An elementary approach to polynomial optimization on polynomial meshes

A polynomial mesh on a multivariate compact set or manifold is a sequence of finite norming sets for polynomials whose norming constant is independent of degree. We apply the recently developed theory of polynomial meshes to an elementary discrete approach for polynomial optimization on nonstandard domains, providing a rigorous (over)estimate of the convergence rate. Examples include surface/solid subregions of sphere or torus, such as caps, lenses, lunes, and slices

Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization

We extend the notion of Dubiner distance from algebraic to trigonometric polynomials on subintervals of the period, and we obtain its explicit form by the Szego variant of Videnskii inequality. This allows to improve previous estimates for Chebyshev-like trigonometric norming meshes, and suggests a possible use of such meshes in the framework of multivariate polynomial optimization on regions defined by circular arcs

Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting

Polynomial Meshes: Computation and Approximation

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

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

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

We show that any compact subset of $\R^d$ which is the closure of a bounded star-shaped Lipschitz domain $\Omega$, such that $\complement \Omega$ 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. Kro\'o on $\mathscr C^ 2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K := \overline\Omega \subset \R^ d$ where $\Omega$ 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.Comment: 29 pages, 3figure

Compressed sampling inequalities by Tchakaloff's theorem

We show that a discrete version of Tchakaloff\u2019s theorem on the existence of positive algebraic cubature formulas, entails that the information required for multivariate polynomial approximation can be suitably compresse