Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for sufficient conditions on the geometry of the environment that ensure standard heat kernel upper bounds hold. We prove that these conditions are satisfied with probability one in the case of the random walk on continuum percolation and use the heat kernel bounds to prove an invariance principle. The random walk on dynamic environment is then considered. It is proven that if the environment evolves ergodically and is, in a certain sense, geometrically d-dimensional then standard on diagonal heat kernel bounds hold. Anomalous lower bounds on the heat kernel are also proven - in particular the random conductance model is shown to be "more anomalous" in the dynamic case than the static. Finally, the reflected random walk amongst random conductances is considered. It is shown in one dimension that under the usual scaling, this walk converges to reflected Brownian motion.

<pre>In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. </pre> <pre><br></pre> <pre>The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements.</pre> <pre><br></pre> <pre>The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition.</pre> <pre><br></pre> <pre>One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter.</pre> <pre><br></pre> <pre>Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.</pre>

thesis submitted to the Faculty of Science in fulfilment of the requirements for the degree of Doctor of Philosophy. March 17, 2015. / A two-level facility location problem (FLP) arose in the transport network of maize crop in Tanzania has been studied. The three layers, namely, production centers (PCs), distribution centers (DCs) and customer points (CPs) are considered in the two-level FLP. The stochastic e ect on the two-level FLP due to rainfall in the network links, between the DCs and CPs, has been studied. The ow of maize crop from PCs to CPs through DCs is designed at a minimum cost under deterministic and stochastic scenarios. The three decisions made simultaneously are: to determine the locations of DCs (including number of DCs), allocation of CPs to the selected DCs, allocation of selected DCs to PCs, and to determine the amount of maize crop transported from PCs to DCs and then from DCs to CPs. We have modelled the problem and generate results by optimizing the model with respect to optimal location-allocation strategies. We have considered two networks, the existing network and an extended network. In the existing network there are four PCs, ve DCs and ninety three CPs. In the extended network three additional DCs are considered. For the modelling purpose we have used the rainfall data from 2007 - 2010 in each week for 17 weeks. The optimized results for the existing network have shown improvements in cost saving compared to the manually operated existing network. In the extended network, the results have shown much more e cient and cost saving distribution system compared to the results of the existing network.

As linhas flexíveis utilizadas em sistemas flutuantes para produção de petróleo são solicitadas por esforços ambientais de natureza aleatória durante toda sua vida útil. Sua resposta dinâmica de longo prazo as solicitações do ambiente, embora não estacionaria pode ser vista como sendo uma sucessão de processos aleatórios estacionários, associados a probabilidades de ocorrência de condições ambientais de curto prazo. Através de simulações da resposta não linear as solicitações de curto prazo, uma descrição probabilística de segunda ordem da resposta das linhas flexíveis pode ser obtida por analise biespectral. A resposta estacionaria não gaussiana fica então sendo caracterizada no domínio da frequência pela identificação de suas energias linear e não linear. A energia linear define um processo gaussiano, e a energia não linear corresponde à parcela não gaussiana da resposta. Uma regressão quadrática da resposta no domínio da frequência permite obter sua função característica, que e a transformada de Fourier da função de densidade de probabilidade. Desta forma, a resposta estacionaria e representada pela soma de parcelas lineares e quadráticas de variáveis aleatórias gaussianas independentes, e a obtenção da função característica faz-se através da solução de um problema de autovalores e autovetores de uma matriz real e simétrica. / Sem resumo em inglês.

In this dissertation, we develop the theory of the periodic Little's law (PLL) as well as discussing one of its applications. As extensions of the famous Little's law, the PLL applies to the queueing systems where the underlying processes are strictly or asymptotically periodic. We give a sample-path version, a steady-state stochastic version and a central-limit-theorem version of the PLL in the first part. We also discuss closely related issues such as sufficient conditions for the central-limit-theorem version of the PLL and the weak convergence in countably infinite dimensional vector space which is unconventional in queueing theory. The PLL provides a way to estimate the occupancy level indirectly. We show how to construct a real-time predictor for the occupancy level inspired by the PLL as an example of its applications, which has better forecasting performance than the direct estimators.

Os aspectos estatísticos inerente à excitação provocada pelas condições ambientais sobre embarcações amarradas são estudados de forma a estabelecer parâmetros aceitáveis para avaliação da qualidade da resposta dinâmica não-linear de longo prazo das linhas de amarração, em termos de vida em fadiga e do risco de ruptura estática. A abordagem proposta subdivide a resposta de longo prazo numa sequência de eventos de curto prazo, nos quais o processo aleatório é considerado estacionário, embora não necessariamente de banda estreita. Nesses termos, o comportamento dinâmico não-linear das linhas de amarração pode ser tomado na sua essência fundamental, sem a necessidade de simplificações de grande vulto. Como parte do procedimento de cálculo, propõe-se que as tensões de curto prazo sejam obtidas por integração direta, no domínio do tempo, do equilíbrio dinâmico de cada linha isolada de amarração, nos moldes já hoje consubstanciados em programas de computador comercialmente disponíveis. Finaliza-se com a apresentação de um exemplo de aplicação do método, para o caso da plataforma semi-submersível GVA 4500 da Petrobrás, que vai operar na Bacia de Campos numa lâmina d\'água ao redor de 1000 m . Atende-se com este trabalho, portanto, alguns dos anseios estabelecidos no \"Programa de Capacitação Tecnológica em Sistemas de Exploração para Águas Profundas (PROCAP)\" da citada empresa, buscando cumprir o objetivo de colocar o conhecimento tecnológico nacional adaptado à realidade do país, tornando-o menos dependente da importação de tecnologias estrangeiras, nem sempre, ou quase nunca, voltadas às nossas reais necessidades. / The inherent statistic aspects due to environmental loads acting on moored vessels are investigates in order to obtain reliable parameters to evaluate the long-term non-linear dynamics response of the mooring lines, as for the fatigue life and the risk of static failure. The proposed approach divides the long-term response into a sequence of short-term events, in which the random process is regarded as stationary, but not narrow band. Thus the non-linear dynamic behavior of the mooring lines can be treated on its fundamental essence, without large simplifications. As part of the calculation procedure, the short-term tensions are obtained by integrating directly the equations of motion for each single mooring line, as performed by the available commercial computer programs. Finally an example of application is presented, for the Petrobrás semisubmersible platform GVA 4500, that will be operating at Bacia de Campos in a water depth of 1000 m. This work goes toward the aims of the \"Programa de Capacitação Tecnológica em Sistemas de Exploração para Águas Profundas (PROCAP)\", in order to take the national technological knowledge adapted to the country actuality, and avoid the importation of foreign technologies, that are, almost never, turned to our actual needs.

Xiong, Ying. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 52-54). / Abstracts in English and Chinese. / Abstract --- p.i / Abstract in Chinese --- p.ii / Acknowledgements --- p.iii / Contents --- p.iv / List of Figures --- p.vi / Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Background and Objective --- p.1 / Chapter 1.1.1. --- Risk Measurement --- p.2 / Value-at-Risk --- p.2 / Expected Shortfall --- p.3 / Computing Method based on Simulation --- p.4 / Chapter 1.1.2. --- Monte-Carlo Simulation --- p.5 / Chapter 1.2. --- Literature Review --- p.6 / Chapter 1.3. --- Structure of This Thesis --- p.8 / Chapter 2. --- Problem Formulation and Review of Past Methods --- p.10 / Chapter 2.1. --- Problem Formulation and Basic Setting --- p.10 / Chapter 2.2. --- Risk Measurement --- p.11 / Chapter 2.3. --- Uniform Sampling --- p.15 / Chapter 2.3.1. --- MSE Estimator --- p.16 / Chapter 2.4. --- Sequential Sampling --- p.17 / Chapter 3. --- Methodology: Our Approach --- p.18 / Chapter 3.1. --- Least-Squares Monte-Carlo Approach --- p.18 / Chapter 3.1.1. --- Framework --- p.19 / Chapter 3.2. --- Stochastic Mesh Method in risk measurement --- p.21 / Chapter 3.2.1. --- Framework --- p.21 / Chapter 3.2.2. --- With a series of cash flows --- p.26 / Chapter 3.2.3. --- Derive Marginal Density and Transition Density --- p.27 / Chapter 4. --- Numerical Experiments --- p.29 / Chapter 4.1. --- Experimental Setting --- p.29 / Chapter 4.2. --- Bias Comparison --- p.31 / Chapter 4.3. --- MSE Comparison --- p.33 / Chapter 4.4. --- Modified Least Square method --- p.44 / Chapter 5. --- Conclusion --- p.47 / Chapter A. --- Appendix A --- p.49 / Chapter A.1. --- Proof of Theorem 3.1 --- p.49 / Chapter A.2. --- Proof of Theorem 3.2 --- p.51

Lin, Minghong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 49-51). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- A Stochastic Framework --- p.5 / Chapter 2.1 --- Model Description --- p.5 / Chapter 2.2 --- Altruistic File Sharing System with Download Con- straint --- p.7 / Chapter 2.2.1 --- Model Formulation --- p.8 / Chapter 2.2.2 --- Steady State Analysis --- p.9 / Chapter 2.3 --- Altruistic File Sharing System with Download and Upload Constraints --- p.14 / Chapter 2.3.1 --- Model Formulation --- p.14 / Chapter 2.3.2 --- Steady State Analysis --- p.15 / Chapter 2.4 --- Incentive File Sharing via Coordinated Matching --- p.18 / Chapter 2.4.1 --- Without Incentive Mechanism --- p.18 / Chapter 2.4.2 --- With Incentive Mechanism --- p.19 / Chapter 2.5 --- Simulation --- p.23 / Chapter 3 --- An ISP-friendly Protocol --- p.28 / Chapter 3.1 --- Simple Mathematical Models --- p.28 / Chapter 3.1.1 --- Assumptions --- p.29 / Chapter 3.1.2 --- Homogeneous Case Analysis --- p.30 / Chapter 3.1.3 --- Heterogeneous Case Analysis --- p.31 / Chapter 3.1.4 --- Flash Crowd Analysis --- p.32 / Chapter 3.2 --- An ISP-friendly BitTorrent Protocol --- p.33 / Chapter 3.3 --- Performance Evaluation & Measurements --- p.36 / Chapter 3.3.1 --- Choice of the BitTorrent Client --- p.37 / Chapter 3.3.2 --- Experimental Setup --- p.37 / Chapter 3.3.3 --- Regular Peer Arrival --- p.38 / Chapter 3.3.4 --- Flash Crowd --- p.41 / Chapter 3.4 --- Black Hole Security Attack --- p.42 / Chapter 4 --- Related Work --- p.46 / Chapter 5 --- Conclusion --- p.48 / Bibliography --- p.49 / Appendix --- p.52

This thesis considers the effect of stochastic skew in the interest rate cap and currency option markets, where we observe obvious stochastic variation of skew of implied volatility curve over time. To develop option pricing models consistent with empirical evidence, we adopt the Wishart process to model both stochastic volatility and stochastic skew of the asset return and to price options in both markets. As an affine model, the model is analytically tractable. Some distributional properties of the models are studied. The key feature of our model is that, when compared with the multi-factor Heston model, which generates stochastic skew through its volatility processes, the Wishart process contains not only volatility processes, but also volatility-unrelated processes which provide extra freedom to model the variation of skew that is not captured by the volatility processes. Numerical experiments demonstrate that the Wishart model has greater flexibility to model stochastic skew than the multi-factor Heston model in both the interest rate cap market and currency option market. Finally, results of calibration to market data and model estimation demonstrate the superiority of the Wishart model to the multi-factor Heston model in the interest rate cap market. / Ng, Hon Yip. / Advisers: Kwai-Sun Leung; Duan Li. / Source: Dissertation Abstracts International, Volume: 73-09(E), Section: A. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 89-98). / 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 dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

The existing theories for predicting longitudinal dispersion in straight open channels have long been recognized as inadequate when applied to natural rivers. These theories tend to grossly underestimate dispersion in real streams since an important mixing mechanism due to nonuniform river cross-section variations is not explicitly taken into account. Recognizing the important role of stream irregularities on solute transport and the analytical difficulties of classical deterministic analysis, we develop a stochastic approach for analyzing solute transport in natural streams. Variations in river width and bed elevation are conveniently represented as one-dimensional random fields, characterized by their autocorrelation functions. Advection and dispersion due to the combined effect of turbulent diffusion and nonuniform flow are described by the stochastic solute transport equation. When boundary variations are small and statistically homogeneous, a stochastic spectral technique is used to obtain closed-form stochastic solutions. In particular, closed-form expressions are obtained for effective mean solute transport velocity and effective dispersion coefficient reflecting mixing due to flow variations both within the river cross-section and in the streamwise direction. The results show that the mean behavior of solute transport in a statistically irregular stream can be described as a gradient dispersion process. The effective mean transport velocity in natural rivers is smaller than that in a corresponding uniform channel, and the effective longitudinal dispersion coefficient in natural rivers can be considerably greater than that of uniform open channels. The discrepancy between uniform channels and natural rivers increases rapidly as the variances of river width and bed elevation increase, especially when the mean flow Froude number is high.