Quadrature Points via Heat Kernel Repulsion

2018/04/06 by Jianfeng Lu, Lu, Jianfeng, Matthias Sachs +3
Engineering · Materials Science · Mathematics · #Advanced Numerical Analysis Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1804.02327

openalex publication_date 2018/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ≃ (1)/(N) ∑_n=1^Na_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 ∑_{i,j =1}^N a_i a_j exp(-(d(x_i,x_j)²)/(4t)) → min, where~t ∼ 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.

Citations

• On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

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