  Obtaining solutions to Optimal Transportation (OT) problems is typically
intractable when the marginal spaces are continuous. Recent research has
focused on approximating continuous solutions with discretization methods based
on i.i.d. sampling, and has proven convergence as the sample size increases.
However, obtaining OT solutions with large sample sizes requires intensive
computation effort, that can be prohibitive in practice. In this paper, we
propose an algorithm for calculating discretizations with a given number of
points for marginal distributions, by minimizing the (entropy-regularized)
Wasserstein distance, and result in plans that are comparable to those obtained
with much larger numbers of i.i.d. samples. Moreover, a local version of such
discretizations which is parallelizable for large scale applications is
proposed. We prove bounds for our approximation and demonstrate performance on
a wide range of problems.
