Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Fulton B. Gonzalez. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 72-73). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

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.

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.

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.

The rapid increase in computational power over the last decade has unlocked the possibility of providing patient-specific healthcare via simulation and data assimilation. In the past 2 decades, computational approaches to simulating cardiovascular flows have advanced significantly due to intense research and adoption of methods in medical device companies. A significant source of friction in porting these tools to the hospital and getting in the hands of surgeons is due to the expertise required to handle the geometry pre-processing and meshing of models. Meshless meth- ods reduce the amount of corner cases which makes it easier to develop robust tools surgeons need. To accurately simulate modifications to a region of vasculature as in surgical planning, the entire system must be modeled. Unfortunately, this is computationally prohibitive even on to- day’s machines. To circumvent this issue, the Radial-Basis Function Finite Difference (RBF-FD) method for solution of the higher-dimensional (2D/3D) region of interest is tightly-coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The incompress- ible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and in- ductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF meshless approach is well-suited for modeling complicated non-Newtonian hemo- dynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of meshless methods and LPMs for fast and accurate hemody- namic simulations. The results show the efficacy of the method to be used in building robust tools to inform surgical decisions and further development is motivated.

We present a unifying framework for the global optimization of functions which are expensive to evaluate. The framework is based on a Bayesian interpretation of radial basis function interpolation which incorporates existing methods such as Kriging, Gaussian process regression and neural networks. This viewpoint enables the application of Bayesian decision theory to derive a sequential global optimization algorithm which can be extended to include existing algorithms of this type in the literature. By posing the optimization problem as a sequence of sampling decisions, we optimize a general cost function at each stage of the algorithm. An extension to multi-stage decision processes is also discussed. The key idea of the framework is to replace the underlying expensive function by a cheap surrogate approximation. This enables the use of existing branch and bound techniques to globally optimize the cost function. We present a rigorous analysis of the canonical branch and bound algorithm in this setting as well as newly developed algorithms for other domains including convex sets. In particular, by making use of Lipschitz continuity of the surrogate approximation, we develop an entirely new algorithm based on overlapping balls. An application of the framework to the integration of expensive functions over rectangular domains and spherical surfaces in low dimensions is also considered. To assess performance of the framework, we apply it to canonical examples from the literature as well as an industrial model problem from oil reservoir simulation.