An Introduction and Summary of Use of Optimal Control Methods for PDE’s

  • Owe AxelssonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)


In optimal control formulations of partial differential equations the aim is to find a control function that steers the solution to a desired form. A Lagrange multiplier, i.e. an adjoint variable is introduced to handle the PDE constraint. One can reduce the problem to a two-by-two block matrix form with square blocks for which a very efficient preconditioner, PRESB can be applied. This method gives sharp and tight eigenvalue bounds, which hold uniformly with respect to regularization, mesh size and problem parameters, and enable use of the second order inner product free Chebyshev iteration method, which latter enables implementation on parallel computers without any need to use global data communications. Furthermore this method is insensitive to round-off errors. It outperforms other earlier published methods. Implementational and spectral properties of the method, and a short survey of applications, are given.


Optimal control Preconditioner Inner product free 


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of GeonicsThe Czech Academy of SciencesOstravaCzech Republic
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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