In this work, we introduce a numerical method for approximating arbitrary differential operators on vector fields in the weak form given point cloud data sampled randomly from a $d$ dimensional manifold embedded in $\mathbb{R}^n$. This method generalizes the local linear mesh method to the local curved mesh method, thus, allowing for the estimation of differential operators with nontrivial Christoffer symbols, such as the Bochner or Hodge Laplacians. In particular, we leverage the potentially small intrinsic dimension of the manifold $(d \ll n)$ to construct local parameterizations that incorporate both local meshes and higher-order curvature information. The former is constructed using low dimensional meshes obtained from local data projected to the tangent spaces, while the latter is obtained by fitting local polynomials with the generalized moving least squares. Theoretically, we prove the weak and spectral convergence rates for the proposed method for the estimation of the Bochner Laplacian. We provide numerical results supporting the theoretical convergence rates for the Bochner and Hodge Laplacians on simple manifolds.

翻译：本文提出了一种数值方法，用于在点云数据（从嵌入在$\mathbb{R}^n$中的$d$维流形上随机采样获得）给定的情况下，以弱形式逼近向量场上的任意微分算子。该方法将局部线性网格方法推广至局部弯曲网格方法，从而能够估计具有非平凡克里斯托费尔符号的微分算子，例如Bochner拉普拉斯算子或Hodge拉普拉斯算子。特别地，我们利用流形可能较小的本征维度$(d \ll n)$来构建局部参数化，该参数化同时融合了局部网格和高阶曲率信息。前者通过将局部数据投影到切空间获得的低维网格构建，而后者则通过广义移动最小二乘法拟合局部多项式获得。理论上，我们证明了所提方法在估计Bochner拉普拉斯算子时的弱收敛和谱收敛速率。我们提供了数值结果，支持该方法在简单流形上对Bochner和Hodge拉普拉斯算子的理论收敛速率。