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The aim of this thesis is to demonstrate how the versatility of radial basis functions can be used to construct algorithms for approximating discrete sets of scattered data. In many cases, these algorithms have been constructed by blending together existing methods or by extending algorithms that exploit certain properties of a particular basis function to include certain radial functions. In the later chapters, we shall see that methods which currently use radial basis functions can be made more efficient by considering a change to the existing methods of solution. In chapter one we introduce radial basis functions (RBFs) and show how they can be used to construct interpolation and approximation models. We examine the uniqueness properties of the interpolation scheme for two specific functions and review some of the methods currently being used to determine the type of function to use and how to choose the number and location of centres. We describe three methods for choosing centres based on data clustering techniques and compare the accuracy of an approximation using two of these schemes. We show through a numerical example how greater accuracy can be achieved by combining these two schemes intelligently to construct a new, hybrid method. Problems that currently exist, for a particular clustering algorithm, when dealing with domain boundaries and which are not covered in great detail in the literature are highlighted and a new method is proposed. We conclude the chapter with an investigation into point distributions on the sphere. Radial basis functions are increasingly being used as a tool for approximating both discrete data and known functions on the sphere. Much of the current research focuses on constructing optimum point distributions for approximations using spherical harmonics. In this section we compare and evaluate these point distributions for RBF approximations and contrast the accuracy of the spherical harmonics with results obtained using the multiquadric function. In chapter two we develop an algorithm for surface approximation by combining the works of Mason & Bennell [40], and Clenshaw & Hayes [18]. Here, the well known method for constructing tensor products on rectangular grids is combined with an algorithm for approximating data collected along curved paths. The method developed in the literature for separable Chebyshev polynomials is extended to include the Gaussian radial function. Since the centres of the Gaussians can be distinct from the data points, we suggest a method for constructing a suitable set of centres to enable the efficiency of the two methods to be preserved. Possibilities for further efficiency using parallel processing are also discussed. We conclude the chapter by reviewing the Gram-Schmidt method and show how the use of orthogonal functions results in a numerically stable computation for evaluating the model parameters. The local support of the Gaussian function is investigated and the method of Mason & Crampton [41] is explained for constructing orthogonalised Gaussian functions. Chapter three introduces a relatively new topic in data approximation called support vector machines (SVMs). The motivation behind using SVMs for constructing regression models to corrupted data is addressed and the use of RBF kernels to map data into feature space is explained. We show how the regression model is formulated and discuss currently used methods of solution. The flexibility of SVMs to adapt to different types and level of noise is demonstrated through some numerical examples. We make use of the techniques developed in SVM regression to show how the algorithm described in chapter two can be extended. Here we make use of SVMs in the early stages of the algorithm to remove the need for further consideration of noise. We complete the discussion of SVMs by explaining their use in the field of data classification through a simple pattern recognition example.Chapter four focuses on a new approach to the solution of an SVM. The new approach taken is one of constructing an entirely linear objective function. This is achieved by changing the regularisation term. We show, in detail, how the changes made to the existing framework affects the construction of the model. We describe the solution method and explain how advantage can be taken of the new linear structure. To determine the model parameters, we show how the solution, in the form of a simplex tableau, can be found extremely efficiently by recognising certain relationships between variables that allow us to employ Lei's algorithm. Examples that show SVM approximants to noisy data for both curves and surfaces are given together with a comparison between Lei's algorithm and a standard simplex solution method. We finish the section by highlighting the link between support vectors and radial basis function centres. The sparsity produced by the method in the coefficient vector is also discussed. The new linearised approach to constructing SVM regression models is used in a new algorithm developed to construct planar curves that model the path of fault lines in a surface. Part of a detection algorithm proposed by Gutzmer & Iske [33] is used to determine points that lie close to a fault line. The new approach is then to model the fault line by constructing an SVM regression curve. The chapter concludes with some examples and remarks. The thesis concludes with Chapter five in which we summarise the main points discussed and point to possibilities for extending the work presented.

This Licentiate Thesis is devoted to the presentation and discussion of some new contributions in applied mathematics directed towards scientific computing in sports engineering. It considers inverse problems of biomechanical simulations with rigid body musculoskeletal systems especially in cross-country skiing. This is a contrast to the main research on cross-country skiing biomechanics, which is based mainly on experimental testing alone. The thesis consists of an introduction and five papers. The introduction motivates the context of the papers and puts them into a more general framework. Two papers (D and E) consider studies of real questions in cross-country skiing, which are modelled and simulated. The results give some interesting indications, concerning these challenging questions, which can be used as a basis for further research. However, the measurements are not accurate enough to give the final answers. Paper C is a simulation study which is more extensive than paper D and E, and is compared to electromyography measurements in the literature. Validation in biomechanical simulations is difficult and reducing mathematical errors is one way of reaching closer to more realistic results. Paper A examines well-posedness for forward dynamics with full muscle dynamics. Moreover, paper B is a technical report which describes the problem formulation and mathematical models and simulation from paper A in more detail. Our new modelling together with the simulations enable new possibilities. This is similar to simulations of applications in other engineering fields, and need in the same way be handled with care in order to achieve reliable results. The results in this thesis indicate that it can be very useful to use mathematical modelling and numerical simulations when describing cross-country skiing biomechanics. Hence, this thesis contributes to the possibility of beginning to use and develop such modelling and simulation techniques also in this context.

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<p>We describe, implement, and analyze a class of staggered grid algorithms for efficient simulation and analysis of time-domain Maxwell systems in the case of heterogeneous, conductive, and nondispersive, isotropic, linear media. We provide the derivation of a continuous mathematical model from the Maxwell equations in vacuum; however, the complexity of this system necessitates the use of computational methods for approximately solving for the physical unknowns. The finite difference approximation has been used for partial differential equations and the Maxwell Equations in particular for many years. We develop staggered grid based finite difference discrete operators as a class of approximations to continuous operators based on second order in time and various order approximations to the electric and magnetic field at staggered grid locations. A generalized parameterized operator which can be specified to any of this class of discrete operators is then applied to the Maxwell system and hence we develop discrete approximations through various choices of parameters in the approximation. We describe analysis of the resulting discrete system as an approximation to the continuous system. Using the comparison of dispersion analysis for the discrete and continuous systems, we derive a third difference approximation, in addition to the known (2, 2) and (2, 4) schemes. We conclude by providing the comparison of these three methods by simulating the Maxwell system for several choices of parameters in the system

<p> We present a detailed analysis of the application of a multi-scale Hierarchical Reconstruction method for solving a family of ill-posed linear inverse problems. When the observations on the unknown quantity of interest and the observation operators are known, these inverse problems are concerned with the recovery of the unknown from its observations. Although the observation operators we consider are linear, they are inevitably ill-posed in various ways. We recall in this context the classical Tikhonov regularization method with a stabilizing function which targets the specific ill-posedness from the observation operators and preserves desired features of the unknown. Having studied the mechanism of the Tikhonov regularization, we propose a multi-scale generalization to the Tikhonov regularization method, so-called the Hierarchical Reconstruction (HR) method. First introduction of the HR method can be traced back to the Hierarchical Decomposition method in Image Processing. The HR method <i> successively</i> extracts information from the previous hierarchical residual to the current hierarchical term at a <i>finer</i> hierarchical <i> scale.</i> As the sum of all the hierarchical terms, the hierarchical sum from the HR method provides an reasonable approximate solution to the unknown, when the observation matrix satisfies certain conditions with specific stabilizing functions. When compared to the Tikhonov regularization method on solving the same inverse problems, the HR method is shown to be able to decrease the total number of iterations, reduce the approximation error, and offer self control of the approximation distance between the hierarchical sum and the unknown, thanks to using a ladder of <i>finitely many</i> hierarchical scales. We report numerical experiments supporting our claims on these advantages the HR method has over the Tikhonov regularization method.</p>

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