Quadrature Points via Heat Kernel Repulsion

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 $$ \frac{1}{|M|}\int_{M}{f(x) dx} \simeq \frac{1}{N} \sum_{n=1}^{N}{a_i f(x_i)}.$$ 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 $$ \sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) } \rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d},$$ $d(x,y)$ 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.