In this thesis, we investigate Riesz bases of wavelets and their applications to numerical solutions of elliptic equations. Compared with the finite difference and finite element methods, the wavelet method for solving elliptic equations is relatively young but powerful. In the wavelet Galerkin method, the efficiency of the numerical schemes is directly determined by the properties of the wavelet bases. Hence, the construction of Riesz bases of wavelets is crucial. We propose different ways to construct wavelet bases whose stability in Sobolev spaces is then established. An advantage of our approaches is their far superior simplicity over many other known constructions. As a result, the corresponding numerical schemes are easily implemented and efficient. We apply these wavelet bases to solve some important elliptic equations in physics and show their effectiveness numerically. Multilevel algorithm based on preconditioned conjugate gradient algorithm is also developed to significantly improve the numerical performance. Numerical results and comparison with other existing methods are presented to demonstrate the advantages of the wavelet Galerkin method we propose. / Mathematics

Surface pasting is an interactive hierarchical modelling technique used to construct surfaces with varying levels of local detail. The concept is similar to that of the physical process of modelling with clay, where features are placed on to a base surface and attached by a smooth join obtained by adjusting the feature. Cylindrical surface pasting extends this modelling paradigm by allowing for two base surfaces to be joined smoothly via a blending cylinder, as in attaching a clay head to the body using a neck. Unfortunately, computer-based pasting involves approximations that can cause cracks to appear in the composite surface. In particular this occurs when the pasted feature boundary does not lie exactly over the user-specified pasting region on the base surface. Determining pasted locations for the feature boundary control points that give a close to exact join is non-trivial, especially in the case of cylinders as their control points can not be defined to lie on their closed curve boundary. I propose and compare six simple methods for positioning a feature cylinder's control points such that the join boundary discontinuities are minimized. The methods considered are all algorithmically simple alternatives having low computational costs. While the results demonstrate an order of magnitude quality improvement for some methods on a convex-only curved base, as the complexity of the base surface increases, all the methods show similar performance. Although unexpected, it turns out that a simple mapping of the control points directly onto the pasting closed curve given on the base surface offers a reasonable cylindrical boundary pasting technique.

Surface pasting is an interactive hierarchical modelling technique used to construct surfaces with varying levels of local detail. The concept is similar to that of the physical process of modelling with clay, where features are placed on to a base surface and attached by a smooth join obtained by adjusting the feature. Cylindrical surface pasting extends this modelling paradigm by allowing for two base surfaces to be joined smoothly via a blending cylinder, as in attaching a clay head to the body using a neck. Unfortunately, computer-based pasting involves approximations that can cause cracks to appear in the composite surface. In particular this occurs when the pasted feature boundary does not lie exactly over the user-specified pasting region on the base surface. Determining pasted locations for the feature boundary control points that give a close to exact join is non-trivial, especially in the case of cylinders as their control points can not be defined to lie on their closed curve boundary. I propose and compare six simple methods for positioning a feature cylinder's control points such that the join boundary discontinuities are minimized. The methods considered are all algorithmically simple alternatives having low computational costs. While the results demonstrate an order of magnitude quality improvement for some methods on a convex-only curved base, as the complexity of the base surface increases, all the methods show similar performance. Although unexpected, it turns out that a simple mapping of the control points directly onto the pasting closed curve given on the base surface offers a reasonable cylindrical boundary pasting technique.

Diss.--Mathematik--Universität Mainz, 1998. / Bibliogr. p. 170-175.

Thèse de doctorat : Informatique : INPL : 2006. / Titre provenant de l'écran-titre. Bibliogr.

Reproduction de : Thèse d'Etat : mathématiques : Grenoble 1 : 1987. / Titre provenant de l'écran-titre. Bibliogr. p. 143-157.

Thèse doctorat : Instrumentation, Système, Signal & Image : Villeurbanne, INSA : 2007. / Titre provenant de l'écran-titre. Bibliogr. p. 157-163.

The incompressible Navier-Stokes equations are among the most important partial differential systems arising from classical physics. They are utilized to model a wide range of fluids, from water moving around a naval vessel to blood flowing through the arteries of the cardiovascular system. Furthermore, the secrets of turbulence are widely believed to be locked within the Navier-Stokes equations. Despite the enormous applicability of the Navier-Stokes equations, the underlying behavior of solutions to the partial differential system remains little understood. Indeed, one of the Clay Mathematics Institute's famed Millenium Prize Problems involves the establishment of existence and smoothness results for Navier-Stokes solutions, and turbulence is considered, in the words of famous physicist Richard Feynman, to be "the last great unsolved problem of classical physics." Numerical simulation has proven to be a very useful tool in the analysis of the Navier-Stokes equations. Simulation of incompressible flows now plays a major role in the industrial design of automobiles and naval ships, and simulation has even been utilized to study the Navier-Stokes existence and smoothness problem. In spite of these successes, state-of-the-art incompressible flow solvers are not without their drawbacks. For example, standard turbulence models which rely on the existence of an energy spectrum often fail in non-trivial settings such as rotating flows. More concerning is the fact that most numerical methods do not respect the fundamental geometric properties of the Navier-Stokes equations. These methods only satisfy the incompressibility constraint in an approximate sense. While this may seem practically harmless, conservative semi-discretizations are typically guaranteed to balance energy if and only if incompressibility is satisfied pointwise. This is especially alarming as both momentum conservation and energy balance play a critical role in flow structure development. Moreover, energy balance is inherently linked to the numerical stability of a method. In this dissertation, novel B-spline discretizations for the generalized Stokes and Navier-Stokes equations are developed. The cornerstone of this development is the construction of smooth generalizations of Raviart-Thomas-Nedelec elements based on the new theory of isogeometric discrete differential forms. The discretizations are (at least) patch-wise continuous and hence can be directly utilized in the Galerkin solution of viscous flows for single-patch configurations. When applied to incompressible flows, the discretizations produce pointwise divergence-free velocity fields. This results in methods which properly balance both momentum and energy at the semi-discrete level. In the presence of multi-patch geometries or no-slip walls, the discontinuous Galerkin framework can be invoked to enforce tangential continuity without upsetting the conservation and stability properties of the method across patch boundaries. This also allows our method to default to a compatible discretization of Darcy or Euler flow in the limit of vanishing viscosity. These attributes in conjunction with the local stability properties and resolution power of B-splines make these discretizations an attractive candidate for reliable numerical simulation of viscous incompressible flows. / text

The main objectives of this master thesis are to explore the effectiveness of nonlinear mixed effects model for longitudinal data. Mixed effect models allow to investigate the nature of relationship between the time-varying covariates and the response while also capturing the variations of subjects. I investigate the robustness of the longitudinal models by building up the complexity of the models starting from multiple linear models and ending up with additive nonlinear mixed models. I use a dataset where firms’ leverage are explained by four explanatory variables in addition to a grouping factor that is the firm factor. The models are compared using comparison statistics such as AIC, BIC and by a visual inspection of residuals. Likelihood ratio test has been used in some nested models only. The models are estimated by maximum likelihood and restricted maximum likelihood estimation. The most efficient model is the nonlinear mixed effects model which has lowest AIC and BIC. The multiple linear regression model failed to explain the relation and produced unrealistic statistics

Functional data analysis (FDA) is a demonstrably effective, practical, and powerful method of data analysis, yet it remains virtually unheard of outside of academic circles and has almost no exposure to industry. FDA adds to the milieu of statistical methods by treating functions of one or more independent variables as data objects, analogous to the way in which discrete points are the data objects we are familiar with in conventional statistics. The first step in functional analysis is to “functionalize” the data, or convert discrete points into a system represented most times by continuous functions. Choosing the type of functions to use is data-dependent and often straightforward – for example, Fourier series lend themselves well to periodic systems, while splines offer great flexibility in approximating more irregular trends, such as chemical spectra. This work explores the question of how B-splines can be rapidly and reliably used to denoised infrared chemical spectra, a difficult problem not only because of the many parameters involved in generating a spline fit, but also due to the disparate nature of spectra in terms of shape and noise intensity. Automated selection of spline parameters is required to support high-throughput analysis, and the heteroscedastic nature of such spectra presents challenges for existing techniques. The heuristic knot placement algorithm of Li et al. (2005) for 1D object contours is extended to spectral fitting by optimizing the denoising step for a range of spectral types and signal/noise ratios, using the following criteria: robustness to types of spectra and noise conditions, parsimony of knots, low computational demand, and ease of implementation in high-throughput settings. Pareto-optimal filter configurations are determined using simulated data from factorial experimental designs. The improved heuristic algorithm uses wavelet transforms and provides improved performance in robustness, parsimony of knots and the quality of functional regression models used to correlate real spectral data with chemical composition. In practical applications, functional principal component regression models yielded similar or significantly improved results when compared with their discrete partial least squares counterparts. / Thesis (Master, Chemical Engineering) -- Queen's University, 2012-10-01 20:18:31.119