Dynamic Optimal Transport with optimal star shaped graphs

TL;DR

This paper introduces a novel framework for dynamic optimal transport in convex sets that couples bulk transport with transport on embedded metric graphs, proving existence of solutions for fixed and star-shaped varying graphs.

Contribution

It develops a new existence theory for optimal transport problems involving embedded graphs, including a penalty term to handle topology preservation in star-shaped graph variations.

Findings

Existence of solutions for fixed embedded graphs.

Existence of minimizers for star-shaped graph variations with topology-preserving penalties.

A new variational approach to coupled bulk and graph-based transport problems.

Abstract

We study an optimal transport problem in a compact convex set $\Omega\subset\mathbb{R}^d$ where bulk transport is coupled to dynamic optimal transport on a metric graph $ \mathsf{G} = (\mathsf{V},\mathsf{E})$ which is embedded in $\Omega$. We prove existence of solutions for fixed graphs. Next, we consider varying graphs, yet only for the case of star-shaped ones. Here, the action functional is augmented by an additional penalty that prevents the edges of the graph to overlap. This allows to preserve the graph topology and thus to rely on standard techniques in Calculus of Variations in order to show existence of minimizers.