In this thesis, we analyse multilevel collocation methods involving compactly supported radial basis functions. We focus on linear second-order elliptic bound- ary value problems as well as Darcy's problem. While in the former case we use scalar-valued positive definite functions for constructing multilevel approximants, in the latter case we use matrix-valued functions that are automatically divergence-free. A similar result is presented for interpolating divergence-free vector fields. Even though it had been observed more than a decade ago that the stationary setting, i.e. when the support radii shrink as fast as the mesh norm, does not lead to convergence, it was up to now an open question how the support radii should depend on the mesh norm to ensure convergence. For each case above, we answer this question here thoroughly. Furthermore, we analyse and improve the stability of the linear systems. And lastly, we examine the case when the approximant does not lie in the same space as the solution to the PDE.

Based on the combined surface and volume RWG (Rao-Wilton-Glisson) basis functions, a simulator of a patch antenna on a finite dielectric substrate using the Method of Moments (MoM) has been implemented in Matlab. The metal surface is divided into planar triangular elements whereas the (inhomogeneous) dielectric volume is divided into tetrahedral elements. The structure under study is comprised of a typical patch antenna consisting of a single patch above a finite ground plane, and a probe feed. The performance of the solver is studied for different mesh configurations. The results obtained are tested by comparison with the commercial ANSOFT HFSS v8.5 and WIPL-D simulators. The former uses a large number of finite elements (up to 30,000) and adaptive mesh refinement, thus providing the reliable data for comparison. Behavior of the most sensitive characteristic ¡V antenna input impedance ¡V is tested, close to the first resonant frequency. The error in the resonant frequency is estimated at different values of the relative dielectric constant ƒÕr, which ranges from 1 to 20. The reported results show reasonable agreement. However, the solver needs to be further improved.

Lui, Fung Yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 83-92). / Abstracts in English and Chinese. / Acknowledgement --- p.iii / Abbreviation --- p.xii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Related Work --- p.6 / Chapter 1.2.1 --- Parametric Optimization Method and Radial Basis Functions --- p.6 / Chapter 1.3 --- Contribution and Organization of the Dissertation --- p.7 / Chapter 2 --- Level Set Method for Structure Shape and Topology Optimization --- p.8 / Chapter 2.1 --- Primary Ideas of Shape and Topology Optimization --- p.8 / Chapter 2.2 --- Level Set models of implicit moving boundaries --- p.11 / Chapter 2.2.1 --- Representation of the Boundary via Level Set Method --- p.11 / Chapter 2.2.2 --- Hamilton-Jacobin Equations --- p.13 / Chapter 2.3 --- Numerical Techniques --- p.13 / Chapter 2.3.1 --- Sign-distance function --- p.14 / Chapter 2.3.2 --- Discrete Computational Scheme --- p.14 / Chapter 2.3.3 --- Level Set Surface Re-initialization --- p.16 / Chapter 2.3.4 --- Velocity Extension --- p.16 / Chapter 3 --- Structure Topology Optimization with Discrete Level Sets --- p.18 / Chapter 3.1 --- A Level Set Method for Structural Shape and Topology Optimization --- p.18 / Chapter 3.1.1 --- Problem Definition --- p.18 / Chapter 3.2 --- Shape Derivative: an Engineering-oriented Deduction --- p.21 / Chapter 3.2.1 --- Sensitivity Analysis --- p.23 / Chapter 3.2.2 --- Optimization Algorithm --- p.28 / Chapter 3.3 --- Limitations of Discrete Level Set Method --- p.30 / Chapter 4 --- RBF based Parametric Level Set Method --- p.32 / Chapter 4.1 --- Introduction --- p.32 / Chapter 4.2 --- Radial Basis Functions Modeling --- p.33 / Chapter 4.2.1 --- Inverse Multiquadric (IMQ) Radial Basis Functions --- p.38 / Chapter 4.3 --- Parameterized Level Set Method in Structure Topology Optimization --- p.39 / Chapter 4.4 --- Parametric Shape and Topology Structure Optimization Method with Radial Basis Functions --- p.42 / Chapter 4.4.1 --- Changing Coefficient Method --- p.43 / Chapter 4.4.2 --- Moving Knot Method --- p.45 / Chapter 4.4.3 --- Combination of Changing Coefficient and Moving Knot method --- p.46 / Chapter 4.5 --- Numerical Implementation --- p.48 / Chapter 4.5.1 --- Sensitivity Calculation --- p.48 / Chapter 4.5.2 --- Optimization Algorithms --- p.49 / Chapter 4.5.3 --- Numerical Examples --- p.52 / Chapter 4.6 --- Summary --- p.65 / Chapter 5 --- Conclusion and Future Work --- p.80 / Chapter 5.1 --- Conclusion --- p.80 / Chapter 5.2 --- Future Work --- p.81 / Bibliography --- p.83

We had previously shown that regularization principles lead to approximation schemes, as Radial Basis Functions, which are equivalent to networks with one layer of hidden units, called Regularization Networks. In this paper we show that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models, Breiman's hinge functions and some forms of Projection Pursuit Regression. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In the final part of the paper, we also show a relation between activation functions of the Gaussian and sigmoidal type.

Thesis (M.S.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 63-67). Also available via World Wide Web.

The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions.

This research develops a memory model capable of performing in a human-like fashion on a maze traversal task. The model is based on and retains the underlying ideas of Minerva 2 but is executed with different mathematical operations and with some added parameters and procedures that enable more capabilities. When applied to the same maze traversal task as was used in a previous experiment with human subjects, the performance of a maze traversal agent with the developed model as its memory emulated the error rates of the human data remarkably well. As well, the maze traversal agent and memory model successfully emulated the human data when it was divided into two groups: fast maze learners and slow maze learners. It was able to account for individual differences in performance, specifically, individual differences in the learning rate. Because forgetting was not applied and therefore all experiences were flawlessly encoded in memory, the model additionally demonstrates that error can be due to interference between memories rather than forgetting. / Thesis (Master, Computing) -- Queen's University, 2008-06-04 13:39:38.179

This thesis aims to acquire, deepen and promote understanding of computing techniques for high dimensional scattered data approximation with radial basis functions. The main contributions of this thesis include sufficient conditions for the sovability of compactly supported radial basis functions with different shapes, near points preconditioning techniques for high dimensional interpolation systems with compactly supported radial basis functions, a heterogeneous hierarchical radial basis function interpolation scheme, which allows compactly supported radial basis functions of different shapes at the same level, an O(N) algorithm for constructing hierarchical scattered data set andan O(N) algorithm for sparse kernel summation on Cartesian grids. Besides the main contributions, we also investigate the eigenvalue distribution of interpolation matrices related to radial basis functions, and propose a concept of smoothness matching. We look at the problem from different perspectives, giving a systematic and concise description of other relevant theoretical results and numerical techniques. These results are interesting in themselves and become more interesting when placed in the context of the bigger picture. Finally, we solve several real-world problems. Presented applications include 3D implicit surface reconstruction, terrain modelling, high dimensional meteorological data approximation on the earth and scattered spatial environmental data approximation.

Mesh free modeling techniques are a promising alternative to traditional meshed methods for solving computational fluid dynamics problems. These techniques aim to solve for the field variable using solely the values of nodes and therefore do not require the generation of a mesh. This results in a process that can be much more reliably automated and is therefore attractive. Radial basis functions (RBFs) are one type of "meshless" method that has shown considerable growth in the past 50 years. Using these RBFs to directly solve a partial differential equation is known as Kansa's method and has been used to successfully solve many flow problems. The problem with Kansa's method is that there is no formal guarantee that its solution matrix will be non-singular. More recently, an expansion on Kansa's method was proposed that incorporates the boundary and PDE operators into the solution of the field variable. This method, known as Hermitian method, has been shown to be non-singular provided certain nodal criteria are met. This work aims to perform a comparison between Kansa and Hermitian methods to aid in future selection of a method. These two methods were used to solve steady and transient one-dimensional convection-diffusion problems. The methods are compared in terms of accuracy (error) and computational complexity (conditioning number) in order to evaluate overall performance. Results suggest that the Hermitian method does slightly outperform Kansa method at the cost of a more ill-conditioned collocation matrix.

Learning algorithms based on the Fourier transform attempt to learn functions by approximating the largest coefficients of their Fourier representations. Nearly all previous work in Fourier-based learning has been in the theoretical realm, where properties of the transform have made it possible to prove many interesting learnability results. The real-world usefulness of Fourier-based methods, however, has not been thoroughly explored. This thesis explores methods for the practical application of Fourier-based learning. The primary contribution of this thesis is a new search algorithm for finding the largest coefficients of a function's Fourier representation. Although the search space is exponentially large, empirical results demonstrate that only a small fraction of the space needs to be explored to find the largest coefficients. Furthermore, the algorithm is applicable to a much wider range of learning scenarios than previous approaches. Results of learning real-world problems with algorithms based on this search technique are also presented. The accuracies of the Fourier-based learning methods are not particularly impressive, however, and analysis and empirical results suggest why the Fourier representation may be a poor choice for typical real-world learning problems. Finally, this thesis shows that the search algorithm can be generalized to explore any basis of functions. Furthermore, it can search multiple bases simultaneously. This greatly enhances the learning techniques, and empirical results demonstrate significantly improved accuracy over the Fourier-based approach.