In this work, we present a rather general class of transport distances over
the space of positive semidefinite matrix valued Radon measures, called the
weighted Wasserstein Bures distance, and consider the convergence property of
their fully discretized counterparts. These distances are defined via a
generalization of Benamou Brenier formulation of the quadratic optimal
transport, based on a new weighted action functional and an abstract matricial
continuity equation. It gives rise to a convex optimization problem. We shall
give a complete characterization of its minimizer (i.e., the geodesic) and
discuss some topological and geometrical properties of these distances. Some
recently proposed models: the interpolation distance by Chen et al. [18] and
the Kantorovich Bures distance by Brenier et al. [11], as well as the well
studied Wasserstein Fisher Rao distance [43, 19, 40], fit in our model. The
second part of this work is devoted to the numerical analysis of the fully
discretization of the new transport model. We reinterpret the convergence
framework proposed very recently by Lavenant [41] for the quadratic optimal
transport from the perspective of Lax equivalence theorem and extend it to our
general problem. In view of this abstract framework, we suggest a concrete
fully discretized scheme inspired by the finite element theory, and show the
unconditional convergence under mild assumptions. In particular, these
assumptions are removed in the case of Wasserstein Fisher Rao distance due to
the existence of a static formulation.Comment: 47 pages, raw draf