ALEX 2018 Workshop: Abstracts

Gromov–Hausdorff convergence of discrete optimal transport

Eva Kopfer${}^{\mathrm{(}\mathrm{1}\mathrm{)}}$, Peter Gladbach${}^{\mathrm{(}\mathrm{2}\mathrm{)}}$, and Jan Maas${}^{\mathrm{(}\mathrm{3}\mathrm{)}}$

(1) Mathematisches Institut, Universität Leipzig (Germany)

(2) Institut für Angewandte Mathematik, Universität Bonn (Germany)

(3) Institute of Science and Technology Austria (Austria)

For a natural class of discretisations of a convex domain in ${ℝ}^n$, we consider the dynamical optimal transport metric for probability measures on the discrete mesh. Although the associated discrete heat flow converges to the continuous heat flow as the mesh size tends to $0$, we show that the transport metric may fail to converge to the $2$-Kantorovich metric. Under a strong additional symmetry condition on the mesh, we show that Gromov–Hausdorff convergence to the $2$-Kantorovich metric holds.