Wasserstein Gradient Flows of semi-discret energies: evolution of urban areas anduniform quantization

TL;DR

This paper investigates the Wasserstein gradient flow of semi-discrete energies relevant to urban planning, proving convergence of the JKO scheme to a coupled PDE-ODE system and exploring qualitative behaviors through numerical simulations.

Contribution

It introduces a rigorous analysis of the gradient flow for semi-discrete energies, including convergence proofs and qualitative property studies, with applications to urban planning models.

Findings

Convergence of the JKO scheme to a coupled PDE-ODE system.

Atoms tend to move towards the centers of their Laguerre cells.

Numerical simulations reveal a crystallization phenomenon in linear diffusion cases.

Abstract

We study the Wasserstein gradient flow of semi-discrete energies in the space of probability measures, that is functionals depending on two measures-one being an absolutely continuous density and the other an atomic measure. These energies appear naturally in the field of urban planning. This is done via the celebrated JKO scheme, for which we prove convergence to a limiting system composed of a parabolic PDE with singular advection coupled with an ODE, also presenting singular dynamics. This is first done under more general assumptions using classical tools, and in a second moment convergence is proven to hold in $L^2_tH^1_x$ for the cases of linear and Porous-Medium type diffusions. We then pass to the study of some qualitative properties of this system, such as the convergence of the atoms towards the baricenters of their corresponding Laguerre cells. We finish this work with extensive numerical simulations that aid in formulating conjectures for the qualitative behavior of this system; in the case of linear diffusion, for instance, we observe a dynamic crystallization phenomenon.