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.

翻译：本文提出了一种新颖的无网格方法，称为约束最小二乘幽灵样本点方法，用于求解以随机生成样本点表示的不规则区域或流形上的偏微分方程。我们的方法包含两项关键创新。首先，我们使用以一组精心选择的、独立于点云样本的\textit{幽灵样本点}为中心的径向基函数的线性组合来局部重构底层函数。其次，与传统最小二乘法（最小化所有样本点的平方差之和）不同，我们通过施加硬约束来正则化局部重构，以确保最小二乘近似精确通过中心点。这一简单而有效的约束显著增强了所得微分矩阵的对角占优性和条件数。我们提供了分析证明，表明该方法能够一致地估计精确的拉普拉斯算子。此外，我们展示了多种数值算例，验证了所提方法在求解拉普拉斯/泊松方程及相关特征值问题上的有效性。