Local Radial Basis Function Methods: Comparison, Improvements, and Implementation

作     者：Scott A. Sarra Scott A. Sarra

作者机构：Department of Mathematics Marshall University Huntington USA 

出 版 物：《Journal of Applied Mathematics and Physics》 (应用数学与应用物理（英文）)

年 卷 期：2023年第11卷第12期

页      面：3867-3886页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：Radial Basis Functions Shape Parameter Selection Quasi-Random Centers Numerical PDEs Scientific Computing Open Source Software Python Programming Language Reproducible Research 

摘      要：Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.