Math @ Duke

Publications [#350396] of Matthias Ernst Sachs
Papers Published
 Lu, J; Sachs, M; Steinerberger, S, Quadrature Points via Heat Kernel Repulsion,
Constructive Approximation, vol. 51 no. 1
(February, 2020),
pp. 2748 [doi]
(last updated on 2021/06/16)
Abstract: © 2019, Springer Science+Business Media, LLC, part of Springer Nature. We discuss the classical problem of how to pick N weighted points on a ddimensional manifold so as to obtain a reasonable quadrature rule 1M∫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∼N2/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.


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