IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v177y2018i2d10.1007_s10957-018-1242-4.html
   My bibliography  Save this article

An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design

Author

Listed:
  • Thomas Carraro

    (Heidelberg University)

  • Simon Dörsam

    (Heidelberg University)

  • Stefan Frei

    (University College London)

  • Daniel Schwarz

    (Behavioural Neurophysiology, Max Planck Institute for Medical Research
    Heidelberg University Hospital
    University of Heidelberg)

Abstract

In this work, we present an adaptive Newton-type method to solve nonlinear constrained optimization problems, in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive strategy is based on a goal-oriented a posteriori error estimation for the discretization and for the iteration error. The iteration error stems from an inexact solution of the nonlinear system of first-order optimality conditions by the Newton-type method. This strategy allows one to balance the two errors and to derive effective stopping criteria for the Newton iterations. The algorithm proceeds with the search of the optimal point on coarse grids, which are refined only if the discretization error becomes dominant. Using computable error indicators, the mesh is refined locally leading to a highly efficient solution process. The performance of the algorithm is shown with several examples and in particular with an application in the neurosciences: the optimal electrode design for the study of neuronal networks.

Suggested Citation

  • Thomas Carraro & Simon Dörsam & Stefan Frei & Daniel Schwarz, 2018. "An Adaptive Newton Algorithm for Optimal Control Problems with Application to Optimal Electrode Design," Journal of Optimization Theory and Applications, Springer, vol. 177(2), pages 498-534, May.
    • Handle: RePEc:spr:joptap:v:177:y:2018:i:2:d:10.1007_s10957-018-1242-4
    DOI: 10.1007/s10957-018-1242-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-018-1242-4
    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/s10957-018-1242-4?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. D. Schwarz & M. Kollo & C. Bosch & C. Feinauer & I. Whiteley & T. W. Margrie & T. Cutforth & A. T. Schaefer, 2018. "Architecture of a mammalian glomerular domain revealed by novel volume electroporation using nanoengineered microelectrodes," Nature Communications, Nature, vol. 9(1), pages 1-14, December.
    2. W. Wollner, 2010. "A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints," Computational Optimization and Applications, Springer, vol. 47(1), pages 133-159, September.
    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. A. Rösch & K. G. Siebert & S. Steinig, 2017. "Reliable a posteriori error estimation for state-constrained optimal control," Computational Optimization and Applications, Springer, vol. 68(1), pages 121-162, September.
    2. Hamdullah Yücel & Peter Benner, 2015. "Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations," Computational Optimization and Applications, Springer, vol. 62(1), pages 291-321, September.
    3. Carles Bosch & Tobias Ackels & Alexandra Pacureanu & Yuxin Zhang & Christopher J. Peddie & Manuel Berning & Norman Rzepka & Marie-Christine Zdora & Isabell Whiteley & Malte Storm & Anne Bonnin & Chris, 2022. "Functional and multiscale 3D structural investigation of brain tissue through correlative in vivo physiology, synchrotron microtomography and volume electron microscopy," Nature Communications, Nature, vol. 13(1), pages 1-16, 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:joptap:v:177:y:2018:i:2:d:10.1007_s10957-018-1242-4. 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.