A Numerical Study of a Compactly-Supported Radial Basis Function Applied with a Collocation Meshfree Scheme for Solving PDEs

Abstract

It is known that all Radial Basis Function-based meshfree methods suffer from a lack of reliable judgement on the choice of shape parameter, appearing in most of the RBFs. While the popularity of meshfree/meshless numerical methods is growing fast over the past decade, the great challenge is still to find an optimal RBF form with its optimal shape parameter. In this work, the main focus is on one type of RBF namely ‘Compactly-Supported (CS-RBF)’ that contains no parameter, and yet has not been explored numerically as much in the past, particularly under the context of data interpolation/approximation and solving partial differential equations (PDEs). To compare the potential advantages of CS-RBF, two most popular choices of RBF widely used; Multiquadric (MQ), and Gaussian (GA) were studied parallelly. The information gathered and presented in this work shall be useful for the future users in making decision on RBF.

Numerical Comparison of Shapeless Radial Basis Function Networks in燩attern Recognition

“…Alternatively to shape-contained RBFs and to avoid the demerits mentioned so far, RBFs containing no shapes, referred to as 'shapefree' or 'shapeless' have recently been paid attention to. Recent work can be found in [19] where the authors attempted to approximate the solution of PDEs using RBF with no shapes. The application in the pattern recognition context, to our knowledge, has not been explored as much and our preliminary investigations have strongly been promising when shapeless RBFs are in use [20,21].…”