Lower Bounds for the Number of Nodes in Cubature Formulae

Möller, H. Michael

doi:10.1007/978-3-0348-6288-2_17

Cited by 60 publications

(65 citation statements)

References 8 publications

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“…Corollary 5.5 matches the Möller bound [18] for cubature on S n , but it is more general because the measure µ need not be centrally symmetric. …”

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confidence: 56%

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Numerical Cubature from Archimedes' Hat-box Theorem

Kuperberg¹

2006

SIAM J. Numer. Anal.

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Dedicated to Krystyna Kuperberg on the occasion of her 60th birthdayArchimedes' hat-box theorem states that uniform measure on a sphere projects to uniform measure on an interval. This fact can be used to derive Simpson's rule. We present various constructions of, and lower bounds for, numerical cubature formulas using moment maps as a generalization of Archimedes' theorem. We realize some well-known cubature formulas on simplices as projections of spherical designs. We combine cubature formulas on simplices and tori to make new formulas on spheres. In particular S n admits a 7-cubature formula (sometimes a 7-design) with O(n 4 ) points. We establish a local lower bound on the density of a PI cubature formula on a simplex using the moment map.Along the way we establish other quadrature and cubature results of independent interest. For each t, we construct a lattice trigonometric (2t + 1)-cubature formula in n dimensions with O(n t ) points. We derive a variant of the Möller lower bound using vector bundles. And we show that Gaussian quadrature is very sharply locally optimal among positive quadrature formulas.

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“…Corollary 5.5 matches the Möller bound [18] for cubature on S n , but it is more general because the measure µ need not be centrally symmetric. …”

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confidence: 56%

“…Section 5 presents a refinement of this well-known lower bound in odd degree. It is similar to the Möller bound [18], but applies to some new cases. Section 6 also establishes that Gaussian quadrature is very sharply locally optimal among all positive quadrature formulas (Theorem 6.3).…”

Section: Introductionmentioning

confidence: 78%

Numerical Cubature from Archimedes' Hat-box Theorem

Kuperberg¹

2006

SIAM J. Numer. Anal.

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“…The lower bound of Möller (1979) for centrally symmetric weight functions is the following: If k is odd, then…”

Section: Problem Main Results and Conjecturementioning

confidence: 99%

Cubature formulas for symmetric measures in higher dimensions with few points

Hinrichs

Novak

2007

Math. Comp.

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Abstract. We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree = 5 or = 7 and large dimension d the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.We also show, for any odd degree = 2k + 1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

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“…To summarize: to find a Radon formula one should: (1) Construct the orthonormal basis there is an associated commuting extension given by (18), where a, b are given by (86) and α, β, λ, µ, ν, ρ by (89). The commuting extensions can be simultaneously diagonalized, using the algorithm in [15], to obtain the nodes and weights of the cubature rule.…”

Section: Radon Type Formulaementioning

confidence: 99%

Commuting extensions and cubature formulae

2005

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Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A 1 , . . . , A d , related to the coordinate operators x 1 , . . . , x d , in R d . We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , . . . , A d , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

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Lower Bounds for the Number of Nodes in Cubature Formulae

Cited by 60 publications

References 8 publications

Numerical Cubature from Archimedes' Hat-box Theorem

Numerical Cubature from Archimedes' Hat-box Theorem

Cubature formulas for symmetric measures in higher dimensions with few points

Commuting extensions and cubature formulae

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