Related papers: Minimizers for an aggregation model with attractiv…
We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper,…
We study the interplay between repulsive interaction and an almost staggered on-site potential in one-dimension. Specifically, we address the Harper model for spinless fermions with nearest-neighbor repulsion, close to half-filling. Using…
The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than…
We present a new approach for computing approximate global minimizers to a large class of non-local pairwise interaction problems defined over probability distributions. The approach predicts candidate global minimizers, with a recovery…
Dynamics of two particles with short range repulsive or attractive interaction is studied numerically in the Harper model. It is shown that interaction leads to appearance of localized states and pure-point spectrum component in the case…
The sparse spike estimation problem consists in estimating a number of off-the-grid impulsive sources from under-determined linear measurements. Information theoretic results ensure that the minimization of a non-convex functional is able…
Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper…
This paper presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. The proposed…
Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by…
In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the…
We propose here a proof of existence of a minimizer of a segmentation functional based on a priori information on target shapes, and formulated with level sets. The existence of a minimizer is very important, because it guarantees the…
The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality…
We consider perimeter perturbations of a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. We prove that there exists curves in the…
We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain…
We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which…
We consider a class of particle systems which appear in various applications such as approximation theory, plasticity, potential theory and space-filling designs. The positions of the particles on the real line are described as a global…
We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1-regularity of minimizers for the Coulombic interaction in…
We report a study of the basins of attraction for potential energy minima defined by different minimisation algorithms for an atomic system. We find that whereas some minimisation algorithms produce compact basins, others produce basins…
In this paper we combine two existing approaches for approximating attractors. One of them approximates the attractors arbitrarily well by sublevel sets related to solutions of infinite dimensional linear programming problems. A downside…
We consider a non-local interaction energy over bounded densities of fixed mass $m$. We prove that under certain regularity assumptions on the interaction kernel these energies admit minimizers given by characteristic functions of sets when…