86 research outputs found

    On the computation of sets of points with low Lebesgue constant on the unit disk

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    In this paper of numerical nature, we test the Lebesgue constant of several available point sets on the disk and propose new ones that enjoy low Lebesgue constant. Furthermore we extend some results in Cuyt (2012), analyzing the case of Bos arrays whose radii are nonnegative Gauss–Gegenbauer–Lobatto nodes with exponent, noticing that the optimal still allows to achieve point sets on with low Lebesgue constant for degrees. Next we introduce an algorithm that through optimization determines point sets with the best known Lebesgue constants for. Finally, we determine theoretically a point set with the best Lebesgue constant for the case

    Numerical hyperinterpolation over nonstandard planar regions

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    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

    Subperiodic trigonometric subsampling: A numerical approach

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    We show that Gauss-Legendre quadrature applied to trigonometric poly- nomials on subintervals of the period can be competitive with subperiodic trigonometric Gaussian quadrature. For example with intervals correspond- ing to few angular degrees, relevant for regional scale models on the earth surface, we see a subsampling ratio of one order of magnitude already at moderate trigonometric degrees

    Numerical cubature and hyperinterpolation over Spherical Polygons

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    The purpose of this work is to introduce a strategy for determining the nodes and weights of a low-cardinality positive cubature formula nearly exact for polynomials of a given degree over spherical polygons. In the numerical section we report the results about numerical cubature over a spherical polygon P\cal P approximating Australia and reconstruction of functions over such P\cal P, also affected by perturbations, via hyperinterpolation and some of its variants. The open-source Matlab software used in the numerical tests is available at the author's homepage

    Caratheodory-Tchakaloff Subsampling

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    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 meshes and Caratheodory-Tchakaloff submeshes on the sphere

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    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

    Discrete norming inequalities on sections of sphere, ball and torus

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    By discrete trigonometric norming inequalities on subintervals of the period, we construct norming meshes with optimal cardinality growth for algebraic polynomials on sections of sphere, ball and torus

    Compression of multivariate discrete measures and applications

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    We discuss two methods for the compression of multivariate discrete measures, with applications to node reduction in numerical cubature and least-squares approximation. The methods are implemented in the Matlab computing environment, in dimension two

    Polynomial fitting and interpolation on circular sections

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    We construct Weakly Admissible polynomial Meshes (WAMs) on circular sections, such as symmetric and asymmetric circular sectors, circular segments, zones, lenses and lunes. The construction resorts to recent results on subperiodic trigonometric interpolation. The paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

    Polynomial-free unisolvence of polyharmonic splines with odd exponent by random sampling

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    In a recent paper almost sure unisolvence of RBF interpolation at random points with no polynomial addition was proved, for Thin-Plate Splines and Radial Powers with noninteger exponent. The proving technique left unsolved the case of odd exponents. In this short note we prove almost sure polynomial-free unisolvence in such instances, by a deeper analysis of the interpolation matrix determinant and fundamental properties of analytic functions.Comment: Keywords: multivariate interpolation, Radial Basis Functions, polyharmonic splines, odd integer exponent, unisolvence, analytic function
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