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12
A Moment Matrix Approach to Multivariable Cubature
"... We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrixbased lower bound on the size of a cubature rule of degree 2n+1; for a planar measure µ, the bound is based on estimating ρ(C): = inf{rank (T−C): T Toepl ..."
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We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrixbased lower bound on the size of a cubature rule of degree 2n+1; for a planar measure µ, the bound is based on estimating ρ(C): = inf{rank (T−C): T Toeplitz and T ≥ C}, where C: = C][µ] is a positive matrix naturally associated with the moments of µ. We use this estimate to construct various minimal or nearminimal cubature rules for planar measures. In the case when C = diag (c1,..., cn) (including the case when µ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps ck> ck+1.
HIGHERDIMENSIONAL INTEGRATION WITH GAUSSIAN WEIGHT FOR APPLICATIONS IN PROBABILISTIC DESIGN
, 2004
"... Higherdimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surface of an ..."
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Cited by 6 (0 self)
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Higherdimensional Gaussian weighted integration is of interest in probabilistic simulations. Motivated by the need for variance calculations with functions being at least quadratic, the family of degree 5 formulae is considered. Using an existing formula for the integration over the surface of an nsphere, an efficient, new formula for Gaussian weighted integration is obtained. Several other formulae that have appeared in the numerical integration literature are also given. The number of function evaluations required by the formulae is compared to a minimal bound result. The degree 5 formulae are applied to simple test problems and the relative errors are compared.
On GaussLobatto integration on the triangle
 SIAM J. Numer. Anal
"... Abstract. A recent result in [2] on the nonexistence of GaussLobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules. A method of constructing Lobatto type cubature rules on the triangle is given and used to construct several exam ..."
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Abstract. A recent result in [2] on the nonexistence of GaussLobatto cubature rules on the triangle is strengthened by establishing a lower bound for the number of nodes of such rules. A method of constructing Lobatto type cubature rules on the triangle is given and used to construct several examples. 1.
Numerical Analysis in the Twentieth Century
 in Numerical Analysis: Historical Developments in the 20th Century, C. Brezinski e L. Wuytack, Editors, North–Holland
, 2001
"... This paper attracted much attention while a similar result obtained by William Karush in his Master's Thesis in 1939 [154] under the supervision of Lawrence M. Graves at the University of Chicago and by Fritz John (19101995) in 1948 [147] were almost totally ignored (John's paper was even ..."
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This paper attracted much attention while a similar result obtained by William Karush in his Master's Thesis in 1939 [154] under the supervision of Lawrence M. Graves at the University of Chicago and by Fritz John (19101995) in 1948 [147] were almost totally ignored (John's paper was even rejected)
Commuting extensions and cubature formulae
"... Abstract Based on a novel point of view on 1dimensional Gaussian quadrature, we present a new approach to ddimensional cubature formulae. It is well known that the nodes of 1dimensional Gaussian quadrature can be computed as eigenvalues of the socalled Jacobi matrix. The ddimensional analog is ..."
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Abstract Based on a novel point of view on 1dimensional Gaussian quadrature, we present a new approach to ddimensional cubature formulae. It is well known that the nodes of 1dimensional Gaussian quadrature can be computed as eigenvalues of the socalled Jacobi matrix. The ddimensional 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 A1,...,Ad, related to the coordinate operators x1,...,xd, in R d. We prove a correspondence between cubature formulae and “commuting extensions ” of A1,...,Ad, 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 briefly describe our attempts at computing them.
A hybridized discontinuous Galerkin method with reduced stabilization
 Journal of Scientific Computing
"... In this paper, we propose a new hybridized discontinuous Galerkin method for the Poisson equation with homogeneous Dirichlet boundary condition. Our method has the advantage that the stability is better than the previous hybridized method. We derive L2 and H1 error estimates of optimal order. Some n ..."
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In this paper, we propose a new hybridized discontinuous Galerkin method for the Poisson equation with homogeneous Dirichlet boundary condition. Our method has the advantage that the stability is better than the previous hybridized method. We derive L2 and H1 error estimates of optimal order. Some numerical results are presented to verify our analysis.
A Robust Simplex CutCell Method for Adaptive HighOrder Finite Element Discretizations of Aerodynamics and MultiPhysics Problems
, 2013
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Orthogonal polynomials of several variables for a radially symmetric weight
, 2007
"... This paper considers tight frame decompositions of the Hilbert space Pn of orthogonal polynomials of degree n for a radially symmetric weight on IR d, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p ∈ Pn with property that each f ∈ Pn ca ..."
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This paper considers tight frame decompositions of the Hilbert space Pn of orthogonal polynomials of degree n for a radially symmetric weight on IR d, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p ∈ Pn with property that each f ∈ Pn can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials pξ obtained from p by moving its pole to ξ ∈ S: = {ξ ∈ IR d: ξ  = 1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for Pn.
CUBATURE FORMULAS ON TRIANGLE
"... Abstract. In this article are presented some cubature formulas on triangle T which are obtained by the product of known quadrature formulas and some formulas obtaining by an approximation formula on triangle. 1. ..."
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Abstract. In this article are presented some cubature formulas on triangle T which are obtained by the product of known quadrature formulas and some formulas obtaining by an approximation formula on triangle. 1.
Abstract ARTICLE IN PRESS Journal of Computational and Applied Mathematics ( ) – Quasiinterpolation projectors for box splines
, 2007
"... We consider box spline quasiinterpolants based on local linear functionals of point evaluator and integral type. The approximations are easy to compute, and reproduce the whole spline space in question. c ○ 2007 Elsevier B.V. All rights reserved. ..."
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We consider box spline quasiinterpolants based on local linear functionals of point evaluator and integral type. The approximations are easy to compute, and reproduce the whole spline space in question. c ○ 2007 Elsevier B.V. All rights reserved.