Immune cells must accurately interpret environmental signals to make robust cell fate decisions and control inflammatory signalling. Many signals (e.g. Tumor Necrosis Factor alpha (TNFα) or interferon gamma (IFNγ)) converge on just a few key signalling systems such as Nuclear Factor kappa B (NF-κB) or Signal Transducers and Activators of Transcription (STAT), which exhibit complex activation dynamics that control patterns of downstream gene expression. Often, seemingly identical cells show heterogeneous or random behaviour to a common stimulus. Therefore, a key question is how can immune cells coordinate inflammatory signalling in the presence of this noise. The NF-kB system dynamics were studied in response to rapidly changing inflammatory signals. It was shown that pulsed TNFa cytokine stimulations induced digital single-cell NF-kB responses, with only a fraction of cells able to respond to repeated pulses. These responses appeared to be reproducible in individual cells, but heterogeneous in the population. Mathematical models of the NF-kB signalling network suggested that single-cell responses were governed through a refractory state potentially encoded via 'extrinsic' noise in the levels of signalling molecules related to the TNFa signal transduction pathway. Such signal processing enabled robust and reproducible single cell responses and maintained acute tissue-level signalling, with fewer cells responding to shorter pulsing intervals. The NF-kB system is involved in effector cytokine propagation in response to pathogen infection. It was shown that in macrophages, the dose of TLR4 stimulation (mimicking the pathogen infection) was encoded in graded (yet heterogeneous) NF-kB dynamics in single cells. This resulted in analogue inflammatory gene expression patterns in the population. However, individual cells substantially differed in their ability to encode TLR4 signal and to regulate TNFa expression, which was explained by extrinsic noise in the NF-kB system. Quantitative mathematical modelling showed that tissue-level environment modulates heterogeneous single-cell TNFa outputs; by effectively removing it from circulation. This may determine the interaction distance between tissue-resident immune cells to enable propagation of cellular inflammation. Heterogeneity of single cell macrophage signalling was also observed in NF-kB and STAT1 system responses to a range of IFN stimulation doses. Although each system showed substantial variability between cells, their responses were surprisingly well correlated in individual cells. It was however apparent (based on gene expression studies) that individual cells may not be able to precisely discriminate different IFNg doses. Overall, this work suggests that heterogeneity in the NF-kB (and other) regulatory networks might be a part of an inherent design motif in the inflammatory response, which enables robust control of the tissue-level inflammatory response by preventing homogeneous and thus potentially harmful activation.

Choy Man Wah Minnie. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 83). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Analysis of APT using SEM --- p.3 / Chapter 2.1 --- The APT model --- p.3 / Chapter 2.2 --- The structural equation model approach --- p.5 / Chapter 3 --- Incorporating stochastic constraints into the SEM analysis of APT --- p.8 / Chapter 3.1 --- Introduction --- p.8 / Chapter 3.2 --- Bayesian analysis of stochastic constraints --- p.9 / Chapter 3.3 --- Three types of structures for T I --- p.10 / Chapter 3.3.1 --- Case 1: T = (σ2Imxm --- p.10 / Chapter 3.3.2 --- "Case 2: r is a diagonal matrix with diagonal elements σ2j for = 1， …,m" --- p.13 / Chapter 3.3.3 --- Case 3: Γ is a general positive definite matrix --- p.14 / Chapter 3.4 --- Estimation of parameters using the Mx program --- p.16 / Chapter 4 --- Empirical study on Hong Kong stock market --- p.17 / Chapter 4.1 --- Information of data --- p.17 / Chapter 4.2 --- Source of data --- p.17 / Chapter 4.3 --- Lisrel model with exact constraints --- p.19 / Chapter 4.3.1 --- The resultant model --- p.20 / Chapter 4.4 --- Lisrel model with stochastic constraints --- p.21 / Chapter 4.4.1 --- Result --- p.22 / Chapter 5 --- Simulation study --- p.35 / Chapter 5.1 --- Simulation design --- p.35 / Chapter 5.2 --- Simulation procedure --- p.40 / Chapter 5.3 --- Simulation result --- p.41 / Chapter 5.3.1 --- Sample size --- p.41 / Chapter 5.3.2 --- Analysis methods (constraints) --- p.42 / Chapter 5.3.3 --- Factor loadings --- p.43 / Chapter 5.3.4 --- Factor correlation matrix --- p.43 / Chapter 5.3.5 --- Risk premia --- p.43 / Chapter 5.3.6 --- Overall result --- p.44 / Chapter 6 --- Conclusion and discussion --- p.45 / Appendices --- p.46 / Chapter A --- Simulation result - Mean --- p.47 / Chapter B --- Simulation result - Bias --- p.56 / Chapter C --- Simulation result - RMSE --- p.65 / Chapter D --- Mx input script --- p.74 / Chapter D.l --- Stochastic constraints Case 1 --- p.74 / Chapter D.2 --- Stochastic constraints Case 2 --- p.77 / Chapter D.3 --- Stochastic constraints Case 3 --- p.80 / Bibliography --- p.83

Inverse problems are mathematically and numerically very challenging due to their inherent ill-posedness in the sense that a small perturbation of the data may cause an enormous deviation of the solution. Regularization methods have been established as the standard approach for their stable numerical solution thanks to the ground-breaking work of late Russian mathematician A.N. Tikhonov. However, existing studies mainly focus on general-purpose regularization procedures rather than exploiting mathematical structures of specific problems for designing efficient numerical procedures. Moreover, the stochastic nature of data noise and model uncertainties is largely ignored, and its effect on the inverse solution is not assessed. This thesis attempts to design some problem-specific efficient numerical methods for the Robin inverse problem and to quantify the associated uncertainties. It consists of two parts: Part I discusses deterministic methods for the Robin inverse problem, while Part II studies stochastic numerics for uncertainty quantification of inverse problems and its implication on the choice of the regularization parameter in Tikhonov regularization. / Key Words: Robin inverse problem, variational approach, preconditioning, Modica-Motorla functional, spectral stochastic approach, Bayesian inference approach, augmented Tikhonov regularization method, regularization parameter, uncertainty quantification, reduced-order modeling / Part I considers the variational approach for reconstructing smooth and nonsmooth coefficients by minimizing a certain functional and its discretization by the finite element method. We propose the L2-norm regularization and the Modica-Mortola functional from phase transition for smooth and nonsmooth coefficients, respectively. The mathematical properties of the formulations and their discrete analogues, e.g. existence of a minimizer, stability (compactness), convexity and differentiability, are studied in detail. The convergence of the finite element approximation is also established. The nonlinear conjugate gradient method and the concave-convex procedure are suggested for solving discrete optimization problems. An efficient preconditioner based on the Sobolev inner product is proposed for justifying the gradient descent and for accelerating its convergence. / Part II studies two promising methodologies, i.e. the spectral stochastic approach (SSA) and the Bayesian inference approach, for uncertainty quantification of inverse problems. The SSA extends the variational approach to the stochastic context by generalized polynomial chaos expansion, and addresses inverse problems under uncertainties, e.g. random data noise and stochastic material properties. The well-posedness of the stochastic variational formulation is studied, and the convergence of its stochastic finite element approximation is established. Bayesian inference provides a natural framework for uncertainty quantification of a specific solution by considering an ensemble of inverse solutions consistent with the given data. To reduce its computational cost for nonlinear inverse problems incurred by repeated evaluation of the forward model, we propose two accelerating techniques by constructing accurate and inexpensive surrogate models, i.e. the proper orthogonal decomposition from reduced-order modeling and the stochastic collocation method from uncertainty propagation. By observing its connection with Tikhonov regularization, we propose two functionals of Tikhonov type that could automatically determine the regularization parameter and accurately detect the noise level. We establish the existence of a minimizer, and the convergence of an alternating iterative algorithm. This opens an avenue for designing fully data-driven inverse techniques. / This thesis considers deterministic and stochastic numerics for inverse problems associated with elliptic partial differential equations. The specific inverse problem under consideration is the Robin inverse problem: estimating the Robin coefficient of a Robin boundary condition from boundary measurements. It arises in diverse industrial applications, e.g. thermal engineering and nondestructive evaluation, where the coefficient profiles material properties on the boundary. / Jin, Bangti. / Adviser: Zou Jun. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3541. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 174-187). / 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.

We study optimization algorithms for the finite sum problems frequently arising in machine learning applications. First, we propose novel variants of stochastic gradient descent with a variance reduction property that enables linear convergence for strongly convex objectives. Second, we study distributed setting, in which the data describing the optimization problem does not fit into a single computing node. In this case, traditional methods are inefficient, as the communication costs inherent in distributed optimization become the bottleneck. We propose a communication-efficient framework which iteratively forms local subproblems that can be solved with arbitrary local optimization algorithms. Finally, we introduce the concept of Federated Optimization/Learning, where we try to solve the machine learning problems without having data stored in any centralized manner. The main motivation comes from industry when handling user-generated data. The current prevalent practice is that companies collect vast amounts of user data and store them in datacenters. An alternative we propose is not to collect the data in first place, and instead occasionally use the computational power of users' devices to solve the very same optimization problems, while alleviating privacy concerns at the same time. In such setting, minimization of communication rounds is the primary goal, and we demonstrate that solving the optimization problems in such circumstances is conceptually tractable.

Weiyan Ge. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 64-68). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction of Power Control Problem --- p.2 / Chapter 1.1.1 --- Classification of Power Control Problem --- p.2 / Chapter 1.1.2 --- Previous Works --- p.7 / Chapter 1.2 --- Scope and Contribution of the Thesis --- p.11 / Chapter 1.3 --- Organization of the Thesis --- p.12 / Chapter 2 --- Background --- p.14 / Chapter 2.1 --- Stochastic Approximation --- p.14 / Chapter 2.2 --- Lognormal Distribution --- p.17 / Chapter 2.2.1 --- Definition and Properties --- p.17 / Chapter 2.2.2 --- Application on Radio Propagation --- p.18 / Chapter 3 --- System Model and Centralized Algorithm --- p.21 / Chapter 3.1 --- System Model --- p.21 / Chapter 3.2 --- Problem Statement and the Centralized Algorithm --- p.25 / Chapter 4 --- Proposed Stochastic Power Control Algorithm --- p.30 / Chapter 4.1 --- Proposed Power Control Algorithm --- p.30 / Chapter 4.2 --- Basic Properties of the Algorithm --- p.33 / Chapter 4.3 --- Convergence Property --- p.38 / Chapter 5 --- Numerical Results --- p.44 / Chapter 5.1 --- Simulation Model --- p.44 / Chapter 5.2 --- Numerical Results --- p.47 / Chapter 6 --- Conclusions And Future Works --- p.58 / Chapter 6.1 --- Conclusions --- p.58 / Chapter 6.2 --- Future Works --- p.60 / Chapter A --- Basic Properties of LOG-Distribution --- p.62 / Bibliography --- p.64

Title: Almost stochastic dominance Author: Adam Štefánik Department: Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Department of Probability and Mathematical Statistics, MFF UK Abstract: In the presented work we study the almost stochastic dominance and it's properties. Almost stochastic dominance is a relaxation of stochastic dominance. Almost stochastic dominance also deals with paradox situations occurring in case of stochastic dominance. This is a situation when stochastic dominance determines indifferent relation- ship between two portfolios, but in fact almost all investors can choose the better one. The original almost stochastic dominance presented by Leshno and Levy (2002) is compu- tationally expensive. Lizyayev and Ruszczy'nski (2012) suggested an alternative approach. This work introduces both approaches. The most interesting part of this work is a search for efficient portfolio with respect to the almost stochastic dominance by the simple linear programming. Lizyayev and Ruszczy'nski (2012) approach is applied to Kopa and Chovanec (2008) quantile approach for portfolio efficiency testing with respect to second order stochastic dominance. Keywords: almost stochastic dominance, efficiency, CVaR

Apresentamos um modelo estocástico para a evolução de uma espécie pelo processo de seleção natural. Compreender bem o processo evolutivo é de fundamental importância para a biologia, pois é através dele que as espécies e a vida se transformaram ao longo do tempo até chegarmos no mundo como conhecemos hoje. Detalhamos um resultado encontrado na literatura, e também introduzimos algumas variações e sugestões para aprimorar a modelagem original. O modelo proposto é interessante por conta de sua simplicidade e capacidade de capturar aspectos qualitativos esperados segundo as teorias biológicas. / We present a stochastic model for the evolution of a species by natural selection. A good understanding of the evolutionary process is fundamental for the biological sciences, since it describes how life and all species developed until we reached the world as we know today. We show in details a result available on the literature, and also introduce some variations and suggestions in order to improve the original modeling. The model presented here is interesting due to its simplicity and ability to reproduce some qualitative aspects expected from the biological theories.

In the cellular networks, to support the increasing data rate requirements, many base stations (BSs) with low transmit power and small coverage area are deployed in addition to classical macro cell BSs. Low power nodes, such as micro, pico, and femto nodes (indoor and outdoor), which complement the conventional macro networks, are placed primarily to increase capacity in hotspots (such as shopping malls and conference centers) and to enhance coverage of macro cells near the cell boundary. Combining macro and small cells results in heterogeneous networks (HetNets). An accurate node (BS or user equipment (UE)) model is important in the research, design, evaluation, and deployment of 5G HetNets. The distance between transmitter (TX), receiver (RX), and interferer determines the received signal power and interference signal power. Therefore, the spatial placement of BSs and UEs greatly impacts the performance of cellular networks. However, the investigation on the spatial distribution of UE is limited, though there is ample research on the topic of the spatial distribution of BS. In HetNets, UEs tend to cluster around BSs or social attractors (SAs). The spatial distribution of these UEs is non-uniform. Therefore, the analysis of the impact of non-uniformity of UE distribution on HetNets is essential for designing efficient HetNets. This thesis presents a non-uniform user distribution model based on the existing K-tier BS distribution. Our proposed non-uniform user distribution model is such that a Poisson cluster process with the cluster centers located at SAs in which SAs have a base station offset with their BSs. There are two parameters (cluster radius and base station offset) the combination of which can cover many possible non-uniformity. The heterogeneity analysis of the proposed nonuniform user distribution model is also given. The downlink performance analysis of the designed non-uniform user model is investigated. The numerical results show that our theoretical results closely match the simulation results. Moreover, the effect of BS parameters of small cells such as BS density, BS cell extension bias factor, and BS transmit power is included. At the same time, the uplink coverage probability by the theoretical derivation is also analyzed based on some simplifying assumptions as a result of the added complexity of the uplink analysis due to the UEs’ mobile position and the uplink power control. However, the numerical results show a small gap between the theoretical results and the simulation results, suggesting that our simplifying assumptions are acceptable if the system requirement is not very strict. In addition to the effect of BS density, BS cell extension bias factor, and BS transmit power, the effect of fractional power control factor in the uplink is also introduced. The comparison between the downlink and the uplink is discussed and summarized at the end. The main goal of this thesis is to develop a comprehensive framework of the non-uniform user distribution in order to produce a tractable analysis of HetNets in the downlink and the uplink using the tools of stochastic geometry