Radial Basis Function Approximations: Comparison and Applications

TL;DR

This paper compares various Radial Basis Function approximation methods, highlighting their stability, accuracy, and efficiency in handling large, scattered datasets in high-dimensional spaces, and demonstrates their practical advantages.

Contribution

It introduces and compares multiple RBF approximation techniques, emphasizing improvements in stability, accuracy, and computational efficiency over existing methods.

Findings

RBF approximation methods are effective for high-dimensional scattered data.

Proposed RBF methods require less memory and provide better approximation quality.

Comparison shows improved stability and accuracy of the new RBF approaches.

Abstract

Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher dimension d>2, because the other methods require the conversion of a scattered dataset to an ordered dataset (i.e. a semi-regular mesh is obtained by using some tessellation techniques), which is computationally expensive. The RBF approximation is non-separable, as it is based on the distance between two points. This method leads to a solution of Linear System of Equations (LSE) Ac=h. In this paper several RBF approximation methods are briefly introduced and a comparison of those is made with respect to the stability and accuracy of computation. The proposed RBF approximation offers lower memory requirements and better quality of approximation.