Convergence of the Point Integral method for Poisson equation on point cloud
This work provides rigorous convergence analysis for a numerical method (PIM) used to discretize the Laplace-Beltrami operator on point clouds, which is important for geometric data analysis and manifold learning.
The authors analyze the convergence of the Point Integral method (PIM) for solving the Poisson equation with Neumann boundary conditions on submanifolds embedded in Euclidean spaces, providing theoretical guarantees for the method's accuracy.
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point clouds \cite{LSS}. In this paper, we analyze the convergence of Point Integral method (PIM) for Poisson equation with Neumann boundary condition on submanifolds isometrically embedded in Euclidean spaces.