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 ${\bf W}\;|\;{\bf \Omega\subset}\mathbb{R}^{n+1}\!\to\mathbb{R}^{n+1}$ over compact sets ${\bf \Omega}$ . We then formulate a coupled gradient descent approach containing separate evolution steps for $\partial{\bf \Omega}$ and ${\bf W}$.