by Cheung Kei Wai. / Parallel title in Chinese. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 119-123). / Acknowledgement --- p.ii / Abstract --- p.iii / List of Abbreviations --- p.vii / List of Figure and Table Captions --- p.viii / List of Publication --- p.xiii / Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Normal Diffusion --- p.4 / Chapter 1.2 --- Statistical Origin of Anomalous Diffusion --- p.10 / Chapter 1.3 --- Various Models of Diffusion --- p.15 / Chapter 2. --- Scaling Theory of Hopping Transport in One Dimension --- p.21 / Chapter 2.1 --- "Scaling Exponents, the Models and their Solutions" --- p.22 / Chapter 2.2 --- Numerical Simulations --- p.32 / Chapter 2.3 --- Relations to Diffusion on Fractals and Hierarchical Structures --- p.37 / Chapter 2.4 --- Non-Markovian Nature of Results and Quasi-localization Effect --- p.42 / Chapter 3. --- Renormalization Group (RG) Analysis of the Problem --- p.43 / Chapter 3.1 --- The Length Scale Renormalization (LSR) --- p.45 / Chapter 3.2 --- Application of the LSR to Random Barrier Model --- p.52 / Chapter 3.2.1 --- The distribution of the renormalized coupling constants / Chapter 3.2.2 --- Behaviour of W and V under the LSR transformation / Chapter 3.3.3 --- The scaling exponents from LSR / Chapter 3.3 --- Scaling Form of Diffusion Front --- p.66 / Chapter 4. --- Diffusion in Hierarchical Systems --- p.72 / Chapter 4.1 --- Scaling and Probability Densities --- p.75 / Chapter 4.2 --- Exact Renormalization and Lattice Green Function --- p.82 / Chapter 4.3 --- The Range of Diffusion --- p.89 / Chapter 5. --- Biased Diffusion --- p.95 / Chapter 5.1 --- Scaling Theory --- p.98 / Chapter 5.1.1 --- Linear Response / Chapter 5.1.2 --- Diffusion-Drift Crossover / Chapter 5.2 --- Crossover Behaviour under an Additional Bias --- p.103 / Chapter 6. --- Physical Realizations and Related Problems --- p.108 / Chapter 6.1 --- Physical Realizations --- p.109 / Chapter 6.2 --- Equivalence with Random Resistor Network (RRN) Problems --- p.113 / Chapter 7. --- Discussion and Conclusion --- p.116 / References --- p.119 / Chapter Appendix A --- Derivation of Eq. (1.3.8) --- p.124 / Chapter Appendix B --- Derivation of Eq. (1.3.10) --- p.126 / Chapter Appendix C --- Sampling from a Distribution --- p.129 / Chapter Appendix D --- Scaling Form of P (s) in Ordered System --- p.130 / Chapter Appendix E --- Explicit Form of the Lattice Green Function --- p.132

擴散和對流擴散問題在高度異質性介質以及對流佔優擴散問題一直是一個具挑戰性的問題。眾所周知，對這些問題的數值計算需要相當數量的計算機內存和時間。然而，這些問題的解決方案通常包含一個粗糙的組成部分，它通常是我們感興趣的的量，而這個量一般可以用一個小數目的自由度代表。 / 現今有許多不同方法的目標是在計算粗糙的組件而不計算解的全部細節。在這篇論文中，我們將使用多尺度間斷伽遼金(Galerkin)方法來解決這問題。我們的方法是一種內部處罰間斷伽遼金方法。內部處罰間斷伽遼金方法已被證明是有效和準確的方法用來計算偏微分方程的數值解。我們的方法的一個顯著特點是解空間包含兩個部分，其中一個是傳統的多項式空間，另一個是計算粗糙的組件必不可少的包含分格結構的多尺度空間。我們在這篇論文中證明了該方法的穩定性。此外計算結果中表明，該方法能夠準確地捕捉高度異質性介質中的問題以及邊界和內部層對流佔優問題的解。 / Diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem has been a challenge problem for a long time. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. / There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. In this thesis, we will investigate a multisacle discontinuous Galerkin method to solve the problems. This method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Leung, Wing Tat. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 63-66). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 1.1 --- Overview --- p.6 / Chapter 1.2 --- Strong Formulation --- p.8 / Chapter 1.3 --- Weak Formulation --- p.8 / Chapter 1.4 --- Finite Element Method --- p.10 / Chapter 2 --- The discontinuous Galerkin (DG) method --- p.12 / Chapter 2.1 --- Introducion --- p.12 / Chapter 2.2 --- Model problem --- p.13 / Chapter 2.3 --- Interior Penalty Method --- p.14 / Chapter 3 --- The Multiscale Method --- p.18 / Chapter 3.1 --- Introducion --- p.18 / Chapter 3.2 --- Multiscale finite element method --- p.19 / Chapter 3.3 --- Homogenization --- p.20 / Chapter 3.4 --- Convergence --- p.23 / Chapter 3.4.1 --- Convergence for H < ε --- p.23 / Chapter 3.4.2 --- Convergence for H > ε --- p.24 / Chapter 3.5 --- Singularly perturbed convection-diffusion problem --- p.25 / Chapter 4 --- The multiscale-enhanced IPDG method --- p.27 / Chapter 4.1 --- Method description --- p.27 / Chapter 4.2 --- Stability --- p.31 / Chapter 4.3 --- Boundedness --- p.32 / Chapter 4.4 --- Convergence for elliptic problem --- p.37 / Chapter 5 --- Numerical Result --- p.46 / Chapter 5.1 --- Accuracy and convergence tests --- p.46 / Chapter 5.2 --- Some other cases with smooth coefficient --- p.49 / Chapter 5.3 --- Performance with internal or boundary layers --- p.52 / Chapter 5.4 --- Time-dependent problems --- p.55 / Chapter 5.5 --- Performance with more general media --- p.59

Online social and information networks, like Facebook and Twitter, exploit the influence of neighbors to achieve effective information sharing and spreading. The process that information is spread via the connected nodes in social and information networks is referred to as diffusion. In the literature, a number of diffusion models have been proposed for different applications like influential user identification and personalized recommendation. However, comprehensive studies to discover the hidden diffusion mechanisms governing the information diffusion using the data-driven paradigm are still lacking. This thesis research aims to design novel diffusion models with the structural and behaviorable dependency of neighboring nodes for representing social networks, and to develop computational algorithms to infer the diffusion models as well as the underlying diffusion mechanisms based on information cascades observed in real social networks. By incorporating structural dependency and diversity of node neighborhood into a widely used diffusion model called Independent Cascade (IC) Model, we first propose a component-based diffusion model where the influence of parent nodes is exerted via connected components. Instead of estimating the node-based diffusion probabilities as in the IC Model, component-based diffusion probabilities are estimated using an expectation maximization (EM) algorithm derived under a Bayesian framework. Also, a newly derived structural diversity measure namely dynamic effective size is proposed for quantifying the dynamic information redundancy within each parent component. The component-based diffusion model suggests that node connectivity is a good proxy to quantify how a node's activation behavior is affected by its node neighborhood. To model directly the behavioral dependency of node neighborhood, we then propose a co-activation pattern based diffusion model by integrating the latent class model into the IC Model where the co-activation patterns of parent nodes form the latent classes for each node. Both the co-activation patterns and the corresponding pattern-based diffusion probabilities are inferred using a two-level EM algorithm. As compared to the component-based diffusion model, the inferred co-activation patterns can be interpreted as the soft parent components, providing insights on how each node is influenced by its neighbors as reflected by the observed cascade data. With the motivation to discover a common set of the over-represented temporal activation patterns (motifs) characterizing the overall diffusion in a social network, we further propose a motif-based diffusion model. By considering the temporal ordering of the parent activations and the social roles estimated for each node, each temporal activation motif is represented using a Markov chain with the social roles being its states. Again, a two-level EM algorithm is proposed to infer both the temporal activation motifs and the corresponding diffusion network simultaneously. The inferred activation motifs can be interpreted as the underlying diffusion mechanisms characterizing the diffusion happening in the social network. Extensive experiments have been carried out to evaluate the performance of all the proposed diffusion models using both synthetic and real data. The results obtained and presented in the thesis demonstrate the effectiveness of the proposed models. In addition, we discuss in detail how to interpret the inferred co-activation patterns and interaction motifs as the diffusion mechanisms under the context of different real social network data sets.

All proposed methods are demonstrated by several 2D and 3D examples which have been extensively studied in the recent literature of topology optimization. / The fourth-order nonlinear parabolic C-H equations with elasticity are solved by a powerful nonlinear implicit mutigrid algorithm. To validate its correctness and efficiency, I first use it for the quadternary C-H equations without elasticity and get good results. To my best knowledge, it is the first simulation for such C-H models composed of more than three phases both in 2D and 3D. / The Optimization of Structural Topology (OST) is a breakthrough in product design because it can optimize size, shape and topology synchronously under different physical constraints. It has promising applications in industry ranging from automobile and aerospace engineering to micro electromechanical system. / Then this dissertation introduces a gradient flow in the norm of H-1 for the problem of multi-material structural topology optimization in 2/3D with a generalized Cahn-Hilliard (C-H) model with elasticity. Unlike the traditional C-H model applied to spinodal separation which only has bulk energy and interface energy, the generalized model couples the macroscopic elastic energy (mean compliance) into the total free energy. As a result, the grain morphology is not random islands or zigzag web-like objects but regular truss or bar structure. Although disturbed by elastic energy, the C-H system still keeps its two most important properties: mass conservation and energy dissipation. Therefore, it is unnecessary to compute the Lagrange multipliers for the volume constraints and make extra effort to minimize the mean compliance (elastic energy) for the optimization of structural topology. On the other hand, when pure phases separate from disordered original state, their boundaries will merge and split resulting in natural and flexible topology variation. Such aforementioned properties make the C-H model especially suitable for the problem of optimization of multi-material structural topology. / This dissertation also extends the famous Solid Isotropic Material with Penalization (SIMP) model from 2D to 3D for topology optimization of the structure with single material. A short 177-line Matlab code including 3D Finite Element Method (FEM), filter technique, Optimality Criteria (OC) algorithm and bisection method is listed in appendix A for clear understanding of this model in 3D. / This dissertation first substitutes the nonlinear diffusion method for filter process in the optimization of structural topology. Filtering has been a major technique used in a homogenization-based method for topology optimization of structures. It plays a key role in regularizing the basic problem into a well-behaved setting. But it has a drawback of smoothing effect around the boundary of material domain. A diffusion technique is presented here as a variational approach to the regularization of the topology optimization problem. A nonlinear or anisotropic diffusion process not only leads to a suitable problem regularization but also exhibits strong "edge"-preserving characteristics. Thus, it shows that the use of the nonlinear diffusions brings desirable effects of boundary preservation and even enhancement of lower-dimensional features such as flow-like structures. The proposed diffusion techniques have a close relationship with the diffusion methods and the phase-field methods of the fields of materials and digital image processing. / Zhou Shiwei. / "December 2005." / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6713. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 140-151). / 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.

Wu, Tao. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 56-61). / Abstracts in English and Chinese. / Chapter 1 --- Background and Organization --- p.7 / Chapter 2 --- Parallel Talbot method for solving partial integro- differential equations --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.2 --- Initial-boundary value problem --- p.11 / Chapter 2.3 --- Spatial discretization and semidiscrete problem --- p.12 / Chapter 2.4 --- Parallel Talbot method --- p.15 / Chapter 2.4.1 --- Φ-functions and Talbot quadrature --- p.15 / Chapter 2.4.2 --- Control on nonnormality and feasibility con- straints --- p.18 / Chapter 2.4.3 --- Optimal parameterization of parabolic Talbot contour --- p.22 / Chapter 2.5 --- Numerical experiments --- p.26 / Chapter 2.6 --- Conclusion --- p.32 / Chapter 3 --- Memory-reduction Monte Carlo method for pricing American options --- p.37 / Chapter 3.1 --- Introduction --- p.37 / Chapter 3.2 --- Exponential Levy processes and the full-storage method --- p.39 / Chapter 3.3 --- Random number generators --- p.41 / Chapter 3.4 --- The memory-reduction method --- p.43 / Chapter 3.5 --- Numerical examples --- p.45 / Chapter 3.5.1 --- Black-Scholes model --- p.46 / Chapter 3.5.2 --- Merton's jump-diffusion model --- p.48 / Chapter 3.5.3 --- Variance gamma model --- p.50 / Chapter 3.5.4 --- Remarks on the efficiency of the memory-reduction method --- p.52 / Chapter 3.6 --- Conclusion --- p.53 / Chapter 3.7 --- Appendix --- p.54

General diffusion processes (GDP), or Ito's processes, are potential candidates for the modeling of asset prices, interest rates and other financial quantities to cope with empirical evidence. This thesis considers the applications of general diffusions in finance and potential extensions. In particular, we focus on financial problems involving (optimal) stopping times. A typical example is the valuation of American options. We investigate the use of Laplace-Carson transform (LCT) in valuing American options, and discuss its strengthen and weaknesses. Homotopy analysis from topology is then introduced to derive closed-form American option pricing formulas under GDP. Another example is taken from optimal dividend policies with bankruptcy procedures, which is closely related to excursion time and occupation time of a general diffusion. With the aid of Fourier transform, we further extend the analysis to the case of multi-dimensional GDP by considering the currency option pricing with mean reversion and multi-scale stochastic volatility. / Zhao, Jing. / Adviser: Hoi-Ying Wong. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 97-105). / 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.