arXiv subjects

Joao Miguel Machado

Publications and source records attributed to Joao Miguel Machado.

3 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2024

One-dimensional approximation of measures in Wasserstein distances

We propose a variational approach to approximate measures with measures uniformly distributed over a 1 dimentional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of the support. As it is challenging to prove existence of solutions to this problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the ambient space is $\mathbb{R}^2$ , under techinical assumptions, any solution to the relaxed problem is a solution to the original one. Finally we manage to prove that any optimal solution to the relaxed problem, and hence also to the original, is Ahlfors regular.

math.AP↗

arXiv2023

The Total Variation-Wasserstein Problem

In this work we analyze the Total Variation-Wasserstein minimization problem. We propose an alternative form of deriving optimality conditions from the approach of Calier\&Poon'18, and as result obtain further regularity for the quantities involved. In the sequel we propose an algorithm to solve this problem alongside two numerical experiments.

math.AP↗