A Constrained Least-Squares Ghost Sample Points (CLS-GSP) Method for Differential Operators on Point Clouds

TL;DR

The paper presents CLS-GSP, a meshless method for solving PDEs on irregular domains using ghost sample points and a constrained least-squares approach, improving matrix conditioning and accuracy.

Contribution

It introduces a novel constrained least-squares ghost sample points method that enhances stability and accuracy in meshless PDE solutions on point clouds.

Findings

Analytical proof of consistent Laplacian estimation.

Effective in solving Laplace and Poisson equations.

Improves matrix conditioning and stability.

Abstract

We introduce a novel meshless method called the Constrained Least-Squares Ghost Sample Points (CLS-GSP) method for solving partial differential equations on irregular domains or manifolds represented by randomly generated sample points. Our approach involves two key innovations. Firstly, we locally reconstruct the underlying function using a linear combination of radial basis functions centered at a set of carefully chosen \textit{ghost sample points} that are independent of the point cloud samples. Secondly, unlike conventional least-squares methods, which minimize the sum of squared differences from all sample points, we regularize the local reconstruction by imposing a hard constraint to ensure that the least-squares approximation precisely passes through the center. This simple yet effective constraint significantly enhances the diagonal dominance and conditioning of the resulting differential matrix. We provide analytical proofs demonstrating that our method consistently estimates the exact Laplacian. Additionally, we present various numerical examples showcasing the effectiveness of our proposed approach in solving the Laplace/Poisson equation and related eigenvalue problems.