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ADMM and Non-convex Variational Problems

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Splitting Methods in Communication, Imaging, Science, and Engineering

Part of the book series: Scientific Computation ((SCIENTCOMP))

Abstract

Our main goal in this chapter is to discuss the application of Alternating Direction Methods of Multipliers (ADMM) to the numerical solution of non-convex (and possibly non-smooth) variational problems. After giving a relatively detailed history of the ADMM methodology, we will discuss its application to the solution of problems from nonlinear Continuum Mechanics, nonlinear Elasticity, in particular. The ADMM solution of the two-dimensional Dirichlet problem for the Monge-Ampère equation will be discussed also. The results of numerical experiments will be reported, in order to illustrate the capabilities of the methodology under consideration

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Acknowledgements

The author wants to thank this chapter referee and his present and former colleagues and collaborators J.F. Bourgat, A. Caboussat, E.J. Dean, J.M. Dumay, P. Le Tallec, A. Quaini, T.W. Pan, V. Pons, and L. Tartar for their invaluable help and suggestions. The support of NSF grants DMS 0412267 and DMS 0913982 is also acknowledged.

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  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Roland Glowinski

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  1. Roland Glowinski
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Correspondence to Roland Glowinski .

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Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, Texas, USA

    Roland Glowinski

  2. Department of Mathematics, UCLA, Los Angeles, California, USA

    Stanley J. Osher

  3. Department of Mathematics, UCLA, Los Angeles, California, USA

    Wotao Yin

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Glowinski, R. (2016). ADMM and Non-convex Variational Problems. In: Glowinski, R., Osher, S., Yin, W. (eds) Splitting Methods in Communication, Imaging, Science, and Engineering. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-319-41589-5_8

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