  Wasserstein barycenter is the centroid of a collection of discrete
probability distributions which minimizes the average of the
$\ell_2$-Wasserstein distance. This paper focuses on the computation of
Wasserstein barycenters under the case where the support points are free, which
is known to be a severe bottleneck in the D2-clustering due to the large-scale
and nonconvexity. We develop an inexact proximal alternating minimization
(iPAM) method for computing an approximate Wasserstein barycenter, and provide
its global convergence analysis. This method can achieve a good accuracy with a
reduced computational cost when the unknown support points of the barycenter
have low cardinality. Numerical comparisons with the 3-block B-ADMM in
\cite{YeWWL17} and an alternating minimization method involving the LP
subproblems on synthetic and real data show that the proposed iPAM can yield
comparable even a little better objective values in less CPU time, and hence
the computed barycenter will render a better role in the D2-clustering.
