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

Abstract

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 ${\mathbb{R}}^{n+1}=\mathbb{R}\times {\mathbb{R}}^n$ and blend the mass density and velocity as well into a space-time solenoidal vector field $\mathbf{W}|\mathbf{\Omega }\subset {\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}$ over compact sets $\mathbf{\Omega }$ . We then formulate a coupled gradient descent approach containing separate evolution steps for $\partial \mathbf{\Omega }$ and $\mathbf{W}$.