Thesis (Ph.D.)--University of Western Australia, 2003.

The recent development of the impulse/summation approach for efficient B-spline computation in the discrete domain should increase the use of B-splines in many applications. Because we show here how the impulse/summation approach can also be used for constructing polynomials, the approach with a search table approach for the inverse square root operation allows an efficient shading algorithm for rendering an image in a computer graphics system. The approach reduces the number of multiplies and makes it possible for the entire rendering process to be implemented using an integer processor. In many applications, Chebyshev approximation with polynomials and splines is useful in representing a stream of data or a function. Because the impulse/summation approach is developed for discrete systems, some aspects of traditional continuous approximation are not applicable. For example, the lack of the continuity concept in the discrete domain affects the definition of the local extrema of a function. Thus, the method of finding the extrema must be changed. Both forward differences and backward differences must be checked to find extrema instead of using the first derivative in the continuous domain approximation. Polynomial Chebyshev approximation in the discrete domain, just as in the continuous domain, forms a Chebyshev system. Therefore, the Chebyshev approximation process always produces a unique best approximation. Because of the non-linearity of free knot polynomial spline systems, there may be more than one best solution and the convexity of the solution space cannot be guaranteed. Thus, a Remez Exchange Algorithm may not produce an optimal approximation. However, we show that the discrete polynomial splines approximate a function using a smaller number of parameters (for a similar minimax error) than the discrete polynomials do. Also, the discrete polynomial spline requires much less computation and hardware than the discrete polynomial for curve generation when we use the impulse/summation approach. This is demonstrated using two approximated FIR filter implementations. / Ph. D.

In electrical engineering, the designer is often presented with the problem of synthesizing a circuit for which the mathematical specifications are unsuitable for physical realization. Hence, the engineer must approximate as well as possible the prescribed network function by another function which is realizable. This paper describes a new approximation method for solving the problem of realizing passive network transfer functions, where the realization is carried out through the use of passive, reciprocal,lumped, linear, and time-invariant elements.

The complexity of numerical ocean circulation models requires careful checking with a variety of test problems. The purpose of this paper is to develop a test problem involving Rossby and gravity waves in a two-layer fluid in a channel. The goal is to compute very accurate solutions to this test problem. These solutions can then be used as a part of the checking process for numerical ocean circulation models. Here, Chebychev pseudo-spectral methods are used to solve the governing equations with a high degree of accuracy. Chebychev pseudo-spectral methods can be described in the following way: For a given function, find the polynomial interpolant at a particular non-uniform grid. The derivative of this polynomial serves as an approximation to the derivative of the original function. This approximation can then be inserted to differential equations to solve for approximate solutions. Here, the governing equations reduce to an eigenvalue problem with eigenvectors and eigenvalues corresponding to the spatial dependences of modal solutions and the frequencies of those solutions, respectively. The results of this method are checked in two ways. First, the solutions using the Chebychev pseudo-spectral methods are analyzed and are found to exhibit the properties known to belong to physical Rossby and gravity waves. Second, in the special case where the two-layer model degenerates to a one-layer system, some analytic solutions are known. When the numerical solutions are compared to the analytic solutions, they show an exponential rate of convergence. The conclusion is that the solutions computed using the Chebychev pseudo-spectral methods are highly accurate and could be used as a test problem to partially check numerical ocean circulation models. / Graduation date: 2012

The study of nano-electronic devices is fundamental to the advancement of the semiconductor industry. As electronic devices become increasingly smaller, they will eventually move into a regime where the classical nature of the electrons no longer applies. As the quantum nature of the electrons becomes increasingly important, classical or semiclassical theories and methods will no longer serve their purpose. For example, the simplest non-classical effect that will occur is the tunnelling of electrons through the potential barriers that form wires and transistors. This results in an increase in noise and a reduction in the device?s ability to function correctly. Other quantum effects include coulomb blockade, resonant tunnelling, interference and diffraction, coulomb drag, resonant blockade and the list goes on. This thesis develops both a theoretical model and computational method to allow nanoelectronic devices to be studied in detail. Through the use of computer code and an appropriate model description, potential problems and new novel devices may be identified and studied. The model is as accurate to the physical realisation of the devices as possible to allow direct comparison with experimental outcomes. Using simple geometric shapes of varying potential heights, simple devices are readily accessible: quantum wires; quantum transistors; resonant cavities; and coupled quantum wires. Such devices will form the building blocks of future complex devices and thus need to be fully understood. Results obtained studying the connection of a quantum wire with its surroundings demonstrate non-intuitive behaviour and the importance of device geometry to electrical characteristics. The application of magnetic fields to various nano-devices produced a range of interesting phenomenon with promising novel applications. The magnetic field can be used to alter the phase of the electron, modifying the interaction between the electronic potential and the transport electrons. This thesis studies in detail the Aharonov-Bohm oscillation and impurity characterisation in quantum wires. By studying various devices considerable information can be added to the knowledge base of nano-electronic devices and provide a basis to further research. The computational algorithms developed in this thesis are highly accurate, numerically efficient and unconditionally stable, which can also be used to study many other physical phenomena in the quantum world. As an example, the computational algorithms were applied to positron-hydrogen scattering with the results indicating positronium formation.

This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning. Given a random couple (X,Y) with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. Prediction rule is constructed from a sequence of observations sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for the performance of active learning methods. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem. The second part of this thesis is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P. Given a collection of functions H, the goal of dictionary learning is to construct a prediction rule for Y given by a linear combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that the proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance.