  Adapted optimal transport (AOT) problems are optimal transport problems for
distributions of a time series where couplings are constrained to have a
temporal causal structure. In this paper, we develop computational tools for
solving AOT problems numerically. First, we show that AOT problems are stable
with respect to perturbations in the marginals and thus arbitrary AOT problems
can be approximated by sequences of linear programs. We further study entropic
methods to solve AOT problems. We show that any entropically regularized AOT
problem converges to the corresponding unregularized problem if the
regularization parameter goes to zero. The proof is based on a novel method -
even in the non-adapted case - to easily obtain smooth approximations of a
given coupling with fixed marginals. Finally, we show tractability of the
adapted version of Sinkhorn's algorithm. We give explicit solutions for the
occurring projections and prove that the procedure converges to the optimizer
of the entropic AOT problem.
