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Cluster analysis is commonly described as the classification of unlabeled observations into groups such that they are more similar to one another than to observations in other groups. Model-based clustering assumes that the data arise from a statistical (mixture) model and typically a group of many models are fit to the data, from which the `best' model is selected by a model selection criterion (often the BIC in mixture model applications). This chosen model is then the only model that is used for making inferences on the data. Although this is common practice, proceeding in this way ignores a large component of model selection uncertainty, especially for situations where the difference between the model selection criterion for two competing models is relatively insignificant. For this reason, recent interest has been placed on selecting a subset of models that are close to the selected best model and using a weighted averaging approach to incorporate information from multiple models in this set. Model averaging is not a novel approach, yet its presence in a clustering framework is minimal. Here, we use Occam's window to select a subset of models eligible for two types of averaging techniques: averaging a posteriori probabilities, and direct averaging of model parameters. The efficacy of these model-based averaging approaches is demonstrated for a family of generalized hyperbolic mixture models using real and simulated data. / Thesis / Master of Science (MSc)

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In this dissertation we present an analysis for the discontinuous Galerkin discretization error of multi-dimensional first-order linear symmetric and symmetrizable hyperbolic systems of conservation laws. We explicitly write the leading term of the local DG error, which is spanned by Legendre polynomials of degree p and p+1 when p-th degree polynomial spaces are used for the solution. For special hyperbolic systems, where the coefficient matrices are nonsingular, we show that the leading term of the error is spanned by (p+1)-th degree Radau polynomials. We apply these asymptotic results to observe that projections of the error are pointwise O(h^p+2)-superconvergent in some cases and establish superconvergence results for some integrals of the error. We develop an efficient implicit residual-based a posteriori error estimation scheme by solving local finite element problems to compute estimates of the leading term of the discretization error. For smooth solutions we obtain error estimates that converge to the true error under mesh refinement. We first show these results for linear symmetric systems that satisfy certain assumptions, then for general linear symmetric systems. We further generalize these results to linear symmetrizable systems by considering an equivalent symmetric formulation, which requires us to make small modifications in the error estimation procedure. We also investigate the behavior of the discretization error when the Lax-Friedrichs numerical flux is used, and we construct asymptotically exact a posteriori error estimates. While no superconvergence results can be obtained for this flux, the error estimation results can be recovered in most cases. These error estimates are used to drive h- and p-adaptive algorithms and assess the numerical accuracy of the solution. We present computational results for different fluxes and several linear and nonlinear hyperbolic systems in one, two and three dimensions to validate our theory. Examples include the wave equation, Maxwell's equations, and the acoustic equation. / Ph. D.

For simple fluids, or Newtonian fluids, the study of the Navier-Stokes equations in the high Reynolds number limit brings about two fundamental research subjects, the Euler equations and the Prandtl's system. The consideration of infinite Reynolds number reduces the Navier-Stokes equations to the Euler equations, both of which are dealing with the entire flow region. Prandtl's system consists of the governing equations of the boundary layer, a thin layer formed at the wall boundary where viscosity cannot be neglected. In this dissertation, we investigate the upper convected Maxwell(UCM) model for complex fluids, or non-Newtonian fluids, in the high Weissenberg number limit. This is analogous to the Newtonian fluids in the high Reynolds number limit. We present two well-posedness results. The first result is on an initial-boundary value problem for incompressible hypoelastic materials which arise as a high Weissenberg number limit of viscoelastic fluids. We first assume the stress tensor is rank-one and develop energy estimates to show the problem is locally well-posed. Then we show the more general case can be handled in the same spirit. This problem is closely related to the incompressible ideal magneto-hydrodynamics (MHD) system. The second result addresses the formulation of a time-dependent elastic boundary layer through scaling analysis. We show the well-posedness of this boundary layer by transforming to Lagrangian coordinates. In contrast to the possible ill-posedness of Prandtl's system in Newtonian fluids, we prove that in non-Newtonian fluids the stress boundary layer problem is well-posed. / Ph. D.

A Flexible Galerkin Finite Element Method (FGM) is a hybrid class of finite element methods that combine the usual continuous Galerkin method with the now popular discontinuous Galerkin method (DGM). A detailed description of the formulation of the FGM on a hyperbolic partial differential equation, as well as the data structures used in the FGM algorithm is presented. Some hp-convergence results and computational cost are included. Additionally, an a posteriori error estimate for the DGM applied to a two-dimensional hyperbolic partial differential equation is constructed. Several examples, both linear and nonlinear, indicating the effectiveness of the error estimate are included. / Ph. D.

In this thesis, we present a simple and efficient \emph{a posteriori} error estimation procedure for a discontinuous finite element method applied to scalar first-order hyperbolic problems on structured and unstructured tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and find basis functions spanning the error for several finite element spaces. We describe an implicit error estimation procedure for the leading term of the discretization error by solving a local problem on each tetrahedron. Numerical computations show that the implicit \emph{a posteriori} error estimation procedure yields accurate estimates for linear and nonlinear problems with smooth solutions. Furthermore, we show the performance of our error estimates on problems with discontinuous solutions. We investigate pointwise superconvergence properties of the discontinuous Galerkin (DG) method using enriched polynomial spaces. We study the effect of finite element spaces on the superconvergence properties of DG solutions on each class and type of tetrahedral elements. We show that, using enriched polynomial spaces, the discretization error on tetrahedral elements having one inflow face, is O(h^{p+2}) superconvergent on the three edges of the inflow face, while on elements with one inflow and one outflow faces the DG solution is O(h^{p+2}) superconvergent on the outflow face in addition to the three edges of the inflow face. Furthermore, we show that, on tetrahedral elements with two inflow faces, the DG solution is O(h^{p+2}) superconvergent on the edge shared by two of the inflow faces. On elements with two inflow and one outflow faces and on elements with three inflow faces, the DG solution is O(h^{p+2}) superconvergent on two edges of the inflow faces. We also show that using enriched polynomial spaces lead to a simpler{a posterior error estimation procedure. Finally, we extend our error analysis for the discontinuous Galerkin method applied to linear three-dimensional hyperbolic systems of conservation laws with smooth solutions. We perform a local error analysis by expanding the local error as a series and showing that its leading term is O( h^{p+1}). We further simplify the leading term and express it in terms of an optimal set of polynomials which can be used to estimate the error. / Ph. D.