A study on radial basis function and quasi-Monte Carlo methods

TL;DR

This paper explores the intrinsic relationship between radial basis function (RBF) and quasi-Monte Carlo (QMC) methods, focusing on their application to high-dimensional numerical integration and establishing RBFs through integral analysis.

Contribution

It introduces a novel connection between RBF and QMC methods based on integral analysis, providing new insights into RBF construction and error bounds.

Findings

RBFs can be constructed using integral analysis techniques.

Error bounds for RBFs are established based on integral properties.

Node placement strategies are discussed for improved RBF performance.

Abstract

The radial basis function (RBF) and quasi Monte Carlo (QMC) methods are two very promising schemes to handle high-dimension problems with complex and moving boundary geometry due to the fact that they are independent of dimensionality and inherently meshless. The two strategies are seemingly irrelevant and are so far developed independently. The former is largely used to solve partial differential equations (PDE), neural network, geometry generation, scattered data processing with mathematical justifications of interpolation theory [1], while the latter is often employed to evaluate high-dimension integration with the Monte Carlo method (MCM) background [2]. The purpose of this communication is to try to establish their intrinsic relationship on the grounds of numerical integral. The kernel function of integral equation is found the key to construct efficient RBFs. Some significant results on RBF construction, error bound and node placement are also presented. It is stressed that the RBF is here established on integral analysis rather than on the sophisticated interpolation and native space analysis.