A Variational Surface-Evolution Perspective for Optimal Transport between Densities with Differing Compact Support

We examine the optimal mass transport problem in ${\mathbb{R}}^n$ between densities having independent compact support by considering the geometry of a continuous interpolating support boundary in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space–time embedding in terms of points, vectors, and sets in ${\mathbb{R}}^{n+1}\text{\hspace{0.17em}⁣}=\mathbb{R}\times {\mathbb{R}}^n$ and blend the mass density and velocity as well into a space-time solenoidal vector field ${\bf W};|;{\bf \Omega\subset}$\mathbb{R}$^{n+1}!\to$\mathbb{R}$^{n+1}$ over compact sets $\mathbf{\Omega }$ . We then formulate a coupled gradient descent approach containing separate evolution steps for $\mathrm{\partial }\mathbf{\Omega }$ and $\mathbf{W}$.