This thesis presents an investigation into sources of variability in activating and processing numerical information. Chapter 1 provides an overview of research literature exploring the ways in which magnitude information can be represented, and how models relating to number information have developed. These theoretical models are addressed in relation to the neural representation of number, and the range of behavioural markers which suggest an association between spatial and numerical processing. Chapter 2 using a dual-task paradigm investigated whether magnitude information is accessed on perceiving numbers, or if this information is linked to response selection or execution. Previous research studies investigating this question produced inconsistent findings (Oriet, Tombu & Jolicoeur, 2005; Sigman & Dehaene, 2005) with regard to the locus of magnitude processing; the findings of Experiments 1-3 reliably support access to magnitude information during response selection. Chapter 3 explored the activation of spatial-numerical response associations, where response-irrelevant magnitude information was not represented by a single stimulus (i.e. an Arabic digit) but by a numerosity representation. Experiments 4-7 found a strong association between spatial-orientation processing and numerical magnitude, but no association with perceptual-colour processing, extending previous work by Fias, Lammertyn and Lauwereyns (2001) regarding the neural overlap between the attended and irrelevant stimulus dimensions. However the strength of this association was found to be inconsistent across the number range. Chapter 4 investigated the impact of healthy aging on the presence of neural-overlap in processing spatial-numerical information, further developing the paradigm used in Chapter 3, and addressed direct predictions from the literature as to how age should influence these associations (Wood, Willmes, Nuerk & Fischer, 2008). Experiments 8-11 found evidence for spatial-numerical associations across the lifespan, but that the strength of these effects were moderated by 5 task instruction. Chapter 5 was designed to assess aging differences in numerical and spatial processing with a battery of tests and the extent to which other sources of individual difference (sex, embodiment) have a measureable impact. A range of standardised measures were used to assess verbal ability, mathematical processing, and spatial working memory alongside behavioural measures of spatial numerical associations. Experiment 12 provided evidence of aging and sex differences in different cognitive tasks and a marginal impact of embodiment on spatial-numerical processing; however the effect of embodiment was not supported in a larger more homogenous sample in Experiment 13. Chapter 6 reflects on the current findings and provides contextual information on how they align with previous research, outlining how evidence from the thesis extend current research paradigms and provides new evidence regarding the maintenance of spatial-numerical associations in healthy aging. Methodologies developed in the thesis are considered with relation to how they may be applied to assess individual differences in early number acquisition in children. Finally the discussion outlines methods and controversies within the field of numerical cognition, with consideration of new methods for measuring the strength of spatial-numerical associations (Pinhas, Tzelgov, & Ganor-Stern, 2012), alongside the potential application of modelling techniques to investigate individual differences in task performance.

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

<p>Multi-scale wave propagation problems are computationally costly to    solve by traditional techniques because the smallest scales must be    represented over a domain determined by the largest scales of the    problem.  We have developed new numerical methods for multi-scale wave    propagation in the framework of heterogeneous multi-scale methods.  The    numerical methods couples simulations on macro and micro scales with    data exchange between models of different scales.  With the new method    we are able to consider a general class of problems including some    problems where a homogenized equation is unknown.  We show that the    complexity of the new method is significantly lower than that of    traditional techniques.  Numerical results are presented from problems    in one, two and three dimensional and for finite and long time.  We also    analyze the method, in one and several dimensions and for finite time,    using Fourier analysis.</p>

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No description available.

The present thesis consists of two parts. The first part is devoted to the analysis of discretizations of a class of basic electronic density functionals. In the second part we suggest and analyze Quasicontinuum Methods for an atomistic interaction potential that is based on a field. We begin by formulating and analyzing a model for the study of finite clusters of atoms or localized defects in infinite crystals based on a version of the classical Thomas{Fermi{Dirac{von Weizs?acker density functional. We show that the resulting constrained optimization problem has a minimizer and we provide a careful analysis of the solvability of the associated Euler{Lagrange equation. Based on these results, and using tools from saddle-point theory and nonlinear analysis, we then show that a Galerkin discretization has a solution that converges to the correct limit (in the case of Dirichlet as well as periodic boundary conditions). Furthermore, we investigate the issue of optimal convergence rates. Using appropriate dual problems, we can show faster convergence for the energy, the Lagrange multiplier of the underlying minimization problem, and the L2-errors of the solutions. We also look at the dependence of the density functional on the nucleus coordinates and show a convergence result for minimizing nucleus configurations. These results are subsequently generalized to the case of discretizations with numerical integration. Existence and convergence of solutions, as well as optimal convergence rates can be established if quadrature rules of sufficiently high order are applied. In the second part of the thesis we consider an atomistic interaction potential in one dimension given through a minimization problem, which gives rise to a field. The forces on atoms are in this case given by local expressions involving this field. A convenient feature of this model is the existence of a weak formulation for the forces, which provides a natural connection point for the coupling with a continuum model. We suggest Quasicontinuum-like coupling mechanisms that are based on a decomposition of the domain into an atomistic and a continuum region. In the continuum region we use an approximation based on the Cauchy{ Born rule. In the atomistic subdomain a version of the atomistic model with Dirichlet boundary conditions is applied. Special attention has to be paid to the dependence of the atomistic subproblem on the boundary and the boundary conditions. Applying concepts from nonlinear analysis we show existence and convergence of solutions to the Quasicontinuum approximation.

The theory and the formulation of the special wave finite elements are discussed, and the special integration schemes for the elements are developed. Then the special wave infinite elements, a new concept of the mapped wave infinite elements with multiple wave directions, are developed. Computational models using these elements coupled together are tested by the applications of wave problems. In the special wave finite elements, the potential at each node is expanded in a discrete series of approximating plane waves propagating in different directions. Because of this a single element can contain many wavelengths, unlike the standard finite elements. This is a great advantage in the reduction of the degree of freedom of the problem, however the computational cost of the numerical integration over an element becomes high due to the oscillatory shape functions. Therefore the special semi-analytical integration schemes for the special wave finite elements are developed. The schemes are independent of wavenumber and efficient for short waves problems. In many cases of wave problems, it is practical to consider the domain as being infinite. However the finite element method can not deal with infinite domains. Infinite elements are an extension of the concept of finite elements in which the element has an infinite extent in one or more directions to address this limitation. In the special wave infinite element developed in this study multiple waves propagating in different directions are considered, in contrast to conventional infinite elements in which only a single wave propagating in the radial direction is considered. The shape functions of the special wave infinite elements contain trigonometric functions to describe multiple waves, and the amplitude decay factor to satisfy the radiation condition. The special wave infinite elements become a straightforward extension to the special wave finite elements for wave problems in an unbounded domain.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2014. / In this thesis we will establish effective numerical schemes appropriate for the solution of a non-linear partial differential equation modelling heat transfer in one dimensional longitudinal fins. We will consider the problem as it stands without any physical simplification. The main methodology is based on balancing the non-linear source term as well as the application of numerical relaxation techniques. In either approach we will incorporate the no-flux condition for singular fins. By doing so, we obtain appropriate numerical schemes which improve results found in literature. To generalize, we will provide a relaxed numerical scheme that is applicable for both integer and fractional order non-linear heat transfer equations for one dimensional longitudinal fins.

by Chung Shun Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 158-170). / Abstracts in English and Chinese. / List of Figures --- p.x / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background and Historical Development --- p.1 / Chapter 1.2 --- Practical Problems --- p.3 / Chapter 1.2.1 --- Statistics --- p.3 / Chapter 1.2.2 --- Aerodynamics --- p.4 / Chapter 1.2.3 --- Factory Allocation Problem --- p.5 / Chapter 1.2.4 --- Parameter Problem --- p.5 / Chapter 1.2.5 --- Chemical Engineering --- p.5 / Chapter 1.2.6 --- Operational Research --- p.6 / Chapter 1.2.7 --- Economics --- p.6 / Chapter 1.3 --- Mathematical Models for Optimization Problems --- p.6 / Chapter 1.4 --- Unconstrained Optimization Techniques --- p.8 / Chapter 1.4.1 --- Direct Method - Differential Calculus --- p.8 / Chapter 1.4.2 --- Iterative Methods --- p.10 / Chapter 1.5 --- Main Objectives of the Thesis --- p.11 / Chapter 2 --- Basic Concepts in Optimizations of Smooth Func- tions --- p.14 / Chapter 2.1 --- Notation --- p.14 / Chapter 2.2 --- Different Types of Minimizer --- p.16 / Chapter 2.3 --- Necessary and Sufficient Conditions for Optimality --- p.18 / Chapter 2.4 --- Quadratic Functions --- p.22 / Chapter 2.5 --- Convex Functions --- p.24 / Chapter 2.6 --- "Existence, Uniqueness and Stability of a Minimum" --- p.29 / Chapter 2.6.1 --- Existence of a Minimum --- p.29 / Chapter 2.6.2 --- Uniqueness of a Minimum --- p.30 / Chapter 2.6.3 --- Stability of a Minimum --- p.31 / Chapter 2.7 --- Types of Convergence --- p.34 / Chapter 2.8 --- Minimization of Functionals --- p.35 / Chapter 3 --- Steepest Descent Method --- p.37 / Chapter 3.1 --- Background --- p.37 / Chapter 3.2 --- Line Search Method and the Armijo Rule --- p.39 / Chapter 3.3 --- Steplength Control with Polynomial Models --- p.43 / Chapter 3.3.1 --- Quadratic Polynomial Model --- p.43 / Chapter 3.3.2 --- Safeguarding --- p.45 / Chapter 3.3.3 --- Cubic Polynomial Model --- p.46 / Chapter 3.3.4 --- General Line Search Strategy --- p.49 / Chapter 3.3.5 --- Algorithm of Steepest Descent Method --- p.51 / Chapter 3.4 --- Advantages of the Armijo Rule --- p.54 / Chapter 3.5 --- Convergence Analysis --- p.56 / Chapter 4 --- Iterative Methods Using Second Derivatives --- p.63 / Chapter 4.1 --- Background --- p.63 / Chapter 4.2 --- Newton's Method --- p.64 / Chapter 4.2.1 --- Basic Concepts --- p.64 / Chapter 4.2.2 --- Convergence Analysis of Newton's Method --- p.65 / Chapter 4.2.3 --- Newton's Method with Steplength --- p.69 / Chapter 4.2.4 --- Convergence Analysis of Newton's Method with Step-length --- p.70 / Chapter 4.3 --- Greenstadt's Method --- p.72 / Chapter 4.4 --- Marquardt-Levenberg Method --- p.74 / Chapter 4.5 --- Fiacco and McComick Method --- p.76 / Chapter 4.6 --- Matthews and Davies Method --- p.79 / Chapter 4.7 --- Numerically Stable Modified Newton's Method --- p.80 / Chapter 4.8 --- The Role of the Second Derivative Methods --- p.89 / Chapter 5 --- Multi-step Methods --- p.92 / Chapter 5.1 --- Background --- p.93 / Chapter 5.2 --- Heavy Ball Method --- p.94 / Chapter 5.3 --- Conjugate Gradient Method --- p.99 / Chapter 5.3.1 --- Some Types of Conjugate Gradient Method --- p.99 / Chapter 5.3.2 --- Convergence Analysis of Conjugate Gradient Method --- p.108 / Chapter 5.4 --- Methods of Variable Metric and Methods of Conju- gate Directions --- p.111 / Chapter 5.5 --- Other Approaches for Constructing the First-order Methods --- p.116 / Chapter 6 --- Quasi-Newton Methods --- p.121 / Chapter 6.1 --- Disadvantages of Newton's Method --- p.122 / Chapter 6.2 --- General Idea of Quasi-Newton Method --- p.124 / Chapter 6.2.1 --- Quasi-Newton Methods --- p.124 / Chapter 6.2.2 --- Convergence of Quasi-Newton Methods --- p.129 / Chapter 6.3 --- Properties of Quasi-Newton Methods --- p.131 / Chapter 6.4 --- Some Particular Algorithms for Quasi-Newton Methods --- p.137 / Chapter 6.4.1 --- Single-Rank Algorithms --- p.137 / Chapter 6.4.2 --- Double-Rank Algorithms --- p.144 / Chapter 6.4.3 --- Other Applications --- p.149 / Chapter 6.5 --- Conclusion --- p.152 / Chapter 7 --- Choice of Methods in Optimization Problems --- p.154 / Chapter 7.1 --- Choice of Methods --- p.154 / Chapter 7.2 --- Conclusion --- p.157 / Bibliography --- p.158

by Cheng Ting. / "August 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 103-107). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.