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 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.
Original language | English |
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Pages (from-to) | 27-48 |
Number of pages | 22 |
Journal | Constructive Approximation |
Volume | 51 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Feb 2020 |
Bibliographical note
Funding Information:This work was partially supported by the National Science Foundation under Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute. The research of J.L. was also supported in part by the National Science Foundation under award DMS-1454939.
Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Keywords
- Heat kernel
- Numerical Integration
- Quadrature
ASJC Scopus subject areas
- Analysis
- General Mathematics
- Computational Mathematics