  We rephrase Monge's optimal transportation (OT) problem with quadratic
cost--via a Monge-Amp\`ere equation--as an infinite-dimensional optimization
problem, which is in fact a convex problem when the target is a log-concave
measure with convex support. We define a natural finite-dimensional
discretization to the problem and associate a piecewise affine convex function
to the solution of this discrete problem. The discrete problems always admit a
solution, which can be obtained by standard convex optimization algorithms
whenever the target is a log-concave measure with convex support. We show that
under suitable regularity conditions the convex functions retrieved from the
discrete problems converge to the convex solution of the original OT problem
furnished by Brenier's theorem. Also, we put forward an interpretation of our
convergence result that suggests applicability to the convergence of a wider
range of numerical methods for OT. Finally, we demonstrate the practicality of
our convergence result by providing visualizations of OT maps as well as of the
dynamic OT problem obtained by solving the discrete problem numerically.
