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A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

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A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

Content Provider	Springer Nature Link
Author	Damelin, S. B.
Copyright Year	2008
Abstract	(A) The celebrated Gaussian quadrature formula on finite intervals tells us that the Gauss nodes are the zeros of the unique solution of an extremal problem. We announce recent results of Damelin, Grabner, Levesley, Ragozin and Sun which derive quadrature estimates on compact, homogenous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. (B) Given $\mathcal{X}$ , some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, $\mathcal{P} \subset \mathcal{X}$ , with a small energy or discrepancy. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that these two measures of quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^2(=\mathcal{X}\times\mathcal{X}) \to \mathbb{R}$ . The error of approximating the integral $\int_{\mathcal{X}} f(\mathbf{\mathit{x}}) \, {\rm d} \mu(\mathbf{\mathit{x}})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms the energy or discrepancy of $\mathcal{P}$ . The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. (C) Let $\mathcal{X}$ be the orbit of a compact, possibly non Abelian group, $\mathcal{G}$ , acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$ . This allows us to calculate explicit representations of equilibrium measures. There is an extensive literature on the topics (A–C). We emphasize that this paper surveys recent work of Damelin, Grabner, Levesley, Hickernell, Ragozin, Sun and Zeng and does not mean to serve as a comprehensive survey of all recent work covered by the topics (A–C).
Ending Page	235
Page Count	23
Starting Page	213
File Format	PDF
ISSN	10171398
e-ISSN	15729265
Journal	Numerical Algorithms
Issue Number	1-3
Volume Number	48
Language	English
Publisher	Springer US
Publisher Date	2008-05-21
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Group Discrepancy Torus Positive definite Theory of Computation Mathematics Spline approximation Sphere Algebra Energy Reflexive manifold Capacity Signed measure Equilibrium measure Group invariant measure Energy minimizer Numerical integration Projection kernels Group invariant kernel Numeric Computing Invariant polynomial Weight Reproducing Hilbert space Multidimensional problems (should also be assigned at least one other classification number in this section) Quadature Interpolation Potential field Algorithms Riesz kernel Cubature Distribution Compact homogeneous manifold Spherical harmonic Invariant kernels Projective space
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space

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A walk through energy, discrepancy, numerical integration and group invariant measures on measurable subsets of euclidean space