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