On the convergence of discrete dynamic unbalanced transport models

Understand

A generalized unbalanced optimal transport distance ${\rm WB}_{\Lambda}$ on matrix-valued measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$ was defined in [arXiv:2011.05845] \`{a} la Benamou-Brenier, which extends the Kantorovich-Bures and the Wasserstein-Fisher-Rao distances.

• In this work, we investigate the convergence properties of the discrete transport problems associated with ${\rm WB}_{\Lambda}$.
• We first present a convergence framework for abstract discretization.
• Then, we propose a specific discretization scheme that aligns with this framework, under the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure.