18 research outputs found
Numerical hyperinterpolation over nonstandard planar regions
We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes
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
Rapid polynomial approximation in -spaces with Freud weights on the real line
The weights form a subclass
of Freud weights on the real line. Primarily from a functional analytic angle,
we investigate the subspace of consisting
of those elements that can be rapidly approximated by polynomials. This
subspace has a natural Fr\'echet topology, in which it is isomorphic to the
space of rapidly decreasing sequences. We show that it consists of smooth
functions and obtain concrete results on its topology. For there is
a complete and elementary description of this topological vector space in terms
of the Schwartz functions.Comment: 18 page
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
Numerical cubature on scattered data by adaptive interpolation
We construct cubature methods on scattered data via resampling on the support
of known algebraic cubature formulas, by different kinds of adaptive
interpolation (polynomial, RBF, PUM). This approach gives a promising
alternative to other recent methods, such as direct meshless cubature by RBF or
least-squares cubature formulas