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

TL;DR

The paper addresses solving PDEs on irregular domains represented by point clouds by introducing the Constrained Least-Squares Ghost Sample Points (CLS-GSP) method, with LS-GSP as a baseline. CLS-GSP centers a local RBF reconstruction at ghost points and imposes a hard center-through constraint, yielding improved conditioning and diagonal dominance of the differential matrix, along with a rigorous consistency analysis for the Laplacian. The authors prove order $k-1$ consistency when using a polynomial basis up to degree $k$, and validate the approach through extensive numerical experiments on local reconstruction, 2D Poisson problems, and Laplace-Beltrami eigenproblems on manifolds, demonstrating stability and accuracy. The method offers a robust, meshless framework for accurate differential-operator discretization on scattered data, with potential applications in diffusion processes on manifolds and geometric data analysis.

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.

Figures (14)

• Figure 1: The graphical representation of ghost sample points. The black dot represents the center point $\mathbf{x}_0$. Blue squares denote the $n$ nearest scattered data points around $\mathbf{x}_0$. Red triangles correspond to the proposed $d$ ghost sample points associated with the center point $\mathbf{x}_0$. Here, we pick $d=8$ ghost sample points uniformly distributed on a ball with a radius given by the mean distance to the sample points.
• Figure 2: (Example \ref{['Ex:ErrorApproximation']}) A random sampling of the function (a) $u(x) = \cos 4x$ and (b) $u(x) = 1 - \text{sgn}(x)x$. The red circles represent the random sample points chosen in the reconstruction.
• Figure 3: (Example \ref{['Ex:ErrorApproximation']}) Reconstructions of (a) a smooth and (b) a nonsmooth function using our CLS-GSP method with different RBF kernels. The solid black lines are the exact function, and the solid red lines are obtained by the original least-squares reconstruction.
• Figure 4: (Section \ref{['Ex:Laplacian_Approximation']}) Error in approximating the Laplacian of (a) $u_1(x,y)=\exp(-x^2-y^2)$ and (b) $u_2(x,y)=3\cos x-4\sin y$ at the origin for our CLS-GSP method with the second, the third and the fourth order polynomial basis.
• Figure 5: (Section \ref{['Ex: Shape_Para']}) Error in the local reconstruction for (a) the RBF-FD method and (b) the CLS-GSP method using IQ-RBF. The white crosses and the green circle present the location of the sample and ghost points, respectively. The red dot is the origin.

Theorems & Definitions (7)

• Definition 1
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Theorem \ref{['Thm:Consistency']}