In this paper, we propose a mesh-free numerical method for solving elliptic PDEs on unknown manifolds, identified with randomly sampled point cloud data. The PDE solver is formulated as a spectral method where the test function space is the span of the leading eigenfunctions of the Laplacian operator, which are approximated from the point cloud data. While the framework is flexible for any test functional space, we will consider the eigensolutions of a weighted Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced by a weak approximation of a weighted Laplacian on an appropriate Hilbert space. Especially, we consider a test function space that encodes the geometry of the data yet does not require us to identify and use the sampling density of the point cloud. To attain a more accurate approximation of the expansion coefficients, we adopt a second-order tangent space estimation method to improve the RBF interpolation accuracy in estimating the tangential derivatives. This spectral framework allows us to efficiently solve the PDE many times subjected to different parameters, which reduces the computational cost in the related inverse problem applications. In a well-posed elliptic PDE setting with randomly sampled point cloud data, we provide a theoretical analysis to demonstrate the convergent of the proposed solver as the sample size increases. We also report some numerical studies that show the convergence of the spectral solver on simple manifolds and unknown, rough surfaces. Our numerical results suggest that the proposed method is more accurate than a graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two approaches are comparable. Due to the flexibility of the framework, we empirically found improved accuracies in both smoothed and unsmoothed Stanford bunny domains by blending the graph Laplacian eigensolutions and RBF interpolator.

翻译：本文提出一种无网格数值方法，用于求解随机采样点云数据表征的未知流形上的椭圆型偏微分方程。该偏微分方程求解器基于谱方法构建，其中检验函数空间由拉普拉斯算子的主特征函数张成，这些特征函数通过点云数据近似得到。尽管该框架适用于任意检验函数空间，我们重点考虑加权拉普拉斯算子的特征解——该算子通过对称径向基函数方法在适当希尔伯特空间上对加权拉普拉斯算子进行弱近似而获得。特别地，我们采用的检验函数空间既能编码数据的几何特征，又无需识别或使用点云的采样密度。为更精确地逼近展开系数，我们采用二阶切空间估计方法提升径向基函数插值在估计切向导数时的精度。该谱框架可高效求解不同参数下的重复偏微分方程，从而降低相关反问题应用的计算成本。针对随机采样点云数据下的适定椭圆型偏微分方程，我们提供了理论分析证明随着样本量增加解算器的收敛性。数值实验展示了该谱求解器在简单流形及未知粗糙表面上的收敛特性，结果表明在光滑流形上本方法较基于图拉普拉斯的求解器具有更高精度，而在粗糙流形上两者性能相当。凭借框架的灵活性，我们通过融合图拉普拉斯特征解与径向基函数插值器，经验性地发现该方法在经平滑与未平滑处理的斯坦福兔子域上均能实现精度提升。