Thesis (Ph. D.)--University of Tennessee, Knoxville, 2010. / Title from title page screen (viewed on July 14, 2010). Thesis advisor: Gary Skolits. Vita. Includes bibliographical references.

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Thesis (MTech (Mechanical Engineering))--Cape Peninsula University of Technology, 2007. / Twelve years into a democracy, South Africa still faces many developmental challenges. Since 2002 Universities of Technology in South Africa have introduced Foundational Programmes/provisions in their Science and Engineering programmes as a key mechanism for increasing throughput and enhancing quality. The Department of Education has been funding these foundational provisions since 2005. This case study evaluates an aspect of a foundational provision in Mechanical Engineering, from the beginning of 2002 to the end of 2005, at a University of Technology, with a view to contributing to its improvement. The Cape Peninsula University of Technology (CPUT), the locus for this case study, is the only one of its kind in a region that serves in excess of 4.5 million people. Further, underpreparedness in Mathematics for tertiary level study is a national and international phenomenon. There is thus a social interest in the evaluation of a Mathematics course that is part of a strategy towards addressing the shortage in Engineering graduates. This Evaluation of integration of the Foundation Mathematics course into Foundation Science, within the Department of Mechanical Engineering at CPUT, falls within the ambit of this social need. An integrated approach to curriculum conception, design and implementation is a widely accepted strategy in South Africa and internationally; this approach formed the basis of the model used for the Foundation programme that formed part of this Evaluation. A review of the literature of the underpinnings of the model provided a theoretical framework for this evaluation study. In essence this involved the use of academic literacy theory together with learning approach theory to provide a lens for this case study. The research methodology used was largely qualitative, with both qualitative and quantitative methods used for purposes of triangulation. The evaluation was conducted of four key aspects of integration of foundation mathematics into foundation science, namely conception, design, implementation and impact. This provided the framework for the main argument of this thesis, namely that conceptual and design flaws in the integration modelled to student learning of Mechanics concepts (in Foundation Science) not being effectively supported. The final section of the study outlines recommendations for improvement of the foundation mathematics course. It also identifies areas for future research. / National Research Foundation (NRF)

本博士论文对于有结构但又有相当一般性的约束条件下的非线性优化问题给出了系统性研究。比较经典的例子包指球面约束下的多重线性函数优化问题。这些模型已被广泛应用于数值线性代数、材料科学、量子物理学、信号处理、语音识别、生物医学工程以及控制论等。本论文着重探讨一类特定的方法来解这些广义模型，即块变量改进方法。具体地说，我们构造了一类块坐标下降型搜索算法来解带块变量结构的非线性优化问题。这类算法通过每次迭代中只更新一块变量以达到最大限度的目标函数值的改进(因而，这一新搜索算法命名为最优块改进算法(简称为MBI)。之后，我们重点研究了该算法在求解众多领域中实际问题的潜在能力。首先，这一算法可以直接应用于生物信息学中聚类基因表达数据的一种新模型的设计及求解。接着，我们把注意力转移到球面约束下的齐次多项式优化问题，此问题与张量的最优秩-1 逼近问题相关。对于这一优化问题， MBI 算法通常可以在较少的计算时间内找到全局最优解。第三，我们继续深入研究多项式优化问题，在双协半正定的新概念下建立了齐次多项式优化问题与其多重线性优化问题关系的一般性结果。最后，我们在Tucker 分解的框架下给出了求解高阶张量的最优多重线性秩的逼近问题的方法，并提出一种新的模型和算法来解未知变量数的Tucker 分解问题。本论文讨论并试验了一些应用实例，数值实验表明所提出的算法分别对于求解以上这些问题是可行并有效的。 / In this thesis we present a systematic analysis for optimization of a general nonlinear function, subject to some fairly general constraints. A typical example includes the optimization of a multilinear tensor function over spherical constraints. Such models have found wide applications in numerical linear algebra, material sciences, quantum physics, signal processing, speech recognition, biomedical engineering, and control theory. This thesis is mainly concerned with a specific approach to solve such generic models: the block variable improvement method. Specifically, we establish a block coordinate descent type search method for nonlinear optimization, which accepts only a block update that achieves the maximum improvement (hence the name of our new search method: maximum block improvement (MBI)). Then, we focus on the potential capability of this method for solving problems in various research area. First, we demonstrate that this method can be directly used in designing a new framework for co-clustering gene expression data in the area of bioinformatics. Second, we turn our attention to the spherically constrained homogeneous polynomial optimization problem, which is related to best rank-one approximation of tensors. The MBI method usually finds the global optimal solution at a low computational cost. Third, we continue to consider polynomial optimization problems. A general result between homogeneous polynomials and multi-linear forms under the concept of co-quadratic positive semidefinite is established. Finally, we consider the problem of finding the best multi-linear rank approximation of a higher-order tensor under the framework of Tucker decomposition, and also propose a new model and algorithms for computing Tucker decomposition with unknown number of components. Some real application examples are discussed and tested, and numerical experiments are reported to reveal the good practical performance and efficiency of the proposed algorithms for solving those problems. / Detailed summary in vernacular field only. / Chen, Bilian. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 86-98). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.vii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.2 --- Notations and Preliminaries --- p.5 / Chapter 1.2.1 --- The Tensor Operations --- p.6 / Chapter 1.2.2 --- The Tensor Ranks --- p.9 / Chapter 1.2.3 --- Polynomial Functions --- p.11 / Chapter 2 --- The Maximum Block Improvement Method --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- MBI Method and Convergence Analysis --- p.14 / Chapter 3 --- Co-Clustering of Gene Expression Data --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- A New Generic Framework for Co-Clustering Gene Expression Data --- p.22 / Chapter 3.2.1 --- Tensor Optimization Model of The Co-Clustering Problem --- p.22 / Chapter 3.2.2 --- The MBI Method for Co-Clustering Problem --- p.23 / Chapter 3.3 --- Algorithm for Co-Clustering 2D Matrix Data --- p.25 / Chapter 3.4 --- Numerical Experiments --- p.27 / Chapter 3.4.1 --- Implementation Details and Some Discussions --- p.27 / Chapter 3.4.2 --- Testing Results using Microarray Datasets --- p.30 / Chapter 3.4.3 --- Testing Results using 3D Synthesis Dataset --- p.32 / Chapter 4 --- Polynomial Optimization with Spherical Constraint --- p.34 / Chapter 4.1 --- Introduction --- p.34 / Chapter 4.2 --- Generalized Equivalence Result --- p.37 / Chapter 4.3 --- Spherically Constrained Homogeneous Polynomial Optimization --- p.41 / Chapter 4.3.1 --- Implementing MBI on Multilinear Tensor Form --- p.42 / Chapter 4.3.2 --- Relationship between Homogeneous Polynomial Optimization over Spherical Constraint and Tensor Relaxation Form --- p.43 / Chapter 4.3.3 --- Finding a KKT point for Homogeneous Polynomial Optimization over Spherical Constraint --- p.45 / Chapter 4.4 --- Numerical Experiments on Randomly Simulated Data --- p.47 / Chapter 4.4.1 --- Multilinear Tensor Function over Spherical Constraints --- p.49 / Chapter 4.4.2 --- Tests of Another Implementation of MBI --- p.49 / Chapter 4.4.3 --- General Polynomial Function over Quadratic Constraints --- p.51 / Chapter 4.5 --- Applications --- p.53 / Chapter 4.5.1 --- Rank-One Approximation of Super-Symmetric Tensors --- p.54 / Chapter 4.5.2 --- Magnetic Resonance Imaging --- p.55 / Chapter 5 --- Logarithmically Quasiconvex Optimization --- p.58 / Chapter 5.1 --- Introduction --- p.58 / Chapter 5.2 --- Logarithmically Quasiconvex Optimization --- p.60 / Chapter 5.2.1 --- A Simple Motivating Example --- p.61 / Chapter 5.2.2 --- Co-Quadratic Positive Semide nite Tensor Form --- p.61 / Chapter 5.2.3 --- Equivalence at Maxima --- p.64 / Chapter 6 --- The Tucker Decomposition and Generalization --- p.68 / Chapter 6.1 --- Introduction --- p.68 / Chapter 6.2 --- Convergence of Traditional Tucker Decomposition --- p.71 / Chapter 6.3 --- Tucker Decomposition with Unknown Number of Components --- p.73 / Chapter 6.3.1 --- Problem Formulation --- p.74 / Chapter 6.3.2 --- Implementing the MBI Method on Tucker Decomposition with Unknown Number of Components --- p.75 / Chapter 6.3.3 --- A Heuristic Approach --- p.79 / Chapter 6.4 --- Numerical Experiments --- p.80 / Chapter 7 --- Conclusion and Recent Developments --- p.83 / Bibliography --- p.86

A detailed literature review on the development and applications of several numerical methods in solid mechanics and electromagnetic field analysis is presented in the thesis. Despite the great achievements in this research area, there are always the needs to develop new numerical methods or to explore alternative techniques for the purpose of solving the complicate problems and improve the efficiency and accuracy of the existing or new numerical methods. This thesis presents the development of a novel numerical method, the moving least square Ritz (MLS-Ritz) method, and its applications for solving science and engineering problems. The MLS-Ritz method is based on the moving least square (MLS) data interpolation technique and the Ritz minimization principle. The MLS technique is utilized to establish the Ritz trial functions for two-dimensional (2-D) and three-dimensional (3-D) cases. A point substitution approach is developed to enforce boundary conditions. The proposed MLS-Ritz method has the ability to expand the applicability of the conventional Ritz method and meshless method for analysing problems with complex geometries and multiple mediums. The applications of the MLS-Ritz method are also extended to the analysis of the electromagnetic field problems. Three cases including electrical potential problems in a uniform trough and with dielectric medium and a waveguide eigenvalue problem are analysed and compared with solutions obtained by other methods. Comparison studies show that excellent agreement is achieved for the three cases when comparing with existing results in the open literature. The future directions in the development of the MLS-Ritz method for science and engineering computations are discussed. / Doctor of Philosophy (PhD)

Thesis (Ed.D.)--University of Central Florida, 2009. / Adviser: Rosa Cintrón. Includes bibliographical references (p. 327-371).

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