An adaptive hierarchical approximation method on the sphere using axisymmetric locally supported basis functions
type: Item
instance of: scholarly article
title: An adaptive hierarchical approximation method on the sphere using axisymmetric locally supported basis functions
zbMATH Open document ID: 0857.65017
DOI: 10.1007/BF02247405
author: Yanyan Li
published in: Computing
publication date: 1997-03-09T00:00:00Z
review text: The authors approximate scattered data on the sphere by singular integrals based on non-orthogonal sets of kernel functions \(\widetilde B_\varrho\) derived from functions \(B_\varrho^{(k)}\) which are \(0\) on \(-1\leq t<\varrho\) and are the \(k\)th Bernstein polynomial on \([\varrho,1]\). They derive a quadrature formula for functions defined on the sphere of the type \[ \int F(\eta)\widetilde B_\varrho(\eta\xi) d\omega(\eta)= d_\Gamma\sum_{\zeta\in \Gamma} F(\zeta)\widetilde B_\varrho(\zeta\xi), \] where \(\Gamma\) is a given pointset, \(d\) an appropriately defined constant, and multiplication is scalar multiplication of unit vectors. They derive error estimates necessary for an adaptive use of the formula for the evaluation of iterated convolutions. Since for \(k\to\infty\) and \(\varrho\to 1\) the basis functions tend to be concentrated at 1, they develop an adaptive algorithm that starts with a relatively small value of \(\varrho\) and then computes error estimates at the points of the grid and refines the computations only in those domains where the errors exceed a given threshold. An example given is a topographic map of Australia that shows generally good results except for one corner where because of the inaccessibility of the terrain the number of data is small and, therefore, no good approximation is to be expected.
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 86-08
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 935213
zbMATH Keywords: axisymmetric locally supported basis functions
zbMATH Keywords: scattered data
zbMATH Keywords: sphere
zbMATH Keywords: singular integrals
zbMATH Keywords: Bernstein polynomial
zbMATH Keywords: quadrature formula
zbMATH Keywords: error estimates
zbMATH Keywords: adaptive algorithm
zbMATH Keywords: topographic map of Australia
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02247405
OpenAlex ID: W1522496282
description: scientific article; zbMATH DE number 935213