A Higher Order Local Mesh Method for Approximating 1-Laplacians on Unknown Manifolds

TL;DR

This paper presents a novel higher order local mesh method for approximating complex differential operators like the Bochner and Hodge Laplacians on unknown manifolds from point cloud data, with proven spectral convergence.

Contribution

It introduces a generalized local curved mesh method that incorporates curvature information and proves spectral convergence for estimating Laplacians on manifolds.

Findings

Spectral convergence proven for the Bochner Laplacian.

Numerical results support convergence rates on simple manifolds.

Method effectively estimates differential operators on unknown manifolds.

Abstract

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 Christoffel 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 spectral convergence 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.