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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-404565 |
| Date | January 2020 |
| Creators | Sundin, Ulrika |
| Publisher | Uppsala universitet, Avdelningen för beräkningsvetenskap, Uppsala universitet, Tillämpad beräkningsvetenskap |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | IT licentiate theses / Uppsala University, Department of Information Technology, 1404-5117 ; 2020-002 |
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