Invariant Integration Formulas for the <i>n</i>-Simplex by Combinatorial Methods

Grundmann, Axel; Möller, H. Michael

doi:10.1137/0715019

Cited by 123 publications

(84 citation statements)

References 2 publications

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“…Several formulas have been derived for the n-simplex, low-order point formulas of various degrees as well as Gaussian product formulas [56,57,[69][70][71][72]. In general, one should prefer that formula which gives the smallest error possible.…”

Section: Numerical Integrationmentioning

confidence: 99%

Approximate calculation of anharmonic densities of vibrational states for very large molecules

Schmatz

2008

Chemical Physics

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Section: Numerical Integrationmentioning

confidence: 99%

Approximate calculation of anharmonic densities of vibrational states for very large molecules

Schmatz

2008

Chemical Physics

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“…The last equation is obtained by using the formula for integrating a polynomial on the canonical tetrahedron [18],…”

Section: Z Chen W Wang B Lévy L Liu and F Sunmentioning

confidence: 99%

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Chen¹,

Wang²,

Lévy³

et al. 2014

SIAM J. Sci. Comput.

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Abstract. This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh. It revisits the class of methods based on optimal Delaunay triangulation (ODT) and proposes a proper way of injecting a background density function into the objective function minimized by ODT. This continuous/analytic point of view leads to an objective function that is continuous and Delaunay consistent, in contrast with the discrete/geometrical point of view developed in previous work. To optimize the objective function, this paper proposes a hybrid algorithm that combines a local search (quasi-Newton) with a global optimization (simulated annealing). The benefits of the method are both improved performances and an improved quality of the result in terms of dihedral angles. This results from the combination of two effects. First, the local search has a faster speed of convergence than previous work due to the better behavior of the objective function, and second, the algorithm avoids getting stuck in a poor local minimum. Experimental results are evaluated and compared using standard metrics.

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“…The value of J d is also calculated in [10,21]. On the basis of Theorem 4.1, we can compute it easily.…”

Section: Examplesmentioning

confidence: 99%

Multivariate splines and polytopes

2011

Journal of Approximation Theory

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In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by using the multivariate truncated power. Moreover, we show that the volume of cube slicing can be considered as the maximum value of the box spline. On the basis of this connection, we give a simple proof for Good's conjecture, which has been settled before by probability methods.

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Invariant Integration Formulas for the n-Simplex by Combinatorial Methods

Cited by 123 publications

References 2 publications

Approximate calculation of anharmonic densities of vibrational states for very large molecules

Approximate calculation of anharmonic densities of vibrational states for very large molecules

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Multivariate splines and polytopes

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