28 research outputs found
On the extremals of the P\'olya-Szeg\H{o} inequality
The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a
translate of its symmetric decreasing rearrangement is controlled by the
measure of the set of critical points.Comment: 17 pages, 3 figure
Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
We analyze under which conditions equilibration between two competing
effects, repulsion modeled by nonlinear diffusion and attraction modeled by
nonlocal interaction, occurs. This balance leads to continuous compactly
supported radially decreasing equilibrium configurations for all masses. All
stationary states with suitable regularity are shown to be radially symmetric
by means of continuous Steiner symmetrization techniques. Calculus of
variations tools allow us to show the existence of global minimizers among
these equilibria. Finally, in the particular case of Newtonian interaction in
two dimensions they lead to uniqueness of equilibria for any given mass up to
translation and to the convergence of solutions of the associated nonlinear
aggregation-diffusion equations towards this unique equilibrium profile up to
translations as
Self-similar minimizers of a branched transport functional
We solve here completely an irrigation problem from a Dirac mass to the
Lebesgue measure. The functional we consider is a two dimensional analog of a
functional previously derived in the study of branched patterns in type-I
superconductors. The minimizer we obtain is a self-similar tree.Comment: Indiana University Mathematics Journal, Indiana University
Mathematics Journal, In pres
Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as t → ∞
Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation
Hierarchical modeling and learning has proven very powerful in the field of Gaussian process regression and kernel methods, especially for machine learning applications and, increasingly, within the field of inverse problems more generally. The classical approach to learning hierarchical information is through Bayesian formulations of the problem, implying a posterior distribution on the hierarchical parameters or, in the case of empirical Bayes, providing an optimization criterion for them. Recent developments in the machine learning literature have suggested new criteria for hierarchical learning, based on approximation theoretic considerations that can be interpreted as variants of cross-validation, and exploiting approximation consistency in data splitting. The purpose of this paper is to compare the empirical Bayesian and approximation theoretic approaches to hierarchical learning, in terms of large data consistency, variance of estimators, robustness of the estimators to model misspecification, and computational cost. Our analysis is rooted in the setting of Matérn-like Gaussian random field priors, with smoothness, amplitude and inverse lengthscale as hierarchical parameters, in the regression setting. Numerical experiments validate the theory and extend the scope of the paper beyond the Matérn setting
Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
We consider a classical system of n charged particles in an external
confining potential, in any dimension d larger than 2. The particles interact
via pairwise repulsive Coulomb forces and the coupling parameter scales like
the inverse of n (mean-field scaling). By a suitable splitting of the
Hamiltonian, we extract the next to leading order term in the ground state
energy, beyond the mean-field limit. We show that this next order term, which
characterizes the fluctuations of the system, is governed by a new
"renormalized energy" functional providing a way to compute the total Coulomb
energy of a jellium (i.e. an infinite set of point charges screened by a
uniform neutralizing background), in any dimension. The renormalization that
cuts out the infinite part of the energy is achieved by smearing out the point
charges at a small scale, as in Onsager's lemma. We obtain consequences for the
statistical mechanics of the Coulomb gas: next to leading order asymptotic
expansion of the free energy or partition function, characterizations of the
Gibbs measures, estimates on the local charge fluctuations and factorization
estimates for reduced densities. This extends results of Sandier and Serfaty to
dimension higher than two by an alternative approach.Comment: Structure has slightly changed, details and corrections have been
added to some of the proof