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 ruleThis 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 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 −∆, 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.

Numerical integration on graphs: Where to sample and how to weigh

“…The second author recently extended some of these results to weighted points and general manifolds [15]. These ideas are shown to be an effective source of good quadrature points in the Euclidean setting in a paper of Lu, Sachs and the second author [12]. The second author recently proved [16] a generalized Delsarte-Goethals-Seidel bound for graph designs (the analogue of spherical t−designs on combinatorial graphs).…”

On the Logarithmic Energy of Points on S^2

“…If s ∼ n −1 , then this is, in some sense, a much simpler problem: it is essentially Gaussian interaction at the scale of nearest neighbor distances. This energy functional already arose in a series of other settings [22,29,30]. The specific problem that governs the behavior of the minimal logarithmic energy at the linear scale is:…”

A low discrepancy sequence on graphs

“…The bounds depend upon the spectral band, the L 2 norm of the target function, and certain powers of the graph Laplacian applied to the vector of the quadrature weights involved. Algorithms to choose these points and weights can be found in [21] for manifolds and [35] for a greedy algorithm of point and weight selection on graphs.…”