Browsing by Subject "B-splines"
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Item Bayesian Semiparametric Density Deconvolution and Regression in the Presence of Measurement Errors(2014-06-24) Sarkar, AbhraAlthough the literature on measurement error problems is quite extensive, solutions to even the most fundamental measurement error problems like density deconvolution and regression with errors-in-covariates are available only under numerous simplifying and unrealistic assumptions. This dissertation demonstrates that Bayesian methods, by accommodating measurement errors through natural hierarchies, can provide a very powerful framework for solving these important measurement errors problems under more realistic scenarios. However, the very presence of measurement errors often renders techniques that are successful in measurement error free scenarios inefficient, numerically unstable, computationally challenging or intractable. Additionally, measurement error problems often have unique features that compound modeling and computational challenges. In this dissertation, we develop novel Bayesian semiparametric approaches that cater to these unique challenges of measurement error problems and allow us to break free from many restrictive parametric assumptions of previously existing approaches. In this dissertation, we first consider the problem of univariate density deconvolution when replicated proxies are available for each unknown value of the variable of interest. Existing deconvolution methods often make restrictive and unrealistic assumptions about the density of interest and the distribution of measurement errors, e.g., normality and homoscedasticity and thus independence from the variable of interest. We relax these assumptions and develop robust and efficient deconvolution approaches based on Dirichlet process mixture models and mixtures of B-splines in the presence of conditionally heteroscedastic measurement errors. We then extend the methodology to nonlinear univariate regression with errors-in-covariates problems when the densities of the covariate, the regression errors and the measurement errors are all unknown, and the regression and the measurement errors are conditionally heteroscedastic. The final section of this dissertation is devoted to the development of flexible multivariate density deconvolution approaches. The methods available in the existing sparse literature all assume the measurement error density to be fully specified. In contrast, we develop multivariate deconvolution approaches for scenarios when the measurement error density is unknown but replicated proxies are available for each subject. We consider scenarios when the measurement errors are distributed independently from the vector valued variable of interest as well as scenarios when they are conditionally heteroscedastic. To meet the significantly harder modeling and computational challenges of the multivariate problem, we exploit properties of finite mixture models, multivariate normal kernels, latent factor models and exchangeable priors in many novel ways. We provide theoretical results showing the flexibility of the proposed models. In simulation experiments, the proposed semiparametric methods vastly outperform previously existing approaches. Our methods also significantly outperform theoretically more flexible possible nonparametric alternatives even when the true data generating process closely conformed to these alternatives. The methods automatically encompass a variety of simplified parametric scenarios as special cases and often outperform their competitors even in those special scenarios for which the competitors were specifically designed. We illustrate practical usefulness of the proposed methodology by successfully applying the methods to problems in nutritional epidemiology. The methods can be readily adapted and applied to similar problems from other areas of applied research. The methods also provide the foundation for many interesting extensions and analyses.Item Divergence-free B-spline discretizations for viscous incompressible flows(2011-12) Evans, John Andrews; Hughes, Thomas J. R.; Babuska, Ivo; Demkowicz, Leszek; Ghattas, Omar; Moser, Robert D.; Bazilevs, YuriThe 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.Item Implementation of B-splines in a Conventional Finite Element Framework(2010-01-16) Owens, Brian C.The use of B-spline interpolation functions in the finite element method (FEM) is not a new subject. B-splines have been utilized in finite elements for many reasons. One reason is the higher continuity of derivatives and smoothness of B-splines. Another reason is the possibility of reducing the required number of degrees of freedom compared to a conventional finite element analysis. Furthermore, if B-splines are utilized to represent the geometry of a finite element model, interfacing a finite element analysis program with existing computer aided design programs (which make extensive use of B-splines) is possible. While B-splines have been used in finite element analysis due to the aforementioned goals, it is difficult to find resources that describe the process of implementing B-splines into an existing finite element framework. Therefore, it is necessary to document this methodology. This implementation should conform to the structure of conventional finite elements and only require exceptions in methodology where absolutely necessary. One goal is to implement B-spline interpolation functions in a finite element framework such that it appears very similar to conventional finite elements and is easily understandable by those with a finite element background. The use of B-spline functions in finite element analysis has been studied for advantages and disadvantages. Two-dimensional B-spline and standard FEM have been compared. This comparison has addressed the accuracy as well as the computational efficiency of B-spline FEM. Results show that for a given number of degrees of freedom, B-spline FEM can produce solutions with lower error than standard FEM. Furthermore, for a given solution time and total analysis time B-spline FEM will typically produce solutions with lower error than standard FEM. However, due to a more coupled system of equations and larger elemental stiffness matrix, B-spline FEM will take longer per degree of freedom for solution and assembly times than standard FEM. Three-dimensional B-spline FEM has also been validated by the comparison of a three-dimensional model with plane-strain boundary conditions to an equivalent two-dimensional model using plane strain conditions.