向量场在凸多面体域上的多项式插值
Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain
AI总结：
本文提出了一种在凸多面体域上基于最小二乘法的多项式向量场重建方法，能够精确满足无穿透边界条件。
摘要：
我们提出了一种计算方法，用于从离散样本重建凸多面体$\mathcal{P} \subset \mathbb{R}^d$上的向量场。我们特别考虑了向量场受无穿透（滑动）边界条件限制的情况，要求其在边界$\partial \mathcal{P}$上切线。给定一个次数上限$k$，我们的算法计算一个次数不超过$k$的多项式向量场，使其在最小二乘意义上拟合观测数据，同时精确满足切线约束。我们方法的核心是显式地对切于$\partial \mathcal{P}$的多项式向量场模块进行表征，该表征利用了超平面排列理论中的代数概念。
英文摘要：
We develop a method for reconstructing polynomial vector fields from discrete velocity samples on a convex polyhedral domain under an exact no-penetration boundary condition. For a prescribed degree bound, the method computes a polynomial vector field that fits the sampled data in the least-squares sense while satisfying the tangency condition identically on every boundary facet. The central ingredient is an explicit algebraic characterization of the space of polynomial vector fields tangent to the boundary, obtained from the theory of logarithmic derivations of hyperplane arrangements. This characterization reduces the constrained reconstruction problem to finite-dimensional linear algebra. We also discuss extensions incorporating additional linear differential constraints, such as incompressibility, and piecewise polynomial constructions on non-convex polyhedral domains with prescribed smoothness across cell interfaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
