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Quadrature Points via Heat Kernel Repulsion

  • Jianfeng Lu
  • Matthias SachsEmail author
  • Stefan Steinerberger
Survey

Abstract

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
$$\begin{aligned} \frac{1}{|M|}\int _{M}{f(x) \mathrm{d}x} \simeq \sum _{n=1}^{N}{a_i f(x_i)}. \end{aligned}$$
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
$$\begin{aligned} \sum _{i,j =1}^{N}{ a_i a_j \exp \left( -\frac{d(x_i,x_j)^2}{4t}\right) } \rightarrow \min , \quad \text{ where }~t \sim N^{-2/d}, \end{aligned}$$
d(xy) 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 \(-\Delta \), 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.

Keywords

Quadrature Heat kernel Numerical Integration 

Mathematics Subject Classification

41A55 65D32 (primary) 35K08 (secondary) 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Jianfeng Lu
    • 1
  • Matthias Sachs
    • 2
    • 3
    Email author
  • Stefan Steinerberger
    • 4
  1. 1.Department of Mathematics, Department of Physics, and Department of ChemistryDuke UniversityDurhamUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.Statistical and Applied Mathematical Sciences Institute (SAMSI)DurhamUSA
  4. 4.Department of MathematicsYale UniversityNew HavenUSA

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