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The preliminary experimental results proved, that there are many local optima and if an iteration process is to be used, no guaranteed global optima are obtained. In this contribution, we present preliminary results of the estimation of a vector of “optimal” shape parameters, which are different for each RBF used in the final formula for RBF approximation. These two classes of RBFs: global and local, mostly having a shape parameter determining the RBF behavior. Generally, the approximation is formed using the principle of unity as a sum of weighed RBFs.

The RBF meshless methods are independent of a dimensionality of the problem solved and they mostly lead to a solution of a linear system of equations.

using Delaunay triangulation, in general. The RBF methods belong to meshless methods, which do not require tessellation of the data domain, i.e. Used as class notes for graduate courses at Northwestern University, Illinois Institute of Technology, and Vanderbilt University, this book will appeal to both mathematics and engineering graduate students.Radial basis functions (RBF) are widely used in many areas especially for interpolation and approximation of scattered data, solution of ordinary and partial differential equations, etc. A good balance is supplied between the necessary theory and implementation in terms of many MATLAB programs, with examples and applications to illustrate key points.

Meshfree approximation methods, such as radial basis function and moving least squares method, are discussed from a scattered data approximation and partial differential equations point of view. Whereas other works focus almost entirely on theoretical aspects or applications in the engineering field, this book provides the salient theoretical results needed for a basic understanding of meshfree approximation methods.The emphasis here is on a hands-on approach that includes MATLAB routines for all basic operations.

Meshfree approximation methods are a relatively new area of research, and there are only a few books covering it at present.