Federico Piazzon

University of Padova | UNIPD · Department of Mathematics

We introduce the transport energy functional E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document} (a variant of the Bouchitté–Buttazzo–Seppec...

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 t...

We show that Lasserre measure-based hierarchies for polynomial optimization can be implemented by directly computing the discrete minimum at a suitable set of algebraic quadrature nodes. The sampling cardinality can be much lower than in other approaches based on grids or norming meshes. All the vast literature on multivariate algebraic quadrature...

We show that the notion of polynomial mesh (norming set), used to provide discretizations of a compact set nearly optimal for certain approximation theoretic purposes, can also be used to obtain finitely supported near G-optimal designs for polynomial regression. We approximate such designs by a standard multiplicative algorithm, followed by measur...

We correct the calculation of the Monge-Amp\`ere measure of a certain extremal plurisubharmonic function for the complex Euclidean ball in C^2.

The Baran metric $\delta_E$ 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 $\delta_E$ is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary con...

We introduce the \emph{transport energy} functional $\mathcal E$ acting on Borel measures, a variant of the Bouchitt\'e-Buttazzo-Seppecher shape optimization functional, and we prove that the Evans-Gangbo optimal transport density $\mu^*$ is the unique minimizer of $\mathcal E.$ We study the gradient flow of $\mathcal E$ showing that $\mu^*$ is the...

We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in Rd, and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a (1-ε)-approximation to the minimum of any polynomial of degree not exceeding n by O(n/ε)αd samples,...

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} $\delta(E),$ and the \emph{pluripotential equilibrium measure} $\mu_E:=\ddcn{V_E^*}.$ The method...

We compute the extremal plurisubharmonic function of the real torus viewed as a compact subset of its natural algebraic complexification.

We discuss the Siciak-Zaharjuta extremal function of pluripotential theory for the unit ball in C^d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated.

Using the approximation theory notions of polynomial mesh and Dubiner distance in a compact set, we derive error estimates for total degree polynomial optimization on Chebyshev grids of the hypercube.

We prove that L∞-norming sets for finite-dimensional multivariate function spaces on compact sets are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials.

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 {\mu} 2 -norms, where {\mu} is a positive finite measure. We consider two variants of BMP for rational functions with restricted poles and compare them with the...

We give a remarkable additional othogonality property of the classical Legendre polynomials on the real interval [-1,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[...

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 spac...

We prove by Bernstein inequality that Gauss-Jacobi(-Lobatto) nodes of suitable order are L ∞ norming meshes for algebraic polynomials, in a wide range of Jacobi parameters. A similar result holds for trigono-metric polynomials on subintervals of the period, by a nonlinear transformation of such nodes and Videnskii inequality. 2000 AMS subject class...

We show that any compact subset of R-d which is the closure of a bounded star-shaped Lipschitz domain Omega, such that subset of Omega has positive reach in the sense of Federer, admits an optimal AM (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroo on l(2) star-shaped...

We prove that L ∞-norming sets for finite-dimensional function spaces on compact sets admitting a Markov-like inequality, are stable under small perturbations. This implies stability of interpolation operator norms (Lebesgue constants), in spaces of algebraic and trigonometric polynomials. 2000 AMS subject classification: 41A10, 41A63, 42A15, 65D05...

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

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs.

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 t...

We construct polynomial norming meshes with optimal cardinality growth, on planar compact starlike domains that satisfy a Uniform Interior Ball Condition (UIBC

We construct norming meshes with cardinality 𝒪(n s ), s = 3, for polynomials of total degree at most n on the closure of bounded planar Lipschitz domains. Such cardinality is intermediate between optimality (s = 2), recently obtained by Kroó on multidimensional C 2 star-like domains, and that arising from a general construction on Markov compact se...

It has been proved in 2008 by Calvi and Levenberg that discrete least squares polynomial approximation performed on \textbf{(Polynomial) Admissible Meshes}, say \textbf{AM}, enjoys a nice property of convergence. \textbf{Optimal AM}s are AMs which cardinality grows with optimal rate w.r.t. the degree of approximation. In Section 2 we show that any...

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.

We obtain good discrete sets for real or complex multivariate polynomial approximation (admissible meshes) on compact sets satisfying a Markov polynomial inequality by analytic transformations. Then we apply the result to the construction of near optimal admissible meshes, and we discuss two examples concerning complex analytic curves and real anal...