  Inspired by the matching of supply to demand in logistical problems, the
optimal transport (or Monge--Kantorovich) problem involves the matching of
probability distributions defined over a geometric domain such as a surface or
manifold. In its most obvious discretization, optimal transport becomes a
large-scale linear program, which typically is infeasible to solve efficiently
on triangle meshes, graphs, point clouds, and other domains encountered in
graphics and machine learning. Recent breakthroughs in numerical optimal
transport, however, enable scalability to orders-of-magnitude larger problems,
solvable in a fraction of a second. Here, we discuss advances in numerical
optimal transport that leverage understanding of both discrete and smooth
aspects of the problem. State-of-the-art techniques in discrete optimal
transport combine insight from partial differential equations (PDE) with convex
analysis to reformulate, discretize, and optimize transportation problems. The
end result is a set of theoretically-justified models suitable for domains with
thousands or millions of vertices. Since numerical optimal transport is a
relatively new discipline, special emphasis is placed on identifying and
explaining open problems in need of mathematical insight and additional
research.
