Variable shape parameter strategies in Radial Basis Function methods.

摘要

The Radial Basis Function (RBF) method is an important tool in the interpolation of multidimensional scattered data. The method has several important properties. One is the ability to handle sparse and scattered data points. Another property is its ability to interpolate in more than one dimension. Furthermore, the Radial Basis Function method provides phenomenal accuracy which has made it very popular in many fields. Some examples of applications using the RBF method are numerical solutions to partial differential equations, image processing, and cartography. This thesis involves researching Radial Basis Functions using different shape parameter strategies. First, we introduce the Radial Basis Function method by stating its history and development in Chapter 1. Second, we explain how Radial Basis Functions work in Chapter 2. Chapter 3 compares RBF interpolation to polynomial interpolation. Chapters 4 and 5 introduce the idea of variable shape parameters. In these chapters we compare and analyze the variable shape parameters in one and two dimensions. In Chapter 6, we introduce the challenges in interpolations due to errors in boundary regions. Here, we try to reduce the error using different shape parameter strategies. Chapter 7 lists the conclusions resulting from the research.;Throughout the thesis we use some abbreviations and acronyms, they are listed below in the Table 1.*;The following notes are highlighted, in order to clarify how things are stated in the thesis:;Note. Matlab does not display scientific notation as we are used to seeing. For example, 1.2x102 is written by Matlab to be 1.2e2. Please be aware of this throughout the thesis. This different form of scientific notation is very common in numerical analysis papers. Do not mistake e to be e ≈ 2.71828183.;Note. Starting at chapter 3, the reader will see the words &egr;Min, &egr;Max, and &egr;Average. &egr;Min, &egr;Max, and &egr;Average correspond to the value associated with the shape parameter in the MQ RBF. The Multiquadric Radial Basis function does not use c as the shape parameter, but &egr;. The reader will see this notation (&egr;) in Table 3 that displays different radial basis functions.;*Please refer to dissertation for diagram.