Numerical Method For Eigenproblems Of Laplace-beltrami Operator On High Dimension Point Cloud

The Laplace-Beltrami operator is a basic differential operator defined on Riemannian manifold.Its eigenproblem is widely used in many fields.However,in practical problems,the eigenvalue and eigenfunction expressions of the Laplace-Beltrami operator are often unable to be obtained,so it is necessary to research the numerical discretization method of the Laplace-Beltrami operator eigenproblem.However,the convergence of existing methods such as the grid method and heat kernel method often lacks theoretical guarantee,and they are difficult to effectively deal with high-dimensional problems.In this paper,we apply the Point Integral Method to the numerical method of eigenproblems of Laplace-Beltrami operator on high-dimensional point cloud for the first time,and design symmetric schemes for eigenproblems with three natural boundary conditions:Dirichlet,Neumann and Robin.The design of the discrete method in this paper consists of two steps.Firstly,according to the point integral formula,we use a solution operator of the integral equation to approximate the solution operator of the Laplace-Beltrami equation,then discrete the integral equation by Monte Carlo,and finally,use the eigenvalues and eigenfunctions of the solution operator of the discrete equation to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator.With the help of compact operator perturbation theory and empirical process theory,we theoretically analyze the convergence and convergence order of these discrete methods in this paper.Finally,we verify the convergence of the numerical scheme designed in this paper through numerical experiments.Compared with traditional methods,the method designed in this paper no longer needs a grid,so it is more flexible and can effectively deal with high-dimensional problems.Under the framework of the point integral method,the techniques of Dirichlet,Neumann and Robin are pretty different.On the one hand,when the integral equation approximates the Laplace-Beltrami equation,the boundary integral term in the point integral formula is equal to 0 under the Neumann boundary condition,while for the Dirichlet and Robin boundary conditions,we need to use additional techniques to deal with.On the other hand,the discrete schemes corresponding to Dirichlet and Robin boundary conditions can maintain the comparison principle of the Laplace-Beltrami operator.Due to the application of the comparison principle,when estimating the discrete error of integral equation,this part of error estimation is more compact under Dirichlet and Robin boundary conditions than Neumann boundary conditions.