In this paper, we study the Radial Basis Function (RBF) approximation to differential operators on smooth tensor fields defined on closed Riemannian submanifolds of Euclidean space, identified by randomly sampled point cloud data. {The formulation in this paper leverages a fundamental fact that the covariant derivative on a submanifold is the projection of the directional derivative in the ambient Euclidean space onto the tangent space of the submanifold. To differentiate a test function (or vector field) on the submanifold with respect to the Euclidean metric, the RBF interpolation is applied to extend the function (or vector field) in the ambient Euclidean space. When the manifolds are unknown, we develop an improved second-order local SVD technique for estimating local tangent spaces on the manifold. When the classical pointwise non-symmetric RBF formulation is used to solve Laplacian eigenvalue problems, we found that while accurate estimation of the leading spectra can be obtained with large enough data, such an approximation often produces irrelevant complex-valued spectra (or pollution) as the true spectra are real-valued and positive. To avoid such an issue,} we introduce a symmetric RBF discrete approximation of the Laplacians induced by a weak formulation on appropriate Hilbert spaces. Unlike the non-symmetric approximation, this formulation guarantees non-negative real-valued spectra and the orthogonality of the eigenvectors. Theoretically, we establish the convergence of the eigenpairs of both the Laplace-Beltrami operator and Bochner Laplacian {for the symmetric formulation} in the limit of large data with convergence rates. Numerically, we provide supporting examples for approximations of the Laplace-Beltrami operator and various vector Laplacians, including the Bochner, Hodge, and Lichnerowicz Laplacians.

翻译：本文研究定义在欧氏空间闭黎曼子流形（由随机采样点云数据标识）上光滑张量场的微分算子的径向基函数（RBF）逼近。本文的公式利用了一个基本事实：子流形上的协变导数是欧氏空间方向导数到子流形切空间的正交投影。为对子流形上的测试函数（或向量场）按欧氏度量求导，我们应用RBF插值将函数（或向量场）延拓至欧氏空间。当流形未知时，我们开发了一种改进的二阶局部奇异值分解技术来估计流形上的局部切空间。使用经典逐点非对称RBF公式求解拉普拉斯特征值问题时发现：尽管足够大的数据量能获得主谱的准确估计，但此类逼近常产生无关的复值谱（即污染），而真实谱应为正实数。为避免该问题，我们引入基于适当希尔伯特空间弱形式的拉普拉斯对称RBF离散逼近。与非对称逼近不同，该公式保证非负实值谱及特征向量的正交性。理论上，我们建立了大数据极限下对称公式中拉普拉斯-贝尔特拉米算子与博赫纳拉普拉斯算子特征对的收敛速率；数值上，我们提供了拉普拉斯-贝尔特拉米算子及多种向量拉普拉斯算子（包括博赫纳、霍奇与利赫涅罗维茨拉普拉斯）逼近的验证算例。