BURA Collection:http://bura.brunel.ac.uk/handle/2438/2352018-05-23T02:27:29Z2018-05-23T02:27:29ZEstimating errors in quantities of interest in the case of hyperelastic membrane deformationArgyridou, Elenihttp://bura.brunel.ac.uk/handle/2438/162052018-05-22T02:01:58Z2018-01-01T00:00:00ZTitle: Estimating errors in quantities of interest in the case of hyperelastic membrane deformation
Authors: Argyridou, Eleni
Abstract: There are many mathematical and engineering methods, problems and experiments which make use of the finite element method. For any given use of the finite element method we get an approximate solution and we usually wish to have some indication of the accuracy in the approximation. In the case when the calculation is done to estimate a quantity of interest the indication of the accuracy is concerned with estimating the difference between the unknown exact value and the finite element approximation. With a means of estimating the error, this can sometimes be used to determine how to improve the accuracy by repeating the computation with a finer mesh. A large part of this thesis is concerned with a set-up of this type with the physical problem described in a weak form and with the error in the estimate of the quantity of interest given in terms of a function which solves a related dual problem. We consider this in the case of modelling the large deformation of thin incompressible isotropic hyperelastic sheets under pressure loading. We assume throughout that the thin sheet can be modelled as a membrane, which gives us a two dimensional description of a three dimensional deformation and this simplifies further to a one space dimensional description in the axisymmetric case when we use cylindrical polar coordinates. In the general case we consider the deformation under quasi-static conditions and in the axisymmetric case we consider both quasi-static conditions and dynamic conditions, which involves the full equations of motion, which gives three different problems. In all the three problems we describe how to get the finite element solution, we describe associated dual problems, we describe how to solve these dual problems and we consider using the dual solutions in error estimation. There is hence a common framework. The details however vary considerably and much of the thesis is in describing each case.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2018-01-01T00:00:00ZSome new developments for quantile regressionLiu, Xihttp://bura.brunel.ac.uk/handle/2438/162042018-05-19T02:01:45Z2018-01-01T00:00:00ZTitle: Some new developments for quantile regression
Authors: Liu, Xi
Abstract: Quantile regression (QR) (Koenker and Bassett, 1978), as a comprehensive extension to standard mean regression, has been steadily promoted from both theoretical and applied aspects. Bayesian quantile regression (BQR), which deals with unknown parameter estimation and model uncertainty, is a newly proposed tool of QR. This thesis aims to make some novel contributions to the following three issues related to QR. First, whereas QR for continuous responses has received much attention in literatures, QR for discrete responses has received far less attention. Second, conventional QR methods often show that QR curves crossing lead to invalid distributions for the response. In particular, given a set of covariates, it may turn out, for example, that the predicted 95th percentile of the response is smaller than the 90th percentile for some values of the covariates. Third, mean-based clustering methods are widely developed, but need improvements to deal with clustering extreme-type, heavy tailed-type or outliers problems. This thesis focuses on methods developed over these three challenges: modelling quantile regression with discrete responses, ensuring non-crossing quantile curves for any given sample and modelling tails for collinear data with outliers. The main contributions are listed as below: * The first challenge is studied in Chapter 2, in which a general method for Bayesian inference of regression models beyond the mean with discrete responses is developed. In particular, this method is developed for both Bayesian quantile regression and Bayesian expectile regression. This method provides a direct Bayesian approach to these regression models with a simple and intuitive interpretation of the regression results. The posterior distribution under this approach is shown to not only be coherent to the response variable, irrespective of its true distribution, but also proper in relation to improper priors for unknown model parameters. * Chapter 3 investigates a new kernel-weighted likelihood smoothing quantile regression method. The likelihood is based on a normal scale-mixture representation of an asymmetric Laplace distribution (ALD). This approach benefits of the same good design adaptation just as the local quantile regression (Spokoiny et al., 2014) does and ensures non-crossing quantile curves for any given sample. * In Chapter 4, we introduce an asymmetric Laplace distribution to model the response variable using profile regression, a Bayesian non-parametric model for clustering responses and covariates simultaneously. This development allows us to model more accurately for clusters which are asymmetric and predict more accurately for extreme values of the response variable and/or outliers. In addition to the three major aforementioned challenges, this thesis also addresses other important issues such as smoothing extreme quantile curves and avoiding insensitive to heteroscedastic errors as well as outliers in the response variable. The performances of all the three developments are evaluated via both simulation studies and real data
analysis.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2018-01-01T00:00:00ZNovel regression models for discrete responsePeluso, Alinahttp://bura.brunel.ac.uk/handle/2438/155812018-01-06T03:00:32Z2017-01-01T00:00:00ZTitle: Novel regression models for discrete response
Authors: Peluso, Alina
Abstract: In a regression context, the aim is to analyse a response variable of interest conditional to a set of covariates. In many applications the response variable is discrete. Examples include the event of surviving a heart attack, the number of hospitalisation days, the number of times that individuals benefit of a health service, and so on. This thesis advances the methodology and the application of regression models with discrete response. First, we present a difference-in-differences approach to model a binary response in a health policy evaluation framework. In particular, generalized linear mixed methods are employed to model multiple dependent outcomes in order to quantify the effect of an adopted pay-for-performance program while accounting for the heterogeneity of the data at the multiple nested levels. The results show how the policy had a positive effect on the hospitalsâ€™ quality in terms of those outcomes that can be more influenced by a managerial activity. Next, we focus on regression models for count response variables. In a parametric framework, Poisson regression is the simplest model for count data though it is often found not adequate in real applications, particularly in the presence of excessive zeros and in the case of dispersion, i.e. when the conditional mean is different to the conditional variance. Negative Binomial regression is the standard model for over-dispersed data, but it fails in the presence of under-dispersion. Poisson-Inverse Gaussian regression can be used in the case of over-dispersed data, Generalised-Poisson regression can be employed in the case of under-dispersed data, and Conway-Maxwell Poisson regression can be employed in both cases of over- or under-dispersed data, though the interpretability of these models is ot straightforward and they are often found computationally demanding. While Jittering is the default non-parametric approach for count data, inference has to be made for each individual quantile, separate quantiles may cross and the underlying uniform random sampling can generate instability in the estimation. These features motivate the development of a novel parametric regression model for counts via a Discrete Weibull distribution. This distribution is able to adapt to different types of dispersion relative to Poisson, and it also has the advantage of having a closed form expression for the quantiles. As well as the standard regression model, generalized linear mixed models and generalized additive models are presented via this distribution. Simulated and real data applications with different type of dispersion show a good performance of Discrete Weibull-based regression models compared with existing regression approaches for count data.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2017-01-01T00:00:00ZBoundary-domain integral equation systems for the stokes system with variable viscosity and diffusion equation in inhomogeneous mediaFresneda-Portillo, Carloshttp://bura.brunel.ac.uk/handle/2438/145212017-05-11T02:00:32Z2016-01-01T00:00:00ZTitle: Boundary-domain integral equation systems for the stokes system with variable viscosity and diffusion equation in inhomogeneous media
Authors: Fresneda-Portillo, Carlos
Abstract: The importance of the Stokes system stems from the fact that the Stokes system is the stationary linearised form of the Navier Stokes system [Te01, Chapter1]. This linearisation is allowed when neglecting the inertial terms at a low Reinolds numbers Re << 1. The Stokes system essentially models the behaviour of a non - turbulent viscous fluid. The mixed interior boundary value problem related to the compressible Stokes system is reduced to two different BDIES which are equivalent to the original boundary value problem. These
boundary-domain integral equation systems (BDIES) can be expressed in terms of surface and volume parametrix-based potential type operators whose properties are also analysed in appropriate Sobolev spaces. The invertibility and Fredholm properties related to the matrix operators that de ne the BDIES are
also presented. Furthermore, we also consider the mixed compressible Stokes system with variable
viscosity in unbounded domains. An analysis of the similarities and differences with regards to the bounded domain case is presented. Furthermore, we outline the mapping properties of the surface and volume parametrix-based potentials in weighted Sobolev spaces. Equivalence and invertibility results still hold under certain decay conditions on the variable coeffi cient The last part of the thesis refers to the mixed boundary value problem for the stationary heat transfer partial di erential equation with variable coe cient. This BVP is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Chkadua, Mikhailov and Natroshvili in the paper [CMN09].
Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed in both bounded and unbounded domains.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2016-01-01T00:00:00Z