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Quadrature Points via Heat Kernel Repulsion

Publication ,  Journal Article
Lu, J; Sachs, M; Steinerberger, S
Published in: Constructive Approximation
February 1, 2020

We discuss the classical problem of how to pick N weighted points on a d-dimensional manifold so as to obtain a reasonable quadrature rule 1|M|∫Mf(x)dx≃∑n=1Naif(xi).This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional ∑i,j=1Naiajexp(-d(xi,xj)24t)→min,wheret∼N-2/d,d(x, y) is the geodesic distance, and d is the dimension of the manifold. This yields point sets that are theoretically guaranteed, via spectral theoretic properties of the Laplacian - Δ , to have good properties. One nice aspect is that the energy functional is universal and independent of the underlying manifold; we show several numerical examples.

Duke Scholars

Published In

Constructive Approximation

DOI

EISSN

1432-0940

ISSN

0176-4276

Publication Date

February 1, 2020

Volume

51

Issue

1

Start / End Page

27 / 48

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics
 

Citation

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Lu, J., Sachs, M., & Steinerberger, S. (2020). Quadrature Points via Heat Kernel Repulsion. Constructive Approximation, 51(1), 27–48. https://doi.org/10.1007/s00365-019-09471-4
Lu, J., M. Sachs, and S. Steinerberger. “Quadrature Points via Heat Kernel Repulsion.” Constructive Approximation 51, no. 1 (February 1, 2020): 27–48. https://doi.org/10.1007/s00365-019-09471-4.
Lu J, Sachs M, Steinerberger S. Quadrature Points via Heat Kernel Repulsion. Constructive Approximation. 2020 Feb 1;51(1):27–48.
Lu, J., et al. “Quadrature Points via Heat Kernel Repulsion.” Constructive Approximation, vol. 51, no. 1, Feb. 2020, pp. 27–48. Scopus, doi:10.1007/s00365-019-09471-4.
Lu J, Sachs M, Steinerberger S. Quadrature Points via Heat Kernel Repulsion. Constructive Approximation. 2020 Feb 1;51(1):27–48.
Journal cover image

Published In

Constructive Approximation

DOI

EISSN

1432-0940

ISSN

0176-4276

Publication Date

February 1, 2020

Volume

51

Issue

1

Start / End Page

27 / 48

Related Subject Headings

  • Numerical & Computational Mathematics
  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0104 Statistics
  • 0103 Numerical and Computational Mathematics
  • 0102 Applied Mathematics