发表机构
LAAS-CNRS（法国国家科学研究中心实验室与系统分析研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种直接通过面复形传播泛函的齐次数值积分方法，无需先形成矩向量，获得带符号顶点导数系数泛函，支持加权后处理，适用于任意维数凸与非凸多面体域。
AI 中文摘要
齐次数值积分（HNI）结合欧拉恒等式和斯托克斯定理，在多胞体 $P$ 上积分齐次多项式。现有算法返回标量积分或可复用的矩族。任何得到的矩表之后都可以重新编码：一旦所有 $q$ 次矩已知，在 $\mathcal H_q$（$q$ 次形式）上的积分就有一个由任意单点支撑的纯 $q$ 阶分布给出的平凡表示。我们的贡献是转而通过面复形传播泛函，对 $(P,q)$ 只进行一次，且不先形成矩向量，从而获得一个在 $\mathcal H_q$ 上的带符号顶点导数系数泛函，该泛函特定于 $P$，并直接由其提供的面几何和所选锚点确定。一个有限维 Riesz 框架描述了其非唯一性，给出了受限公式的内在 $L^2(P)$ 误差，并支持加权后处理。独立生成的 $q$ 次和 $2q$ 次系数向量满足显式的相容性关系。顶点锚点可以剪枝传播的表示，如一些多边形示例所示，一个离线基准量化了这种效果，但不声称全局基数最优性或改进的条件数。该构造覆盖任意维数的凸多胞体和有向非凸多面体域。
英文摘要
Homogeneous numerical integration (HNI) combines Euler's identity and Stokes' theorem to integrate homogeneous polynomials over a polytope $P$. Existing algorithms return scalar integrals or reusable moment families. Any resulting moment table can be recoded afterward: once all degree-$q$ moments are known, integration on $\mathcal H\_q$ (degree-$q$ forms) has a trivial representation by a pure order-$q$ distribution supported at any single arbitrary point. Our contribution is instead to propagate functionals through the face complex, once for $(P,q)$ and without first forming the moment vector, to obtain a signed vertex-derivative coefficient functional on $\mathcal H\_q$, specific to $P$ and determined directly by its supplied face geometry and chosen anchors. A finite-dimensional Riesz framework describes its nonuniqueness, gives an intrinsic $L^2(P)$ error for restricted formulas, and supports weighted post-processing. The independently generated degree-$q$ and degree-$2q$ coefficient vectors satisfy an explicit compatibility relation. Vertex anchors can prune the propagated representation as shown in some polygon examples and an offline benchmark quantify this effect without claiming global cardinality optimality or improved conditioning. The construction covers convex polytopes and oriented non-convex polyhedral domains in arbitrary dimension.
