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Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severly ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in acurved waveguide geometry

In this paper, we have derived a radial basis function (RBF) based method for the pricing of financial contracts by solving the Black-Scholes partial differential equation. As an example of a financial contract that can be priced with this method we have chosen the multi-dimensional European basket call option. We have shown numerically that our scheme is second order accurate in time and spectrally accurate in space for constant shape parameter. For other, non-optimal choices of shape parameter values, the resulting convergence rate is algebraic. We propose an adaptive node point placement that improves the accuracy compared with a uniform distribution. Compared with an adaptive finite difference method, the RBF method is 20-40 times faster in one and two space dimensions and has approximately the same memory requirements.

Radial basis function (RBF) methods are meshfree, i.e., they can operate on unstructured node sets. Because the only geometric information required is the pairwise distance between the node points, these methods are highly flexible with respect to the geometry of the computational domain. The RBF approximant is a linear combination of translates of a radial function, and for PDEs the coefficients are found by applying the PDE operator to the approximant and collocating with the right hand side data. Infinitely smooth RBFs typically result in exponential convergence for smooth data, and they also have a shape parameter that determines how flat or peaked they are, and that can be used for accuracy optimization. In this thesis the focus is on global RBF collocation methods for PDEs, i.e., methods where the approximant is constructed over the whole domain at once, rather than built from several local approximations. A drawback of these methods is that they produce dense matrices that also tend to be ill-conditioned for the shape parameter range that might otherwise be optimal. One current trend is therefore to use over-determined systems and least squares approximations as this improves stability and accuracy. Another trend is to use localized RBF methods as these result in sparse matrices while maintaining a high accuracy. Global RBF collocation methods together with RBF interpolation methods, however, form the foundation for these other versions of RBF--PDE methods. Hence, understanding the behaviour and practical aspects of global collocation is still important. In this thesis an overview of global RBF collocation methods is presented, focusing on different versions of global collocation as well as on method properties such as error and convergence behaviour, approximation behaviour in the small shape parameter range, and practical aspects including how to distribute the nodes and choose the shape parameter value. Our own research illustrates these different aspects of global RBF collocation when applied to the Helmholtz equation and the Black-Scholes equation.