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On Block Triangular Preconditioners for the Interior Point Solution of PDE-Constrained Optimization Problems

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Domain Decomposition Methods in Science and Engineering XXIV (DD 2017)

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 125))

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Abstract

We consider the numerical solution of saddle point systems of equations resulting from the discretization of PDE-constrained optimization problems, with additional bound constraints on the state and control variables, using an interior point method. In particular, we derive a Bramble–Pasciak Conjugate Gradient method and a tailored block triangular preconditioner which may be applied within it. Crucial to the usage of the preconditioner are carefully chosen approximations of the (1, 1)-block and Schur complement of the saddle point system. To apply the inverse of the Schur complement approximation, which is computationally the most expensive part of the preconditioner, one may then utilize methods such as multigrid or domain decomposition to handle individual sub-blocks of the matrix system.

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Acknowledgements

JWP gratefully acknowledges financial support from the Engineering and Physical Sciences Research Council (EPSRC) Fellowship EP/M018857/2. JG gratefully acknowledges support from the EPSRC Grant EP/N019652/1. The authors thank an anonymous referee for their helpful comments.

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Authors and Affiliations

  1. School of Mathematics, The University of Edinburgh, Edinburgh, UK

    John W. Pearson & Jacek Gondzio

Authors
  1. John W. Pearson
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  2. Jacek Gondzio
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Corresponding authors

Correspondence to John W. Pearson or Jacek Gondzio .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Petter E. Bjørstad

  2. Department of Mathematics, Louisana State University, Baton Rouge, LA, USA

    Susanne C. Brenner

  3. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Lawrence Halpern

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea (Republic of)

    Hyea Hyun Kim

  5. FB Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Ralf Kornhuber

  6. Western Norway University of Applied Sciences, Bergen, Norway

    Talal Rahman

  7. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Olof B. Widlund

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Pearson, J.W., Gondzio, J. (2018). On Block Triangular Preconditioners for the Interior Point Solution of PDE-Constrained Optimization Problems. In: Bjørstad, P., et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_48

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