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We present a primal-dual method to solve L 1 -type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L 1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 × 4096 in a few minutes without parallelization.

We present a method to efficiently solve the optimal transportation problem for a general class of strictly convex costs. Given two probability measures supported on a discrete grid with n points we compute the optimal transport map between them in O(n log(n)) operations and O(n) storage space. Our approach allows us to solve optimal transportation problems on spatial grids as large as 4096 × 4096 and 384 × 384 × 384 in a matter of minutes.

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.

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