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An Efficient Primal-Dual Method for the Obstacle Problem

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Abstract

We solve the non-linearized and linearized obstacle problems efficiently using a primal-dual hybrid gradients method involving projection and/or \(L^1\) penalty. Since this method requires no matrix inversions or explicit identification of the contact set, we find that this method, on a variety of test problems, achieves the precision of previous methods with a speed up of 1–2 orders of magnitude. The derivation of this method is disciplined, relying on a saddle point formulation of the convex problem, and can be adapted to a wide range of other constrained convex optimization problems.

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Notes

  1. Code available at http://www.math.montana.edu/dzosso/code.

  2. None of these sources explicitly specifies the boundary values other than claiming \(f=0\) on “some” (undisclosed) location, clearly different from the square border \(\partial \varOmega \).

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Acknowledgements

The authors thank W. Feldman, I. Kim, and G. Tran for helpful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Montana State University, Bozeman, MT, 59717-2400, USA

    Dominique Zosso

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Braxton Osting

  3. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, 90095-1555, USA

    Mandy(Mengqi) Xia & Stanley J. Osher

Authors
  1. Dominique Zosso
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  2. Braxton Osting
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  3. Mandy(Mengqi) Xia
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  4. Stanley J. Osher
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Corresponding author

Correspondence to Dominique Zosso.

Additional information

DZ and BO gratefully acknowledge support from NSF DMS-1461138. DZ was also supported by UC Lab Fees Research Grant 12-LR-236660 and the W. M. Keck Foundation. SJO is supported by NSF DMS-1118971, UC Lab Fees Grant, and ONR Grant N00014-14-0444.

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Zosso, D., Osting, B., Xia, M. et al. An Efficient Primal-Dual Method for the Obstacle Problem. J Sci Comput 73, 416–437 (2017). https://doi.org/10.1007/s10915-017-0420-0

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  • DOI: https://doi.org/10.1007/s10915-017-0420-0

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