Moving Least Squares (MLS) as the Foundation of Meshless Approximation

In this text, you have referenced the Moving Least Squares (MLS) method. MLS serves as the fundamental basis for meshless approximation techniques, which are primarily employed for solving high-dimensional computational problems. The implementation typically involves constructing local approximation functions using weighted least squares fitting, where the weighting functions depend on spatial distance from evaluation points. However, it is important to note that while MLS represents an effective computational technique, it does not constitute the exclusive solution approach. Under certain circumstances, MLS may encounter limitations such as high computational costs in large-scale implementations or suboptimal performance on specific dataset configurations. Therefore, it becomes necessary to consider alternative numerical methods, including Finite Element Methods (FEM) or Finite Difference Methods (FDM), for addressing high-dimensional problems. These alternative techniques can also deliver high-quality approximation results through different discretization approaches and may demonstrate superior computational efficiency compared to MLS in certain application scenarios.