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Under the architecture of a neural network, this work proposes and applies three multiquadric radial basis function (MQ-RBF) interpolation schemes; The Common Local Radial Basis Function Scheme (CLRBF), The Iterative Local Radial Basis Function Scheme (ILRBF), and The Radius Local Radial Basis Function Scheme (RLRBF). The schemes are designed to perform locally to overcome drawbacks normally encountered when using a global one. The famous Franke function in two dimensions is numerically tackled. It is revealed in this work that all three local methods outperform the traditional MQ interpolation in terms of both CPU-time and condition number, while the accuracy is overall acceptable, particularly when the number of nodes increases. This finding indicates their potential for dealing with bigger datasets and more complex problems.

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.

The main purpose of this work is to shed more light into the inverse quadratic radial basis function (RBF) in the application of interpolation. This RBF contains a parameter that plays a crucial role in determining the final result quality. Five strategies of variable shape parameters are numerically investigated. Both types of node distribution; normally and scattered, are considered. It is discovered that good results can be obtained when using some strategies particularly in 1D problem. Challenges become appearing when dealing with 2D and deserves further study.