  We present the Sequential Ensemble Transform (SET) method, an approach for
generating approximate samples from a Bayesian posterior distribution. The
method explores the posterior distribution by solving a sequence of discrete
optimal transport problems to produce a series of transport plans which map
prior samples to posterior samples. We prove that the sequence of Dirac mixture
distributions produced by the SET method converges weakly to the true posterior
as the sample size approaches infinity. Furthermore, our numerical results
indicate that, when compared to standard Sequential Monte Carlo (SMC) methods,
the SET approach is more robust to the choice of Markov mutation kernels and
requires less computational efforts to reach a similar accuracy when used to
explore complex posterior distributions. Finally, we describe adaptive schemes
that allow to completely automate the use of the SET method.
