Quadrature Points via Heat Kernel Repulsion

TL;DR

This paper introduces a method for selecting quadrature points on manifolds by minimizing a heat kernel-based energy functional, resulting in theoretically sound and universal point distributions for numerical integration.

Contribution

The authors propose a novel energy minimization approach using heat kernel repulsion to select quadrature points on manifolds, independent of the manifold's specific geometry.

Findings

Point sets have guaranteed spectral properties for accurate quadrature.

The energy functional is universal and does not depend on manifold details.

Numerical examples demonstrate the effectiveness of the method.

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 $-\Delta$, 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.