{"paper":{"title":"Quadrature Points via Heat Kernel Repulsion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Jianfeng Lu, Matthias Sachs, Stefan Steinerberger","submitted_at":"2018-04-06T15:49:30Z","abstract_excerpt":"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 poi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02327","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}