In this paper, we introduce a novel and extremely fast algorithm to compute continuous transport maps between 2D probability densities discretized on uniform grids. The core of our method is a novel iterative solver computing the L 2 optimal transport map from a grid to the uniform density in the 2D Euclidean plane. A transport map between arbitrary densities is then recovered through numerical inversion and composition. In this case, the resulting map is only approximately optimal, but it is continuous and density preserving. Our solver is derivative-free, and it converges in a few cheap iterations. We demonstrate interactive performance in various applications such as adaptive sampling, feature sensitive remeshing, and caustic design.