Quadrature Points via Heat Kernel Repulsion

Journal Article (Journal Article)

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.

Full Text

Duke Authors

Cited Authors

  • Lu, J; Sachs, M; Steinerberger, S

Published Date

  • February 1, 2020

Published In

Volume / Issue

  • 51 / 1

Start / End Page

  • 27 - 48

Electronic International Standard Serial Number (EISSN)

  • 1432-0940

International Standard Serial Number (ISSN)

  • 0176-4276

Digital Object Identifier (DOI)

  • 10.1007/s00365-019-09471-4

Citation Source

  • Scopus