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

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.

Figures (9)

• Figure 1: If $x_{k,i}$ belongs in the interior of $\Omega$ and $x_{k+1,i}$ is in the boundary $\partial \Omega$, we can produce a better competitor by projecting $x_{k+1,i}$ to the convex set $\Omega_\delta$ of points whose distance to the boundary is at least $\delta$.
• Figure 2: Construction of the invariant region $\mathscr{D}$. The uniform integrability of $\varrho_t$ ensures that the barycenter $b_i(t,y)$ remains inside $\Omega_\delta$, allowing the construction of a smooth and convex invariant set $\mathscr{D}$ between the boundaries of $\Omega_\delta$ and $\Omega_{\delta/2}$.
• Figure 3: A small variation of the atoms does not impact the barycenter too much, unless they are too close. In this situation, where the points are very close, swapping their labels implies a small variation in their positions, but exchanges the barycenters that were very far apart.
• Figure 4: Crystallization phenomenon for the model with linear diffusion and speed $\alpha = \sqrt{N}$. Long-time configurations for $N = 50,100,200,300,400,500$ after $200$ time steps with $\tau = 0.01$. The density $\varrho$ stabilizes to an almost uniform profile, leading to a static triangular lattice of Laguerre cells.
• Figure 5: Crystallization phenomenon for the model with PME-type diffusion, i.e. $\Delta \varrho^m$ with $m = 10$ and speed $\alpha = \sqrt{N}$. Shown are steady configurations for $N = 50,100,200,500,800,1000$ after $200$ time steps with $\tau = 0.01$. The nonlinear diffusion enhances localization and produces sharper crystalline regions in high-density zones, though small fluctuations remain visible even at long times.

Theorems & Definitions (35)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Proposition 2.4: Chap.8 of santambrogio2015optimal
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3