Thesis (Ph. D.)--University of Chicago, Dept. of Statistics, June 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2007. / In this thesis, we study the numerical approximation of indefinite integrals with algebraic or logarithmic end-point singularities. We show the derivation of the two quadrature formulas proposed by Haber based on the sinc method, as well as, on the basis of error analysis, by means of variable transformations (Single and Double Exponential), the derivation of two other formulas: Stenger’s Single Exponential (SE) formula and Tanaka et al.’s Double Exponential (DE) sinc method. Important tools for our work are residue calculus, functional analysis and Fourier analysis from which we state some standard results, and give the proof of some of them. Next, we introduce the Paley-Wiener class of functions, define the sinc function, cardinal function, when a function decays single and double exponentially, and prove some of their interesting properties. Since the four formulas involve a conformal transformation, we show how to transform from the interval (−¥,¥) to (−1, 1). In addition, we show how to implement the four formulas on two computational examples which are our test problems, and illustrate our numerical results by means of tables and figures. Furthermore, from an application of the four quadrature formulas on two test problems, a plot of the maximum absolute error against the number of function evaluations, reveals a faster convergence to the exact solution by Tanaka et al.’s DE sinc method than by the other three formulas. Next, we convert the indefinite integrals (our test problems) into ordinary differential equations (ODE) with suitable initial values, in the hope that ODE solvers such as Matlabr ode45 or Mathematicar NDSolve will be able to solve the resulting IVPs. But they all failed because of singularities in the initial value. In summary, of the four quadrature formulas, Tanaka et al.’s DE sinc method gives more accurate results than the others and it will be noted that all the formulas are applicable to both singular and non-singular integrals.

D.Phil. (Applied mathematics) / Please refer to full text to view abstract

No / In this paper we present a constrained inverse kinematics algorithm for real-time motion capture in virtual environments, that has its origins in the simulation of multi-body systems. We apply this algorithm to an articulated human skeletal model using an electromagnetic motion tracking system with a small number of sensors to create avatar postures. The method offers efficient inverse kinematics computation and it is also generalised for the configurations of an articulated skeletal model. We investigate the possibility of capturing fast gestures by analysing the convergence patterns of the algorithm with the motion tracking sampling frequency for a range of actions.

Optical and spectroscopic molecular properties are key to characterizing the behavior of molecules interacting with an applied electromagnetic field of light. Response theory has been used for a long time to calculate such properties in the frequency domain. Real-time (RT) methods solve for the frequency-dependent properties in the time domain by explicitly propagating the time-dependent wave function. Various quantum chemical methods can be incorporated with the RT formalism, including Hartree-Fock, density functional theory, configurational interaction, coupled cluster, etc. Among these, coupled cluster (CC) methods provide high accuracy for systems with strong electron correlation, making RT-CC implementations intriguing. All applications of CC methods face a substantial challenge due to their high-order polynomial scaling. For RT-CC methods, two aspects may be explored to improve the efficiency, the numerical techniques regarding the RT propagation and the reduced-scaling methods regarding CC itself. In this work, we start with the exploration of the hardware used for the calculations and the numerical integration methods for propagating the wave function parameters. Firstly, a GPU-enabled Python implementation has been developed by conducting the tensor contractions on GPUs utilizing PyTorch, a machine learning package, that has similar syntax as NumPy for tensor operations. A speedup of a factor of 14 is obtained for the RT-CCSD/cc-pVDZ absorption spectrum calculation of the water tetramer. Furthermore, to optimize the performance on GPUs, single-precision arithmetic is added to the implementation to achieve an additional speedup of a factor of two. Lastly, a group of integrators for solving differential equations are introduced to the RT framework, including regular explicit integrators, adaptive integrators, and a mixed-step-size approach customized for strong-field simulations. The optimal choice of the integrator depends on the requiring accuracy, stability and efficiency. In addition to being highly accurate, CC methods are also systematically improvable and provide a hierarchy of accuracy. Based upon the RT-CCSD implementation, the coupled cluster singles, doubles and approximate triples (CC3) method, favorable for calculating frequency-dependent properties, is tailored to the RT framework for high excitation and approximate orbital relaxation. The calculation is tested on both CPUs and GPUs, with a significant speedup gained from GPUs for the water cluster test cases. To further expand the range of applications of our RT-CC implementation, dynamic polarizabilities, first hyperpolarizabilities, and the G' tensor are calculated from induced electric and magnetic dipole moments using finite-difference methods. A discussion has also been conducted to compare RT-CC3 with RT-CCSD, and time-dependent nonorthogonal orbital-optimized coupled cluster doubles (TDNOCCD) method. Additionally, electron dynamics, including the Rabi oscillation and exited state to excited state transitions, have also been explored utilizing the well-developed RT-CC framework. / Doctor of Philosophy / Theoretical studies aim to match experiments, but more importantly, provide insights to interpret and predict experimental data. Calculating optical properties related to light-matter interactions is one of the most crucial tasks for characterizing molecular properties. In experiments, electromagnetic radiation in the form of light is applied to the system. The absorption or emission of light can be measured to identify, for example, the electronic structure of the molecule. In theoretical simulations, this applied radiation is represented by a perturbation operator that is added to the Hamiltonian in the Schrödinger equation. Quantum chemists are dedicated to developing methods that provide a better description of the spectroscopy. In the current work, the frequency, shape and the intensity of the radiation can all be finely-tuned, similar to experimental setups. The framework for extracting optical properties from time-dependent trajectories of induced dipole moments is established for accurate and efficient simulations. To improve efficiency and make the method feasible for real-world applications, a strong understanding of light-matter interactions on a quantum level and proper utilization of computational resources are both necessary. Improvements achieved and presented in this dissertation demonstrate a powerful tool for a better understanding of the nature of the interaction between the system and the electromagnetic radiation.

No description available.

A new method of numerical integration is presented here, the variable-step Stormer-Cowell method. The method uses error control to regulate the step size, so larger step sizes can be taken when possible, and is double-integration, so only one evaluation per step is necessary when integrating second-order differential equations. The method is not variable-order, because variable-order algorithms require a second evaluation. The variable-step Stormer-Cowell method is designed for space surveillance applications,which require numerical integration methods to track orbiting objects accurately. Because of the large number of objects being processed, methods that can integrate the equations of motion as fast as possible while maintaining accuracy requirements are desired. The force model used for earth-orbiting objects is quite complex and computationally expensive, so methods that minimize the force model evaluations are needed. The new method is compared to the fixed-step Gauss-Jackson method, as well as a method of analytic step regulation (s-integration), and the variable-step variable-order Shampine-Gordon integrator. Speed and accuracy tests of these methods indicate that the new method is comparable in speed and accuracy to s-integration in most cases, though the variable-step Stormer-Cowell method has an advantage over s-integration when drag is a significant factor. The new method is faster than the Shampine-Gordon integrator, because the Shampine-Gordon integrator uses two evaluations per step, and is biased toward keeping the step size constant. Tests indicate that both the new variable-step Stormer-Cowell method and s-integration have an advantage over the fixed-step Gauss-Jackson method for orbits with eccentricities greater than 0.15. / Ph. D.

There are many problems that lead to analysis of dynamical systems with phase variables of two types, slow and fast ones. Such systems are called slow-fast systems. The dynamics of such systems is usually described by means of different versions of perturbation theory. Many questions about accuracy of this description are still open. The difficulties are related to presence of resonances. The goal of the proposed thesis is to establish some estimates of the accuracy of the perturbation theory for slow-fast systems in the presence of resonances. We consider slow-fast Hamiltonian systems and study an accuracy of one of the methods of perturbation theory: the averaging method. In this thesis, we start with the case of slow-fast Hamiltonian systems with two degrees of freedom. One degree of freedom corresponds to fast variables, and the other degree of freedom corresponds to slow variables. Action variable of fast sub-system is an adiabatic invariant of the problem. Let this adiabatic invariant have limiting values along trajectories as time tends to plus and minus infinity. The difference of these two limits for a trajectory is known to be exponentially small in analytic systems. We obtain an exponent in this estimate. To this end, by means of iso-energetic reduction and canonical transformations in complexified phase space, we reduce the problem to the case of one and a half degrees of freedom, where the exponent is known. We then consider a quasi-linear Hamiltonian system with one and a half degrees of freedom. The Hamiltonian of this system differs by a small, ~ε, perturbing term from the Hamiltonian of a linear oscillatory system. We consider passage through a resonance: the frequency of the latter system slowly changes with time and passes through 0. The speed of this passage is of order of ε. We provide asymptotic formulas that describe effects of passage through a resonance with an improved accuracy O(ε3/2). A numerical verification is also provided. The problem under consideration is a model problem that describes passage through an isolated resonance in multi-frequency quasi-linear Hamiltonian systems. We also discuss a resonant phenomenon of scattering on resonances associated with discretisation arising in a numerical solving of systems with one rotating phase. Numerical integration of ODEs by standard numerical methods reduces continuous time problems to discrete time problems. For arbitrarily small time step of a numerical method, discrete time problems have intrinsic properties that are absent in continuous time problems. As a result, numerical solution of an ODE may demonstrate dynamical phenomena that are absent in the original ODE. We show that numerical integration of systems with one fast rotating phase leads to a situation of such kind: numerical solution demonstrates phenomenon of scattering on resonances, that is absent in the original system.

Um dos principais problemas que o Método de Elementos de Contorno (MEC) apresenta encontra-se na avaliação de integrais singulares e quase-singulares que envolvem as soluções fundamentais de Kelvin em deslocamento e força. O processo de integração numérica em MEC tem sido o objetivo de inúmeras pesquisas, pois dele depende a qualidade das respostas quando se deseja obter uma excelente precisão numérica em uma análise. Este trabalho apresenta uma nova proposta de integração numérica para análise tridimensional com MEC. Esta técnica possui três características importantes. A primeira é a determinação da parcela efetiva de singularidade que ocorre na função raio, distância entre o ponto fonte e o elemento de contorno bidimensional. A correta obtenção desta parcela permite representar sem aproximações o comportamento da singularidade da função raio, que é a verdadeira fonte de singularidade e quase-singularidade nas soluções fundamentais. A segunda característica da técnica proposta é que ela baseia-se em um Método Semi-Analítico de avaliação de integrais, onde, para cada parcela efetiva de singularidade, utiliza-se uma quadratura numérica cujos pesos específicos são calculados analiticamente. A terceira característica da técnica proposta é apresentar um tratamento unificado para todos os tipos de integrais singulares, quasesingulares e regulares. Esta técnica foi implementada na plataforma computacional desenvolvida pelo grupo GoBEM, utilizando o conceito de Programação Orientada a Objetos através da Linguagem de programação Java. Com a implementação da nova técnica de integração na plataforma computacional torna-se possível realizar o desenvolvimento de vários tipos de pesquisa sobre análise tridimensional com o MEC como, por exemplo: visualização de isosuperfícies em análise tridimensional sem discretização do domínio, automatização do cálculo elasto-plástico, modelagem de problemas geotécnicos de forma precisa, etc. Para a validação da técnica proposta três procedimentos foram considerados: análise da eficiência da parcela efetiva de singularidade, testes de convergência da integração numérica específica e exemplos numéricos utilizando o MEC em problemas de engenharia. / One of the main problems with the Boundary Elements Method (BEM) is the evaluation of singular and quasi-singular integrals due to the Kelvin\'s Fundamental Solutions in displacement and traction. Today there is an increasing body of research work that focuses on numerical evaluation of BEM integrals, for this is a crucial issue in order to achieve highly accurate results. This work presents an innovative numerical integration procedure for threedimensional analysis with BEMs. The proposed technique encompasses three important features. First, it corresponds to an accurate representation of the effective term of singularity in the radius function, which measures the distance between the source point and a twodimensional boundary element. The correct evaluation of this term leads without approximation to the actual singularity behavior of the radius function, which is the true source of singularity and quasi-singularity in the fundamental solutions. Second, the proposed technique is based on a semi-analytical procedure, i.e, for each singularity effective term it uses a quadrature scheme with specific weights analytically evaluated. Last, the proposed technique represents a unified procedure to singular, quasi-singular and regular integrals. This technique was implemented in the computational platform developed by the group GoBEM, using Object Oriented Programming and the Java programming language. The implementation of the proposed technique into this computational plataform opens new possibilities for future researches on three-dimensional BEM, e.g.: visualization of isosurfaces in three-dimensional analysis without any domain discretization, automatic elasto-plastic analysis, accurate modeling of geomechanical problems, etc. Validation of the proposed technique was carried out using three procedures: efficiency analysis of the singularity effective term, convergence tests for the specific numerical integration and numerical examples using BEM in engineering problems.

High order multistep methods, run at constant stepsize, are very effective for integrating the Newtonian solar system for extended periods of time. I have studied the stability and error growth of these methods when applied to harmonic oscillators and two-body systems like the Sun-Jupiter pair. I have also tried to design better multistep integrators than the traditional Stormer and Cowell methods, and I have found a few interesting ones.