A lower bound for the worst-case cubature error on spheres of arbitrary dimension

By Kerstin Hesse. 2006

50 citations recorded by the source corpus.

This paper is concerned with numerical integration on the unit sphere Sr of dimension r≥2 in the Euclidean space ℝr+1. We consider the worst-case cubature error, denoted by E(Qm;Hs(Sr)), of an arbitrary m-point cubature rule Qm for functions in the unit ball of the Sobolev space Hs(Sr), where s>, and show that The positive constant cs,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space Hs(Sr). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Qm a `bad' function fm, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere Sr with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.

Subfields: Computational Mechanics, Numerical Analysis

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Citations in the corpus

On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, methodology): “It is thus equal to m − 2 t/d , and we recover the result from Hesse (2006).”
On the Equivalence between Quadrature Rules and Random Features (2015, methodology): “In this paper, we recover these results using only the decay of eigenvalues of the associated integral operator Σ, thus allowing straightforward extensions to many situations, like Sobolev spaces on manifolds such as hyperspheres (Hesse, 2006).”
Numerical Integration over Spheres of Arbitrary Dimension (2007, background): “In [6] for the case ofS2 and in [ 4 ] for the case of Sd, where d ‚ 2, it is shown that there exists a positive constant ~c such that for an arbitrary m-point numerical integration rule Qm on Sd the worst-case error in Hs(Sd), s > d2,”