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In this work, we introduce a numerical method to approximate differential operators and integrals on point clouds sampled from a two dimensional manifold embedded in $\mathbb{R}^n$. Global mesh structure is usually hard to construct in this case. While our method only relies on the local mesh structure at each data point, which is constructed through local triangulation in the tangent space obtained by local principal component analysis (PCA). Once the local mesh is available, we propose numerical schemes to approximate differential operators and define mass matrix and stiffness matrix on point clouds, which are utilized to solve partial differential equations (PDEs) and variational problems on point clouds. As numerical examples, we use the proposed local mesh method and variational formulation to solve the Laplace-Beltrami eigenproblem and solve the Eikonal equation for computing distance map and tracing geodesics on point clouds.

A local mesh method for solving PDEs on point clouds