Dynamic L1 Optimal Transport and Energy

Venue: Povo Zero, via Sommarive, 14 (Povo) – Sala Seminari "-1"
At: 2:30 p.m.

Speaker:

• Mario Putti (Università di Padova)

Abstract:

Recently we have introduced a dynamic formulation of the PDE-based optimal transport problem with linear cost (L1), the so-called Monge-Kantorovich equations proposed by Evans&Gangbo (1999). The formulation couples an elliptic equation (pure diffusion) for the transport potential u and a dynamic equation for the transport density µ. We conjecture that the solution of our model converges as t → ∞ to the solution of the M-K equations. In partial support of this conjecture, we propose a Lyapunov-candidate functional. Formal calculations show that this functional is strictly decreasing along the µ(t)-trajectories and becomes stationary only for µ = µ∗, i.e., the optimal transport density of Evans&Gangbo. We develope a Finite-Element based solver for this dynamical system and look at the numerical solution of complex two- and three-dimensional transport problems. Several experiments show that the proposed solver is very efficient in calculating the Wassertstein-1 distance between two densities. An application to the numerical calculation of the cut-locus of compact surfaces is also shown. Finally, we study a variant of the Lyapunov-candidate functional, which we call the transport energy. We rigorously show that there exists a gradient flow for the transport energy whose asymptotic solution is unique and coincides with µ∗.

Contact person: Gian Paolo Leonardi