Proceedings

Inserted: 21 jul 2016
Last Updated: 9 apr 2024

Journal: Radon Series on Computational and Applied Mathematics
Volume: 17
Pages: 20
Year: 2017
Doi: 10.1515/9783110430417

From the book "Topological Optimization and Optimal Transport".

Abstract:

We present the two classical models of Branched Transport: the Lagrangian model and its Eulerian counterpart, with an emphasis on the last one, for which we give a complete proof of existence of minimizers in a --hopefully-- simplified manner. We also treat in detail some $\sigma$-finiteness and rectifiability issues to yield rigorously the energy formula connecting the irrigation cost $\mathbf{I}_\alpha$ to the Gilbert Energy $\mathbf{E}_\alpha$. Our main purpose is to use this energy formula and exploit a Smirnov decomposition of vector flows, which was proved via the Dacorogna-Moser approach, to establish the equivalence between the Lagrangian and Eulerian models.

Keywords: Optimal transport, Branched transport, Rectifiability, Smirnov decomposition

Download:

• branched_equivalence.pdf