Thesis (Ph. D.)--Ohio State University, 2004. / Title from first page of PDF file. Document formatted into pages; contains v, 88 p. Includes bibliographical references (p. 87-88). Available online via OhioLINK's ETD Center

Thesis (PhD)--Montana State University--Bozeman, 2009. / Typescript. Chairperson, Graduate Committee: Tomas Gedeon. Includes bibliographical references (leaves 234-237).

Thesis (M. S.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009. / Committee Chair: David Anderson; Committee Co-Chair: Alexander Gray; Committee Member: Anthony Yezzi; Committee Member: Hongyuan Zha; Committee Member: Justin Romberg; Committee Member: Ronald Schafer.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1968. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--Industrial and Systems Engineering, Georgia Institute of Technology, 2006. / Xiaoming Huo, Committee Chair ; C. F. Jeff Wu, Committee Member ; Brani Vidakovic, Committee Member ; Liang Peng, Committee Member ; Ming Yuan, Committee Member.

Thesis (M.S.)--Brigham Young University. Dept. of Mathematics, 2005. / Includes bibliographical references (p. 51).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 120-124).

This thesis explores the algebraic and geometric structure of the Legendre transform and its application in various field of mathematics and physics. Specifically linear transformation as a mathematical process and motivating it in terms related to phenomena in mathematics and physics. The Legendre transform provides a change of variables to express equations of the motion or other physical relationships in terms of most convenient dynamical quantities for a given experiment or theoretical analysis. In classical mechanics the Legendre transform generates the Hamiltonian function of a system from the Lagrangian function or vice versa. In thermodynamics the Legendre transform allows thermodynamic relationships to be written in terms of alternative sets of independent variables. Here we review the properties of Legendre transform and why it is so important in mathematics and physics.

Many applications arising in various areas can be well modeled as convex optimization models with separable objective functions and linear coupling constraints. Such areas include signal processing, image processing, statistical learning, wireless networks, etc. If these well-structured convex models are treated as generic models and their separable structures are ignored in algorithmic design, then it is hard to effectively exploit the favorable properties that the objective functions possibly have. Therefore, some operator splitting methods have regained much attention from different areas for solving convex optimization models with separable structures in different contexts. In this thesis, some new operator splitting methods are proposed for convex optimiza- tion models with separable structures. We first propose combining the alternating direction method of multiplier with the logarithmic-quadratic proximal regulariza- tion for a separable monotone variational inequality with positive orthant constraints and propose a new operator splitting method. Then, we propose a proximal version of the strictly contractive Peaceman-Rachford splitting method, which was recently proposed for the convex minimization model with linear constraints and an objective function in form of the sum of two functions without coupled variables. After that, an operator splitting method suitable for parallel computation is proposed for a convex model whose objective function is the sum of three functions. For the new algorithms, we establish their convergence and estimate their convergence rates measured by the iteration complexity. We also apply the new algorithms to solve some applications arising in the image processing area; and report some preliminary numerical results. Last, we will discuss a particular video processing application and propose a series of new models for background extraction in different scenarios; to which some of the new methods are applicable. Keywords: Convex optimization, Operator splitting method, Alternating direction method of multipliers, Peaceman-Rachford splitting method, Image processing

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