Au Siu Chun Brian. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 54-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Related sampling methods --- p.4 / Chapter 2.1 --- Introduction --- p.4 / Chapter 2.2 --- Gibbs sampler --- p.4 / Chapter 2.3 --- Importance sampling --- p.5 / Chapter 2.4 --- Sampling-importance resampling (SIR) --- p.7 / Chapter 2.5 --- Inverse Bayes formulae sampling (IBF sampling) --- p.10 / Chapter 3 --- Resampling weights in the SIR algorithm --- p.13 / Chapter 3.1 --- Resampling weights --- p.13 / Chapter 3.2 --- Problem in IBF sampling --- p.18 / Chapter 3.3 --- Adaptive finite mixture of distributions --- p.18 / Chapter 3.4 --- Allowing general distribution of 9 --- p.21 / Chapter 3.5 --- Examples and graphical comparison --- p.24 / Chapter 4 --- Resampling weight in Gibbs sampling --- p.32 / Chapter 4.1 --- Introduction --- p.32 / Chapter 4.2 --- Use Gibbs sampler to obtain ISF --- p.33 / Chapter 4.3 --- How many iterations? --- p.36 / Chapter 4.4 --- Applications --- p.41 / Chapter 4.4.1 --- The genetic linkage model --- p.41 / Chapter 4.4.2 --- Example --- p.43 / Chapter 4.4.3 --- The probit binary regression model --- p.44 / Chapter 5 --- Conclusion and discussion --- p.49 / Appendix A: Exact bias of the SIR --- p.52 / References --- p.54

Hui Ka Ming = 非對稱Hubbard模型之量子蒙地卡羅與精準對角化研究 / 許嘉明. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 85-88). / Text in English; abstracts in English and Chinese. / Hui Ka Ming = Fei dui cheng Hubbard mo xing zhi liang zi Mengdi Kaluo yu jing zhun dui jiao hua yan jiu / Xu Jiaming. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Asymmetric Hubbard Model And Its Physical Background / Chapter 2.1 --- Physical Motivation of the Study --- p.4 / Chapter 2.2 --- Symmetry Properties --- p.5 / Chapter 2.2.1 --- Particle-Hole Transformation --- p.5 / Chapter 2.2.2 --- U(l) Group --- p.5 / Chapter 2.2.3 --- SU(2) Group --- p.6 / Chapter 2.3 --- Relations with Other Models --- p.7 / Chapter 2.3.1 --- Falicov-Kimball Model --- p.7 / Chapter 2.3.2 --- Asymmetric t-J and Heisenberg Model --- p.7 / Chapter 2.3.3 --- Anderson and Kondo Model。 --- p.8 / Chapter 3 --- Exact Diagonalization and Density Matrix Renormalization Group --- p.11 / Chapter 3.1 --- Exact Diagonalization --- p.12 / Chapter 3.1.1 --- Second Quantization Representation --- p.13 / Chapter 3.1.2 --- Binary Representation Of Quantum States --- p.15 / Chapter 3.1.3 --- Two-Table Method --- p.15 / Chapter 3.1.4 --- Generation of Matrix Elements --- p.18 / Chapter 3.1.5 --- Lanczos Method --- p.20 / Chapter 3.1.6 --- Continued Fraction Dynamics At T=0 --- p.22 / Chapter 3.2 --- Density Matrix Renormalization Group --- p.23 / Chapter 3.2.1 --- Infinite system DMRG --- p.24 / Chapter 3.2.2 --- DMRG and Quantum Entanglement --- p.27 / Chapter 4 --- Determinant Quantum Monte Carlo And Finite Temperature Green Function --- p.29 / Chapter 4.1 --- Introduction --- p.29 / Chapter 4.2 --- General Scenario of Fermionic Monte Carol Method --- p.30 / Chapter 4.3 --- Review on Monte Carlo Method for Ising Model --- p.31 / Chapter 4.3.1 --- Ising Model --- p.31 / Chapter 4.3.2 --- Metropolis Algorithm --- p.31 / Chapter 4.3.3 --- Measurement --- p.32 / Chapter 4.3.4 --- Near The Critical Points --- p.33 / Chapter 4.4 --- Determinant Quantum Monte Carlo --- p.33 / Chapter 4.4.1 --- Suzuki-Trotter Decomposition --- p.34 / Chapter 4.4.2 --- Hubbard-St rant onovich Transformation --- p.34 / Chapter 4.5 --- Green Functions in DQMC And Wick's Theorem --- p.37 / Chapter 4.5.1 --- Equal-'Time' Green Functions --- p.37 / Chapter 4.5.2 --- Unequal-'Time' Green Functions --- p.38 / Chapter 4.5.3 --- Wick's Theorem --- p.38 / Chapter 4.6 --- Practical Consideration of DQMC --- p.39 / Chapter 4.6.1 --- Metropolis for DQMC --- p.39 / Chapter 4.6.2 --- Updating the Equal-time Green function --- p.41 / Chapter 4.6.3 --- Matrix Multiplication Stabilization --- p.42 / Chapter 4.6.4 --- A Survey In Negative Sign Problem --- p.43 / Chapter 4.6.5 --- Insight from Feynman --- p.44 / Chapter 4.6.6 --- Sign Problem As A NP-hard problem --- p.45 / Chapter 4.6.7 --- Personal Account on the Problem --- p.46 / Chapter 5 --- Dynamical Mean Field Theory --- p.47 / Chapter 5.1 --- Classical Mean Field Theory . --- p.49 / Chapter 5.2 --- DMFT as an Impurity Problem --- p.52 / Chapter 5.3 --- Scaling of Hopping Integral and Self-consistent Condition --- p.53 / Chapter 5.4 --- A QMC Impurity Solver --- p.55 / Chapter 5.5 --- Further development of DMFT --- p.56 / Chapter 6 --- Results and Discussion --- p.57 / Chapter 6.1 --- Physical Observable --- p.57 / Chapter 6.2 --- Methodology of the Studies --- p.58 / Chapter 6.3 --- Results --- p.59 / Chapter 6.4 --- Discussion and Suggestion --- p.83 / Chapter A --- DMRG of tight-binding model --- p.89 / Chapter B --- Scaling and Density of State of D-dimensional Hubbard Model --- p.94

To value any fixed income security one needs to evaluate the discounted expected cash flows according to an arbitrage free interest rate model. In the case of mortgage-backed securities the future cash flows are uncertain due to mortgagors exercise of their prepayment options. The present project considers prepayments which result from interest rate dependent complete refinancing of mortgages in a pool. The rate of refinancing is modeled as an arbitrary, user defined function of current and past interest rates. This enables the inclusion of refinancing rates that depend on not only on the current level of interest rates but also on the trend of the interest rates and that may also exhibit burnout effects due to past periods of low interest rates. The resulting cash flows depend on the entire past of the path that the interest rates took to get to the current level. The Black-Derman-Toy arbitrage free binomial tree is used to model the underlying interest rates. This is a single-factor market price consistent model which also allows the specification of the observed volatilities. Monte Carlo methodology is used to simulate random paths in the interest rate tree to evaluate the cash flows along the path. A computer program written in MAPLE implements the entire process.

This paper aims to calculate the all-inclusive European option price based on XVA model numerically. For European type options, the XVA can be calculated as so- lution of a BSDE with a specific driver function. We use the FT scheme to find a linear approximation of the nonlinear BSDE and then use linear regression Monte Carlo method to calculate the option price.

本論文主要研究了磁性雜質在三種不同的石墨烯體系中的性質。這三個體系分別是：伯爾納堆垛（Bernal stacked）結構的雙層石墨烯(bilayer graphene)，包含空位的雙層石墨烯和zigzag 型石墨烯納米帶(graphene nanoribbon)。本文中主要運用的數值方法為Hirsch-Fye 量子蒙特卡羅方法 （quantum Monte Carlo method）和貝葉斯最大熵方法（Bayesian Maximum Entropy method）。 / 在論文的第二章會詳細介紹這兩種方法。 我們用量子蒙特卡羅方法得到的結果是準確的，因為原則上我們可以計算無窮大的體系， 並且不需要采用任何近似去處理多體問題。 / 第三章，我們討論磁性雜質在伯爾納堆垛結構的雙層石墨烯中的性質。我們主要考慮兩種情況：磁性雜質分別位於A 亞晶格和B 亞晶格上。我們發現類似于單層石墨烯中的情況，隨著化學勢的變化，雜質原子的磁矩在一定的能量範圍內可調。但是由於雙層石墨烯中的兩套亞晶格不等價，雜質的性質很大程度上取決於雜質的位置，並且其區別隨著s-d 混合強度的增加和d 電子的關聯能的減少而變得更加明顯。我們也討論了雜質原子的譜密度和s-d 電子之間的關聯函數。我們的所有結果都體現雜質性質對空間位置的相同依賴關係。 / 在第四章，我們研究在伯爾納堆垛結構的雙層石墨烯中磁性原子在空位附近的性質。在雙層石墨烯系統中，空位引起的局域態的性質依舊取決於空位所屬的亞晶格。這也是由於兩套晶格的不等價性。我們討論雜質在空位周圍時雜質性質對空位所屬亞晶格的依賴關係，以及兩種缺陷之間的相對位移對雜質磁性的影響。當磁性原子附著在空位的最近鄰格點上時，A 亞晶格上的空位會對雜質原子的磁矩有更強的抑制。而隨著磁性原子和空位之間的距離增加，局域態對磁性原子的影響迅速衰減，並且B 亞晶格上的空位對雜質的影響相對長程一些。 / 在第五章，我們討論兩個磁性雜質在zigzag 型石墨烯納米帶邊緣上的間接互作用隨化學式的變化。由於在zigzag 型石墨烯納米帶邊緣有局域的零能態，雜質的磁矩被嚴重抑制，兩個雜質原子之間的自旋-自旋關聯函數也與石墨烯中的行為有很大區別。有趣的是，當兩個雜質附著在兩個最近鄰的碳原子上時，隨著化學式的降低，雜質之間的反鐵磁關聯會有一個顯著增強再降低的過程。我們也討論了自旋關聯隨著距離的變化，并發現關聯強度隨著距離的變化迅速衰減。 / 最後再第六章，我們會本論文中的內容進行總結和討論。 / In this research thesis, we study the magnetic properties of Anderson impurities in three different graphene based systems: pure Bernal stacked bilayer graphene, Bernal stacked bilayer graphene with a vacancy and graphene ribbon with zigzag edges. Quantum Monte Carlo method based on the Hirsch-Fye algorithms are used to obtain the basic thermodynamic quantities, and the Bayesian maximum entropy method is used to obtain the spectral densities of the impurity sites. We discuss the numerical methods in chapter 2. / In chapter 3, we investigate the local moment formation of a magnetic impurity in Bernal stacked bilayer graphene. In the two cases we study, impurity is placed on top of the two different sublattices in bilayer system. We find that similar to the monolayer case, magnet moment of the impurity could still be tuned in a wide range through changing the chemical potential. However, the property of the impurity depends strongly on its location due to the broken symmetry between the two sublattices. This difference becomes more apparent with the increase in the hybridization and decrease in the on-site Coulomb repulsion. Additionally, we calculate the impurity spectral densities and the correlation functions between the impurity and the conduction-band electrons. All the computational results show the same spatial dependence on the location of the impurity. / In chapter 4, we address the issue of single magnetic adatom located in the vicinity of a vacancy in bilayer graphene with Bernal stacking. In bilayer system, the property of vacancy induced localized states depends on whether the vacancy belongs to A or B sublattice. The magnetic impurity is placed in the vicinity of the vacancy, and the dependence of its magnetic property on the location of the vacancy is discussed. We switch the values of the chemical potential and study the basic thermodynamic quantities and the correlation functions between the magnetic adatom and the carbon sites. When the magnetic adatom is located on the nearest site of the vacancy, the local moment is more strongly suppressed if the vacancy belongs to the sublattice A. The influence of the zero-energy localized states decays fast as the displacement between the two defects increases, and the effect of B vacancy on the local moment of magnetic adatom is relatively long ranged. / In chapter 5, we examine the behavior of the indirect magnetic interactions of two magnetic impurities on the zigzag edge of graphene ribbon as a function of chemical potential. We find that the spin-spin correlation between two adatoms located on the nearest sites are drastically suppressed at the Dirac energy, and as we lower the chemical potential, the antiferromagnetic correlation is first enhanced and then decreased in values. If the two adatoms are adsorbed on the sites belong to the same sublattice, we find similar behavior of spin-spin correlation except for a cross over from ferromagnetic to antiferromagentic correlation in the vicinity of zero-energy. We also calculated the weight of the wave functions and basic thermodynamic quantities for various values of parameters, and compare the results with their counterpart in bulk graphene. / Finally in chapter 6, we summarize our results. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Sun, Jinhua = 石墨烯體系中磁性雜質的量子蒙特卡羅研究 / 孫金華. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 92-[100]). / Abstract also in Chinese. / Sun, Jinhua = Shi mo xi ti xi zhong ci xing za zhi de liang zi Mengte Kaluo yan jiu / Sun Jinhua. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Electronic properties of graphene --- p.1 / Chapter 1.1.1 --- Band structure --- p.1 / Chapter 1.1.2 --- Dirac fermions --- p.4 / Chapter 1.1.3 --- Mesoscopic effects --- p.4 / Chapter 1.1.4 --- Superconductivity in graphene --- p.6 / Chapter 1.2 --- Potential applications --- p.7 / Chapter 1.2.1 --- In semiconductors --- p.8 / Chapter 1.2.2 --- In spintronics --- p.10 / Chapter 1.3 --- Magnetic impurities in graphene based systems --- p.11 / Chapter 1.4 --- Outline of this thesis --- p.13 / Chapter 2 --- Numerical method --- p.15 / Chapter 2.1 --- Quantum Monte Carlo method based on Hirsch-Fye algorithm . --- p.15 / Chapter 2.1.1 --- Anderson impurity model Hamiltonian --- p.16 / Chapter 2.1.2 --- Key steps of Hirsch-Fye algorithm --- p.16 / Chapter 2.1.3 --- Basic thermodynamic quantities --- p.23 / Chapter 2.1.4 --- Extended Hirsch-Fye algorithm --- p.24 / Chapter 2.1.5 --- Two-impurity case --- p.25 / Chapter 2.1.6 --- Relation to the determinant quantum Monte Carlo method based on BSS algorithm --- p.27 / Chapter 2.2 --- Bayesian maximum entropy method --- p.30 / Chapter 2.2.1 --- Bayesian inference --- p.31 / Chapter 2.2.2 --- Maximum entropy analysis --- p.34 / Chapter 3 --- Magnetic impurity in Bernal stacked bilayer graphene --- p.37 / Chapter 3.1 --- Introduction --- p.37 / Chapter 3.2 --- Density of states and local density of states --- p.38 / Chapter 3.3 --- Results --- p.43 / Chapter 3.3.1 --- Basic thermodynamic quantities --- p.43 / Chapter 3.3.2 --- Spectral densities --- p.48 / Chapter 3.3.3 --- Correlation functions --- p.50 / Chapter 3.4 --- Summary --- p.52 / Chapter 4 --- Magnetic impurity in the vicinity of a vacancy in bilayer graphene --- p.54 / Chapter 4.1 --- Introduction --- p.54 / Chapter 4.2 --- Zero-energy localized states in the vicinity of vacancy --- p.56 / Chapter 4.2.1 --- Monolayer graphene with a vacancy --- p.57 / Chapter 4.2.2 --- Bilayer graphene with vacancy --- p.58 / Chapter 4.3 --- Results --- p.60 / Chapter 4.3.1 --- Basic thermodynamic quantities --- p.61 / Chapter 4.3.2 --- Spin-spin and charge-charge correlations --- p.68 / Chapter 4.4 --- Summary --- p.69 / Chapter 5 --- Indirect exchange of magnetic impurities in zigzag graphene ribbon --- p.71 / Chapter 5.1 --- Introduction --- p.71 / Chapter 5.2 --- Density of states and local density of states --- p.73 / Chapter 5.3 --- Results --- p.76 / Chapter 5.3.1 --- Basic thermodynamic quantities --- p.77 / Chapter 5.3.2 --- Spin-spin correlation --- p.80 / Chapter 5.3.3 --- Weight of different components of the d electron wave function --- p.84 / Chapter 5.3.4 --- Comparison : two magnetic impurities in bulk graphene sheet --- p.85 / Chapter 5.4 --- Summary --- p.88 / Chapter 6 --- Summary and discussions --- p.89 / Bibliography --- p.92 / Chapter A --- Derivation of the input Green’s function G0(τ ) in Bernal stacked bilayer graphene --- p.101 / Chapter B --- Input Green’s function for U = 0 in bilayer graphene in the presence of a vacancy --- p.106

In this thesis we make four contributions to the state of the art in numerical lattice simulations of quantum chromodynamics (QCD). First, we present the most detailed investigation yet of the autocorrelations of topological observations in hybrid Monte Carlo simulations of QCD and of the effects of the boundary conditions on these autocorrelations. This results in a numerical criterion for deciding when open boundary conditions are useful for reducing these autocorrelations, which are a major barrier to reliable calculations at fine lattice spacings. Second, we develop a dislocation-enhancing determinant, and demonstrate that it reduces the autocorrelation time of the topological charge. This alleviates problems with slow topological tunneling at fine lattice spacings, enabling simulations on fine lattices to be completed with much less computational effort. Third, we show how to apply the recently developed zMöbius technique to hybrid Monte Carlo evolutions with domain wall fermions, achieving nearly a factor of two speedup in the the light quark determinant, the single most expensive part of the calculation. The dislocation-enhancing determinant and the zMöbius technique have enabled us to begin simulations of fine ensembles with four flavors of dynamical domain wall quarks. Finally, we show how to include the previously-neglected G1 operator in nonperturbative renormalization of the ∆S = 1 effective weak Hamiltonian on the lattice. This removes an important systematic error in lattice calculations of weak matrix elements, in particular the important K → ππ decay.

In this thesis, we analyze the computational problem of estimating financial risk in nested Monte Carlo simulation. An outer simulation is used to generate financial scenarios, and an inner simulation is used to estimate future portfolio values in each scenario. Mean squared error (MSE) for standard nested simulation converges at the rate $k^{-2/3}$, where $k$ is the computational budget. In the first part of this thesis, we focus on one risk measure, the probability of a large loss, and we propose a new algorithm to estimate this risk. Our algorithm sequentially allocates computational effort in the inner simulation based on marginal changes in the risk estimator in each scenario. Theoretical results are given to show that the risk estimator has an asymptotic MSE of order $k^{-4/5+\epsilon}$, for all positive $\epsilon$, that is faster compared to the conventional uniform inner sampling approach. Numerical results consistent with the theory are presented. In the second part of this thesis, we introduce a regression-based nested Monte Carlo simulation method for risk estimation. The proposed regression method combines information from different risk factor realizations to provide a better estimate of the portfolio loss function. The MSE of the regression method converges at the rate $k^{-1}$ until reaching an asymptotic bias level which depends on the magnitude of the regression error. Numerical results consistent with our theoretical analysis are provided and numerical comparisons with other methods are also given. In the third part of this thesis, we propose a method based on weighted regression. Similar to the unweighted regression method, the MSE of the weighted regression method converges at the rate $k^{-1}$ until reaching an asymptotic bias level, which depends on the size of the regression error. However, the weighted approach further reduces MSE by emphasizing scenarios that are more important to the calculation of the risk measure. We find a globally optimal weighting strategy for general risk measures in an idealized setting. For applications, we propose and test a practically implementable two-pass method, where the first pass uses an unweighted regression and the second pass uses weights based on the first pass.

Leung Man-Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 37-41). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Basic concepts of survival data --- p.3 / Chapter 1.2 --- Missing Complete at Random (MCAR) --- p.8 / Chapter 1.3 --- Missing at Random (MAR) --- p.9 / Chapter 2 --- The maximaization of the marginal likelihood --- p.11 / Chapter 2.1 --- Survival function --- p.11 / Chapter 2.2 --- Missing covariate pattern --- p.13 / Chapter 2.3 --- Set of survival time with rank restrictions --- p.13 / Chapter 2.4 --- Marginal likelihood --- p.14 / Chapter 2.5 --- Score function --- p.15 / Chapter 3 --- The MCMC stochastic approximation approach --- p.17 / Chapter 4 --- Simulations Studies --- p.22 / Chapter 4.1 --- MCAR : Simulation 1 --- p.23 / Chapter 4.2 --- MCAR : Simulation 2 --- p.24 / Chapter 4.3 --- MAR : Simulation 3 --- p.26 / Chapter 4.4 --- MAR : Simulation 4 --- p.27 / Chapter 5 --- Example --- p.30 / Chapter 6 --- Discussion --- p.33 / Appendix --- p.35 / Bibliography --- p.37

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