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This thesis is about computational theory in the setting of the mathematics of information. The first goal is to demonstrate that many commonly considered problems in optimisation theory cannot be solved with an algorithm if the input data is only known up to an arbitrarily small error (modelling the fact that most real numbers are not expressible to infinite precision with a floating point based computational device). This includes computing the minimisers to basis pursuit, linear programming, lasso and image deblurring as well as finding an optimal neural network given training data. These results are somewhat paradoxical given the success that existing algorithms exhibit when tackling these problems with real world datasets and a substantial portion of this thesis is dedicated to explaining the apparent disparity, particularly in the context of compressed sensing. To do so requires the introduction of a variety of new concepts, including that of a breakdown epsilon, which may have broader applicability to computational problems outside of the ones central to this thesis. We conclude with a discussion on future research directions opened up by this work.

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Subsea pipelines form an integral part of the infrastructure associated with offshore oil and gas developments. These pipelines fulfill a range of functions from linking extraction wells to other subsea infrastructure to transporting products onshore, or to a central processing facility. Ancillary pipelines may also be present for gas or water injection to the reservoir or transporting additives. Pipelines are typically installed directly onto the seabed and, in the absence of significant drivers to undertake burial operations, they may remain on the seabed for the remainder of their design life. This is typically the case for deepwater developments. Subsea pipelines are subjected to a wide range of load cases including, self weight, installation loads, thermal and pressure driven expansion and hydrodynamic loading. Design of pipeline systems to accommodate these load cases requires an understanding of pipe-soil interaction. This thesis reports the results of a research study investigating pipe-soil interaction on a clay seabed, as relevant to the design of subsea pipeline systems. This study has utilised numerical analysis techniques based on the finite difference code FLAC to investigate a range of problem definitions. These problem definitions include pipelines subject to both vertical loading (V) and combined vertical and horizontal (V-H) loading. Factors such as variation in interface conditions, large strain and large displacement effects, soil unit weight effects and variation in shear strength conditions were considered in these problem definitions. Reliability based analysis techniques have also been used to investigate both V and V-H loading problem definitions. The analyses and investigations undertaken as part of this study generally achieved the following; reproduction and validation of earlier research with additional interpretation, extension of problem definitions to deeper pipeline embedment depths and investigation of pipe-soil interaction problem definitions that have not previously been considered. Reliability based analysis techniques have also provided some interesting insights into the impact of soil shear strength variation as well as providing a fundamental link between safety factors and probability of failure. Application to design practice of this, and similar studies, has been considered as part of this thesis and potential areas for future research have also been suggested.

Tsang, Ka Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves [121]-123). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Inverse problems and formulations --- p.6 / Chapter 3 --- Review of some existing methods --- p.8 / Chapter 4 --- Trust Region Method --- p.16 / Chapter 4.1 --- Some Auxiliary Tools --- p.18 / Chapter 4.2 --- Trust Region Algorithm --- p.23 / Chapter 4.3 --- Convergence of trust region method --- p.28 / Chapter 4.3.1 --- Notations and Assumptions --- p.28 / Chapter 4.3.2 --- Convergence for exact data --- p.29 / Chapter 4.3.3 --- Regularity For Inexact Data --- p.36 / Chapter 4.4 --- Experiment On Trust Region Method --- p.39 / Chapter 4.4.1 --- Problem Setting --- p.39 / Chapter 4.4.2 --- Algorithm --- p.40 / Chapter 4.4.3 --- Experiment Results --- p.42 / Chapter 4.5 --- Trust Region Conjugate Gradient Method --- p.46 / Chapter 4.5.1 --- Notations and Assumptions --- p.49 / Chapter 4.5.2 --- Convergence Properties for Exact Data --- p.52 / Chapter 4.5.3 --- Regularity for Inexact Data --- p.57 / Chapter 5 --- Parameter Identification Problems --- p.60 / Chapter 5.1 --- Introduction --- p.60 / Chapter 5.1.1 --- Computation of VJ(x) --- p.67 / Chapter 5.2 --- Algorithm for Parameter Identification Problems --- p.72 / Chapter 5.2.1 --- "Finite Element Method in Two Dimensions:Ω =[0,1] x [0,1]" --- p.75 / Chapter 5.3 --- Experiments on Trust Region-CG Method for Parameter Identification Problems --- p.82 / Chapter 5.3.1 --- One Dimension Problem --- p.82 / Chapter 5.3.2 --- Two Dimensions Problem --- p.95 / Chapter 5.4 --- Conclusion --- p.119 / Bibliography --- p.121

Liu, Keji. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 97-98). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Background and models of inverse scattering --- p.1 / Chapter 1.1 --- Model I --- p.1 / Chapter 1.2 --- Model II --- p.3 / Chapter 1.3 --- Model III --- p.3 / Chapter 2 --- Direct and Inverse Problems of Three Models --- p.6 / Chapter 2.1 --- First Model: Direct Problem --- p.6 / Chapter 2.2 --- First Model: Inverse Problem --- p.10 / Chapter 2.2.1 --- Linear Sampling Method --- p.10 / Chapter 2.2.2 --- Strengthened Linear Sampling Method --- p.12 / Chapter 2.2.3 --- Multilevel Linear Sampling Method --- p.15 / Chapter 2.3 --- Second Model: Direct Problem --- p.19 / Chapter 2.4 --- Second Model: Inverse Problem --- p.21 / Chapter 2.4.1 --- Contrast Source Inversion Method --- p.21 / Chapter 2.4.2 --- Subspace-based Optimization Method --- p.25 / Chapter 2.4.3 --- Multiple Signal Classification Method --- p.31 / Chapter 2.5 --- Third Model: Direct Problem --- p.33 / Chapter 2.6 --- Third Model: Inverse Problem --- p.41 / Chapter 2.6.1 --- Generalized Dual Space Indicator Method --- p.41 / Chapter 3 --- Numerical Simulations --- p.44 / Chapter 3.1 --- Numerical Simulations of First Model --- p.44 / Chapter 3.1.1 --- Linear Sampling Method --- p.44 / Chapter 3.1.2 --- Strengthened Linear Sampling Method --- p.51 / Chapter 3.1.3 --- Multilevel Linear Sampling Method --- p.58 / Chapter 3.2 --- Numerical Simulations of Second Model --- p.68 / Chapter 3.2.1 --- Contrast Source Inversion Method --- p.68 / Chapter 3.2.2 --- Subspace-based Optimization Method --- p.74 / Chapter 3.2.3 --- Twofold Subspace-based Optimization Method --- p.79 / Chapter 3.2.4 --- Multiple Signal Classification Method --- p.85 / Chapter 3.3 --- Numerical Simulations of Third Model --- p.85 / Chapter 3.3.1 --- Boundary Integral Method --- p.86 / Chapter 3.3.2 --- Generalized Dual Space Indicator Method --- p.89 / Chapter 4 --- Conclusion

Orientador: Cleonice Fátima Bracciali / Banca: Eliana Xavier Linhares de Andrade / Banca: Gilcilene Sanchez de Paulo / Resumo: Utilizando os conceitos da representação hipergeométrica dos polinômios ortogonais em uma variável, da fórmula de Rodrigues e da função geratriz, pode-se obter polinômios em várias variáveis. Neste trabalho, detalhamos, especificamente, os polinômios de Jacobi em duas variáveis, os polinômios de Legendre e de Gegenbauer em várias variáveis, mostrando suas representações como função hipergeométrica, as fórmulas de Rodrigues, as relações de recorrência, a ortogonalidade, entre outras propriedades. Estes resultados são obtidos generalizando-se os conceitos e propriedades dos polinômios ortogonais em uma variável / Abstract: By using the concepts about hypergeometric representation of orthogonal polynomials in one variable, Rodrigues formula and generating function, one can obtain orthogonal polynomials of several variables. In this work, we detail, speci cally, the Jacobi polynomials in two variables, the Legendre and Gegenbauer polynomials in several variables, by presenting their representations in terms of hypergeometric functions, by Rodrigues formulae, recurrence relations, orthogonality, among many others. These results are obtained by generalizing the concepts and properties of orthogonal polynomials in one variable / Mestre

Robots have been a revolutionizing force in manufacturing in the 20th and 21st century but have proven too dangerous around humans to be used in many other fields including medicine. We describe a new control algorithm for robots developed by the Brigham Young University Robotics and Dynamics and Robotics Laboratory that has shown potential to make robots less dangerous to humans and suitable to work in more applications. We analyze the computational complexity of this algorithm and find that it could be a feasible control for even the most complicated robots. We also show conditions for a system which guarantee local stability for this control algorithm.

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers. In order to show mastery of the subject matter this paper gives a somewhat different organization of material and modified or more detailed proofs of theorems. In situations where proofs become rather routine re pet it ions of previously noted techniques the proofs are omitted. The following symbols and notation are used. Natural numbers are denoted by lower case letters such as a,b,c, ... x,y,z. Sets are denoted by upper case letters such as M, N, ... X, Y, Z. If a is an element of M, this will be written atM, The denial of this is written at M. The symbol 3 /x is read "There exists an unique x". If x and y are names for the same number we write x=y. It is assumed that the relation= is an equivalence relation; i.e., (1) x=x, (2) if x=y, then y=x, (3) u x=y and y=z, then x=z. Throughout this paper there will be no special attempt to distinguish between the name of a number and the number itself. For example, the phrase" if xis a number" will be used in place of "if x is the name of a number."