. "A general theorem on error estimates with application to a quasilinear elliptic optimal control problem," Computational Optimization and Applications, Springer, vol. 53(1), pages 173-206, September.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michael Hintermüller & Tao Wu, 2014. "A superlinearly convergent R-regularized Newton scheme for variational models with concave sparsity-promoting priors," Computational Optimization and Applications, Springer, vol. 57(1), pages 1-25, January.
    2. Malte Braack & Martin F. Quaas & Benjamin Tews & Boris Vexler, 2018. "Optimization of Fishing Strategies in Space and Time as a Non-convex Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 950-972, September.
    3. Roland Herzog & Johannes Obermeier & Gerd Wachsmuth, 2015. "Annular and sectorial sparsity in optimal control of elliptic equations," Computational Optimization and Applications, Springer, vol. 62(1), pages 157-180, September.
    4. Richard C. Barnard & Christian Clason, 2017. "$$L^1$$ L 1 penalization of volumetric dose objectives in optimal control of PDEs," Computational Optimization and Applications, Springer, vol. 67(2), pages 401-419, June.
    5. Moritz Keuthen & Michael Ulbrich, 2015. "Moreau–Yosida regularization in shape optimization with geometric constraints," Computational Optimization and Applications, Springer, vol. 62(1), pages 181-216, September.
    6. Francisco Fuica & Felipe Lepe & Enrique Otárola & Daniel Quero, 2023. "An Optimal Control Problem for the Navier–Stokes Equations with Point Sources," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 590-616, February.
    7. Christian Clason & Karl Kunisch, 2012. "A measure space approach to optimal source placement," Computational Optimization and Applications, Springer, vol. 53(1), pages 155-171, September.
    8. 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.
    9. Mattia Bongini & Massimo Fornasier & Francesco Rossi & Francesco Solombrino, 2017. "Mean-Field Pontryagin Maximum Principle," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 1-38, October.
    10. Pedro Merino, 2019. "A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs," Computational Optimization and Applications, Springer, vol. 74(1), pages 225-258, September.
    11. Xiaotong Chen & Xiaoliang Song & Zixuan Chen & Lijun Xu, 2023. "A Multilevel Heterogeneous ADMM Algorithm for Elliptic Optimal Control Problems with L 1 -Control Cost," Mathematics, MDPI, vol. 11(3), pages 1-21, January.
    12. Xiaoliang Song & Bo Chen & Bo Yu, 2018. "An efficient duality-based approach for PDE-constrained sparse optimization," Computational Optimization and Applications, Springer, vol. 69(2), pages 461-500, March.
    13. J. C. De Los Reyes & E. Loayza & P. Merino, 2017. "Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization," Computational Optimization and Applications, Springer, vol. 67(2), pages 225-258, June.
    14. Francesca C. Chittaro & Laura Poggiolini, 2019. "Strong Local Optimality for Generalized L1 Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 207-234, January.
    15. Li, Hongyi & Wang, Chaojie & Zhao, Di, 2020. "Preconditioning for PDE-constrained optimization with total variation regularization," Applied Mathematics and Computation, Elsevier, vol. 386(C).
    16. Sergio González-Andrade & Sofía López-Ordóñez & Pedro Merino, 2021. "Nonsmooth exact penalization second-order methods for incompressible bi-viscous fluids," Computational Optimization and Applications, Springer, vol. 80(3), pages 979-1025, December.
    17. Dante Kalise & Karl Kunisch & Zhiping Rao, 2017. "Infinite Horizon Sparse Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 481-517, February.
    18. Daria Ghilli & Karl Kunisch, 2019. "On monotone and primal-dual active set schemes for $$\ell ^p$$ ℓ p -type problems, $$p \in (0,1]$$ p ∈ ( 0 , 1 ]," Computational Optimization and Applications, Springer, vol. 72(1), pages 45-85, January.
    19. Bui Trong Kien & Arnd Rösch & Nguyen Hai Son & Nguyen Van Tuyen, 2023. "FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 130-173, April.
    20. Christian Clason & Thi Bich Tram Do & Frank Pörner, 2018. "Error estimates for the approximation of multibang control problems," Computational Optimization and Applications, Springer, vol. 71(3), pages 857-878, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:70:y:2018:i:1:d:10.1007_s10589-018-9979-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.