Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size

Matt Jacobs, Flavien Léger, Wuchen Li, Stanley J. Osher

Abstract: We present a primal-dual method to solve $L^1$-type nonsmooth 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 x 4096 in a few minutes without parallelization.