Thesis (Ph. D. in Mathematics)--Vanderbilt University, Aug. 2006. / Title from title screen. Includes bibliographical references.

Collocation with cubic splines is used as a method for solving Linear second order parabolic partial differential equations. The collocation method is shown to be equivalent to a finite difference method that is consistent with the differential equation and stable in the sense of Von Neumann. Results of numerical computations are given, as well as an application of the method to a moving boundary problem for the heat equation. / Science, Faculty of / Mathematics, Department of / Graduate

Orientador: Sérgio Antonio Tozoni / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-22T06:19:51Z (GMT). No. of bitstreams: 1 Lopes_RaquelVieira_M.pdf: 1141005 bytes, checksum: 8091e673668abab031630ae7fd1b7436 (MD5) Previous issue date: 2013 / Resumo: Os sk-splines são uma generalização natural dos splines polinomiais, os quais foram introduzidos e tiveram sua teoria básica desenvolvida por Alexander Kushpel nos anos de 1983-1985. Estas funções são importantes em várias aplicações e seu espaço é gerado por translações discretas de uma única função núcleo. Neste trabalho, estudamos condições necessárias e suficientes para a existência e unicidade de sk-splines interpolantes de funções periódicas. Além disso, estudamos a aproximação de funções de determinadas classes por sk-splines nos espaços Lp. Como aplicação estudamos a aproximação de funções infinitamente diferenciáveis e finitamente diferenciáveis por sk- splines / Abstract: The sk-splines are a natural generalization of polynomial splines. They were introduced and their basic theory developed by Alexander Kushpel between 1983 and 1985. These functions are important in many applications and the space of sk-splines is the linear span of shifts of a single kernel K. In this work, we study necessary and sufficient conditions for the existence and uniqueness of sk-splines interpolants of periodic functions. Furthermore, we study the approximation in several classes of functions by sk-splines in the Lp spaces. As an application we study the approximation of infinitely and finitely differentiable functions by sk-splines / Mestrado / Matematica Aplicada / Mestra em Matemática Aplicada

This dissertation discusses a new approach to spline approximation. A periodic spline approximation 𝑓_M,m,N(x) = Σ_k=1^Nα_kΦ_M,k(x) to a periodic function 𝑓(x) is determined by requiring < Φ_m,j, 𝑓 - 𝑓_M,m,N > = 0 for j = 1,...,N, where the Φ_L,k's are the unique periodic spline basis functions of order 𝐿. Error estimates, examples and some relationships to wavelets are given for the case M - m = 2μ. The case M - m = 2µ + 1 is briefly discussed but not completely explored. / Ph. D.

Thesis. 1978. Elec.E.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / Elec.E.

In medical, behavioral, and social-psychological sciences, latent variable models are useful in handling variables that cannot be directly measured by a single observed variable, but instead are assessed through a number of observed variables. Traditional latent variable models are usually based on parametric assumptions on both relations between outcome and explanatory latent variables, and error distributions. In this thesis, semiparametric models with Bayesian P-splines are developed to relax these rigid assumptions. / In the fourth part of the thesis, the methodology developed in the third part is further extended to a varying coefficient model with latent variables. Varying coefficient model is a class of flexible semiparametric models in which the effects of covariates are modeled dynamically by unspecified smooth functions. A transformation varying coefficient model can handle arbitrarily distributed dynamic data. A simulation study shows that our proposed method performs well in the analysis of this complex model. / In the last part of the thesis, we propose a finite mixture of varying coefficient models to analyze dynamic data with heterogeneity. A simulation study demonstrates that our proposed method can explore possible existence of different groups in a dynamic data, where in each group the dynamic influences of covariates on the response variables have different patterns. The proposed method is applied to a longitudinal study concerning the effectiveness of heroin treatment. Distinct patterns of heroin use and treatment effect in different patient groups are identified. / In the second part of the thesis, a latent variable model is proposed to relax the first assumption, in which unknown additive functions of latent variables in the structural equation are modeled by Bayesian P-splines. The estimation of nonparametric functions is based on powerful Markov chain Monte Carlo (MCMC) algorithm with block update scheme. A simulation study shows that the proposed method can handle much wider situation than traditional models. The proposed semiparametric latent variable model is applied to a study on osteoporosis prevention and control. Some interesting functional relations, which may be overlooked by traditional parametric latent variable models, are revealed. / In the third part of the thesis, a transformation model is developed to relax the second assumption, which usually assumes the normality of observed variables and random errors. In our proposed model, the nonnormal response variables are transformed to normal by unknown functions modeled with Bayesian P-splines. This semiparametric transformation model is shown to be applicable to a wide range of statistical analysis. The model is applied to a study on the intervention treatment of polydrug use in which the traditional model assumption is violated because many observed variables exhibit serious departure from normality. / Lu, Zhaohua. / Adviser: Xin-Yuan Song. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 119-130). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

In the brain mapping case, the geodesic closest points are used as the anatomical constraints for the inter-subject non-rigid registration. The method uses the anatomical constraint in the non-rigid registration which is much more reasonable for the anatomical correspondence. The registration result shows significant improvement comparing with the iterative closest points based method. / In the third application, we use the non-rigid registration method to register the different sweeps of freehand ultrasound images. We setup a 3D freehand ultrasound imaging system to capture images of a human anatomy such as liver, prostate, brain tumor and fetus. The arbitrary scanned image slices are reconstructed and resliced into volumetric dataset. We use a B-spline based non-rigid registration method to compounding different freehand ultrasound sweeps. This technique can be used to make 3D ultrasound models of fetus and other organs. / Medical image registration is an active research area during the last two decades. The registration technique can be widely used in the applications of the computer aided surgery, brain mapping and pathological detection and analysis. With the development of the computing power, fast and accurate registration techniques have been developed into necessary tools for quantitative analysis of the medical image. / Non-rigid registration methods can be used in atlas based image segmentation, inter-subject brain image registration and 3D freehand ultrasound modeling. In one of our proposed novel segmentation methods, we interleave the segmentation and the registration processes by using the segmentation to provide the anatomical constraints for registration to improve the atlas based non-rigid registration. This updated registration can be used to improve the new segmentation. This process is repeated until a good result in segmentation is obtained. / The registration methods can be classified into rigid and non-rigid registrations according to whether the anatomy is locally deformed or not. According to the sensor by which the images are taken, the registration will be divided into mono-modal and multi-modal image registration. Since the invention of the medical imaging devices, great diversity of medical imaging sensors have been developed with different physical principles. In practice we have to face the problem of multi-modal registration. In medical image analysis, we often have to consider the images of the human anatomy with deformable characteristics. In order to achieve this goal we need to use the voxel based registration method which considers all of the voxel information of the images in matching. There are several non-rigid registration approaches. However, the variational approach of non-rigid registration can represent the registration problem into a well-posed problem with a well-founded mathematical base. In our work, we considered the forward and backward deformation functions and proposed a variational approach for a new consistent multi-modal non-rigid registration method. By this way, we will find the forward and backward transform to be close to the inverse of each other. This makes the correspondence between two images more consistent and accurate. We use both explicit and implicit difference method to solve the Euler-Lagrange equation and the results show significant improvements in the transformation inverse consistency. Although variational approach for multi-modal non-rigid registration can solve the non-rigid registration problem well, generally speaking, it is slow. The displacement of each voxel has to be calculated and the iteration time is very long since the number of the unknowns are large. Although a multi-resolution strategy can be used to speed up, the registration problem is still slow when registering large medical datasets. The 3D B-spline based method has been used as an efficient method to register medical images since only a small number of control points are used to manipulate the local deformation field. In our work, we developed a 3D B-spline based consistent multi-modal non-rigid registration method with an explicit representation of derivatives. The conventional optimization methods can be used to find the optimal parameters. We use a hierarchical B-spline refinement method to approximate the deformation function from larger to smaller scale. Since the derivatives of the cost function is represented in an explicit way, the computing is reduced. It is more efficient than directly computing the derivative of the cost function by using a numerical evaluation method. The method can be considered as a multi-grid method for solving the consistent variational registration problem. The computing speed is increased by several times. The B-spline based method needs far less iterations to converge as its number of unknowns is small. / Zhang Zhijun. / "October 2005." / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6645. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 209-233). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis. 1976. M.S.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / Microfiche copy available in Archives and Engineering. / Includes bibliographical references. / M.S.

This thesis consists of three chapters. The first chapter focuses on adaptive smoothing splines for fitting functions with varying roughness. In the first part of the first chapter, we study an asymptotically optimal procedure to choose the value of a discretized version of the variable smoothing parameter in adaptive smoothing splines. With the choice given by the multivariate version of the generalized cross validation, the resulting adaptive smoothing spline estimator is shown to be consistent and asymptotically optimal under some general conditions. In the second part, we derive the asymptotically optimal local penalty function, which is subsequently used for the derivation of the locally optimal smoothing spline estimator. In the second chapter, we propose a Lipschitz regularity based statistical model, and apply it to coordinate measuring machine (CMM) data to estimate the form error of a manufactured product and to determine the optimal sampling positions of CMM measurements. Our proposed wavelet-based model takes advantage of the fact that the Lipschitz regularity holds for the CMM data. The third chapter focuses on the classification of functional data which are known to be well separable within a particular interval. We propose an interval based classifier. We first estimate a baseline of each class via convex optimization, and then identify an optimal interval that maximizes the difference among the baselines. Our interval based classifier is constructed based on the identified optimal interval. The derived classifier can be implemented via a low-order-of-complexity algorithm.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 114-117). Also available on the Internet.