We solve the non-linearized and linearized obstacle problems efficiently using a primal-dual hybrid gradients method involving projection and/or ?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.

The code provided here was used to produce all figures of the following paper:
Zosso, D., Osting, B., Xia, M., and Osher, S., "An Efficient Primal-Dual Method for the Obstacle Problem", J Sci Comput (2017) 73(1):416-437.
https://doi.org/10.1007/s10915-017-0420-0