  We establish numerical methods for solving the martingale optimal transport
problem (MOT) - a version of the classical optimal transport with an additional
martingale constraint on transport's dynamics. We prove that the MOT value can
be approximated using linear programming (LP) problems which result from a
discretisation of the marginal distributions combined with a suitable
relaxation of the martingale constraint. Specialising to dimension one, we
provide bounds on the convergence rate of the above scheme. We also show a
stability result under only partial specification of the marginal
distributions. Finally, we specialise to a particular discretisation scheme
which preserves the convex ordering and does not require the martingale
relaxation. We introduce an entropic regularisation for the corresponding LP
problem and detail the corresponding iterative Bregman projection. We also
rewrite its dual problem as a minimisation problem without constraint and solve
it by computing the concave envelope of scattered data.
