Date of Award
Doctor of Philosophy (PhD)
Committee Chair Department
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Committee Member 2 Department
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Meshless methods utilizing Radial Basis Functions~(RBFs) are a numerical method that require no mesh connections within the computational domain. They are useful for solving numerous real-world engineering problems. Over the past decades, after the 1970s, several RBFs have been developed and successfully applied to recover unknown functions and to solve Partial Differential Equations (PDEs).
However, some RBFs, such as Multiquadratic (MQ), Gaussian (GA), and Matern functions, contain a free variable, the shape parameter, c. Because c exerts a strong influence on the accuracy of numerical solutions, much effort has been devoted to developing methods for determining shape parameters which provide accurate results. Most past strategies, which have utilized a trail-and-error approach or focused on mathematically proven values for c, remain cumbersome and impractical for real-world implementations.
This dissertation presents a new method, Residue-Error Cross Validation (RECV), which can be used to select good shape parameters for RBFs in both interpolation and PDE problems. The RECV method maps the original optimization problem of defining a shape parameter into a root-finding problem, thus avoiding the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned.
With minimal computational time, the RECV method provides shape parameter values which yield highly accurate interpolations. Additionally, when considering smaller data sets, accuracy and stability can be further increased by using the shape parameter provided by the RECV method as the upper bound of the c interval considered by the LOOCV method. The RECV method can also be combined with an adaptive method, knot insertion, to achieve accuracy up to two orders of magnitude higher than that achieved using Halton uniformly distributed points.
2015, Lei-Hsin Kuo
Kuo, Lei-Hsin, "On the Selection of a Good Shape Parameter for RBF Approximation and Its Application for Solving PDEs" (2015). Dissertations. 142.
Aerodynamics and Fluid Mechanics Commons, Applied Mechanics Commons, Computational Engineering Commons, Computer-Aided Engineering and Design Commons, Numerical Analysis and Computation Commons, Partial Differential Equations Commons