  This work presents an algorithm to sample from the Wasserstein barycenter of
absolutely continuous measures. Our method is based on the gradient flow of the
multimarginal formulation of the Wasserstein barycenter, with an additive
penalization to account for the marginal constraints. We prove that the minimum
of this penalized multimarginal formulation is achieved for a coupling that is
close to the Wasserstein barycenter. The performances of the algorithm are
showcased in several settings.
