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Federico Piazzon

Publications and source records attributed to Federico Piazzon.

14 recordsLinked to original sources
arXiv2025

Sampling, approximation, and interpolation of differential forms by admissible integral k-meshes

In this work we introduce the concept of admissible integral $k$-mesh for sampling differential forms with contiuous coefficients on a real body $E\subset \R^n$, and provide two techniques for the construction of admissible integral $k$-meshes on real bodies enjoying the Markov or the Bernstein inequality. Admissible integral $k$-meshes allow for the construction of robust approximation schemes, and are used to extract interpolation sets with high stability properties. To this end, the concepts of Fekete currents and Leja sequences of currents are formalized, and a numerical scheme for their approximation is proposed.

math.NA↗
arXiv2025

A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the proposed modification a vector valued weight is considered, allowing to partially extend the main asymptotic results holding for interpolation of scalar valued functions to the case of vector valued ones. As motivating example and main application we specialize our results to interpolation of differential forms by differential forms with polynomial coefficients.

math.CV↗
arXiv2023

Computing the L1 optimal transport density: a FEM approach

The $L^1$ optimal transport density $μ^*$ is the unique $L^\infty$ solution of the Monge-Kantorovich equations. It has been recently characterized also as the unique minimizer of the $L^1$ -transport energy functional E. In the present work we develop and we prove convergence of a numerical approxi- mation scheme for $μ^*$ . Our approach relies upon the combination of a FEM- inspired variational approximation of E with a minimization algorithm based on a gradient flow method.

math.NA↗
arXiv2022

Computing optimal experimental designs on finite sets by log-determinant gradient flow

Optimal experimental designs are probability measures with finite support enjoying an optimality property for the computation of least squares estimators. We present an algorithm for computing optimal designs on finite sets based on the long-time asymptotics of the gradient flow of the log-determinant of the so called information matrix. We prove the convergence of the proposed algorithm, and provide a sharp estimate on the rate its convergence. Numerical experiments are performed on few test cases using the new matlab package OptimalDesignComputation.

math.NA↗
arXiv2020

Transport Energy

We introduce the \emph{transport energy} functional $\mathcal E$ (a variant of the Bouchitté-Buttazzo-Seppecher shape optimization functional) and we prove that its unique minimizer is the optimal transport density $μ^*$, i.e., the solution of Monge-Kantorovich equations. We study the gradient flow of $\mathcal E$ showing that $μ^*$ is the unique global attractor of the flow. We introduce a two parameter family $\{\mathcal E_{λ,δ}\}_{λ,δ>0}$ of strictly convex functionals approximating $\mathcal E$ and we prove the convergence of the minimizers $μ_{λ,δ}^*$ of $\mathcal E_{λ,δ}$ to $μ^*$ as we let $δ\to 0^+$ and $λ\to 0^+.$ We derive an evolution system of fully non-linear PDEs as gradient flow of $\mathcal E_{λ,δ}$ in $L^2$, showing existence and uniqueness of solutions. All the trajectories of the flow converge in $W^{1,p}_0$ to the unique minimizer $μ_{λ,δ}^*$ of $\mathcal E_{λ,δ}.$ Finally, we characterize $μ_{λ,δ}^*$ by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.

math.AP↗
arXiv2017

Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of $\R^d$

We show that any compact subset of $\R^d$ which is the closure of a bounded star-shaped Lipschitz domain $Ω$, such that $\complement Ω$ has positive reach in the sense of Federer, admits an \emph{optimal AM} (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on $\mathscr C^ 2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K := \overlineΩ\subset \R^ d$ where $Ω$ is a bounded $\mathscr C^{ 1,1}$ domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.

math.NA↗
arXiv2017

Laplace Beltrami operator in the Baran metric and pluripotential equilibrium measure: the ball, the simplex and the sphere

The Baran metric $δ_E$ is a Finsler metric on the interior of $E\subset \R^n$ arising from Pluripotential Theory. We consider the few instances, namely $E$ being the ball, the simplex, or the sphere, where $δ_E$ is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary conditions) are the orthogonal polynomials with respect to the pluripotential equilibrium measure $μ_E$ of $E.$ We conjecture that this may hold in a wider generality. The considered differential operators have been already introduced in the framework of orthogonal polynomials and studied in connection with certain symmetry groups. In this work instead we highlight the relationships between orthogonal polynomials with respect to $μ_E$ and the Riemaniann structure naturally arising from Pluripotential Theory

math.SP↗
arXiv2017

Pluripotential Numerics

We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the \emph{extremal plurisubharmonic function} $V_E^*$ of a compact $\mathcal L$-regular set $E\subset \C^n$, its \emph{transfinite diameter} $δ(E),$ and the \emph{pluripotential equilibrium measure} $μ_E:=\ddcn{V_E^*}.$ The methods rely on the computation of a \emph{polynomial mesh} for $E$ and numerical orthonormalization of a suitable basis of polynomials. We prove the convergence of the approximation of $δ(E)$ and the uniform convergence of our approximation to $V_E^*$ on all $\C^n;$ the convergence of the proposed approximation to $μ_E$ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein Markov measures. Numerical tests are presented for some simple cases with $E\subset \R^2$ to illustrate the performances of the proposed methods.

math.NA↗
arXiv2016

Caratheodory-Tchakaloff Subsampling

We present a brief survey on the compression of discrete measures by Caratheodory-Tchakaloff Subsampling, its implementation by Linear or Quadratic Programming and the application to multivariate polynomial Least Squares. We also give an algorithm that computes the corresponding Caratheodory-Tchakaloff (CATCH) points and weights for polynomial spaces on compact sets and manifolds in 2D and 3D.

math.NA↗
arXiv2015

Some results on the rational Bernstein Markov property in the complex plane

The Bernstein Markov Property, shortly BMP, is an asymptotic quan- titative assumption on the growth of uniform norms of polynomials or rational functions on a compact set with respect to L μ 2 -norms, where μ is a positive finite measure. We consider two variants of BMP for rational functions with restricted poles and compare them with the polynomial BMP finding out some sufficient condi- tions for the latter to imply the former. Moreover, we recover a sufficient mass- density condition for a measure to satisfy the rational BMP on its support.

math.CV↗
arXiv2015

Bernstein-Markov: a survey

We give a survey of recent results, due mainly to the authors, concerning Bernstein-Markov type inequalities and connections with potential theory.

math.CV↗
arXiv2015

A weighted extremal function and equilibrium measure

Let $K={\bf R}^n\subset {\bf C}^n$ and $Q(x):=\frac{1}{2}\log (1+x^2)$ where $x=(x_1,...,x_n)$ and $x^2 = x_1^2+\cdots +x_n^2$. Utilizing extremal functions for convex bodies in ${\bf R}^n\subset {\bf C}^n$ and Sadullaev's characterization of algebraicity for complex analytic subvarieties of ${\bf C}^n$ we prove the following explicit formula for the weighted extremal function $V_{K,Q}$: $$V_{K,Q}(z)=\frac{1}{2}\log \bigl( [1+|z|^2] + \{ [1+|z|^2]^2-|1+z^2|^2\}^{1/2})$$ where $z=(z_1,...,z_n)$ and $z^2 = z_1^2+\cdots +z_n^2$. As a corollary, we find that the Alexander capacity $T_ω({\bf R} {\bf P}^n)$ of ${\bf R} {\bf P}^n$ is $1/\sqrt 2$. We also compute the Monge-Ampère measure of $V_{K,Q}$: $$(dd^cV_{K,Q})^n = n!\frac{1}{(1+x^2)^{\frac{n+1}{2}}}dx.$$

math.CV↗
arXiv2015

An Orthogonality Property of the Legendre Polynomials

We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the degree-$n$ normalized Christoffel function.

math.CA↗