The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon: In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents. As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process. In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

This thesis develops methodologies for the construction of various types of optimal designs with applications in maximum likelihood estimation and factorial structure design. The methodologies are applied to some real data sets throughout the thesis. We start with a broad review of optimal design theory including various types of optimal designs along with some fundamental concepts. We then consider a class of optimization problems and determine the optimality conditions. An important tool is the directional derivative of a criterion function. We study extensively the properties of the directional derivatives. In order to determine the optimal designs, we consider a class of multiplicative algorithms indexed by a function, which satisfies certain conditions. The most important and popular design criterion in applications is D-optimality. We construct such designs for various regression models and develop some useful strategies for better convergence of the algorithms. The remaining thesis is devoted to some important applications of optimal design theory. We first consider the problem of determining maximum likelihood estimates of the cell probabilities under the hypothesis of marginal homogeneity in a square contingency table. We formulate the Lagrangian function and remove the Lagrange parameters by substitution. We then transform the problem to one of maximizing some functions of the cell probabilities simultaneously. We apply this problem to some real data sets, namely, a US Migration data, and a data on grading of unaided distance vision. We solve another estimation problem to determine the maximum likelihood estimation of the parameters of the latent variable models such as Bradley-Terry model where the data come from a paired comparisons experiment. We approach this problem by considering the observed frequency having a binomial distribution and then replacing the binomial parameters in terms of optimal design weights. We apply this problem to a data set from American League Baseball Teams. Finally, we construct some optimal structure designs for comparing test treatments with a control. We introduce different structure designs and establish their properties using the incidence and characteristic matrices. We also develop methods of obtaining optimal R-type structure designs and show how such designs are trace, A- and MV-optimal. / October 2016

Doctor of Philosophy / Department of Industrial & Manufacturing Systems Engineering / Chih-Hang Wu / The topic of infectious disease epidemics has recently attracted substantial attentions in research communities and it has been shown that the changes of human behaviors have significant impacts on the dynamics of disease transmission. However, the study and understanding of human reactions into spread of infectious disease are still in the very beginning phase and how human behaviors change during the spread of infectious disease has not been systematically investigated. Moreover, the study of human behaviors includes not only various enforced measures by public authorities such as school closure, quarantine, vaccination, etc, but also the spontaneous self-protective actions which are triggered by risk perception and fear of diseases. Hence, the goal of this research is to study the impacts of human behaviors to the epidemic from these two perspectives: spontaneous behavioral changes and public intervention strategies. For the sake of studying spontaneous changes of human behaviors, this research first time applied evolutionary spatial game into the study of human reactions to the spread of infectious disease. This method integrated contact structures and epidemics information into the individuals’ decision processes, by adding two different types of information into the payoff functions: the local information and global information. The new method would not only advance the field of game theory, but also the field of epidemiology. In addition, this method was also applied to a classic compartmental dynamic system which is a widely used model for studying the disease transmission. With extensive numerical studies, the results first proved the consistency of two models for the sake of validating the effectiveness of the spatial evolutionary game. Then the impacts of changes of human behaviors to the dynamics of disease transmission and how information impacts human behaviors were discussed temporally and spatially. In addition to the spontaneous behavioral changes, the corresponding intervention strategies by policy-makers played the key role in process of mitigating the spread of infectious disease. For the purpose of minimizing the total lost, including the social costs and number of infected individuals, the intervention strategies should be optimized. Sensitivity analysis, stability analysis, bifurcation analysis, and optimal control methods are possible tools to understand the effects of different combination of intervention strategies or even find an appropriate policy to mitigate the disease transmission. One zoonotic disease, named Zoonotic Visceral Leishmaniasis (ZVL), was studied by adopting different methods and assumptions. Particularly, a special case, backward bifurcation, was discussed for the transmission of ZVL. Last but not least, the methodology and modeling framework used in this dissertation can be expanded to other disease situations and intervention applications, and have a broad impact to the research area related to mathematical modeling, epidemiology, decision-making processes, and industrial engineering. The further studies can combine the changes of human behaviors and intervention strategies by policy-makers so as to seek an optimal information dissemination to minimize the social costs and the number of infected individuals. If successful, this research should aid policy-makers by improving communication between them and the public, by directing educational efforts, and by predicting public response to infectious diseases and new risk management strategies (regulations, vaccination, quarantine, etc.).

Metabolites consumed and produced by microorganisms for mass and energy conservation may cause changes in a microorganism’s environment. The microorganisms are unable to tolerate a particular environment for a long period. They may leave their old existence to find a new environment to sustain life. Essentially, organisms need to maintain their metabolic processes to survive in the new environment. Limitations of experimental studies to explore cell functions and regulations in detail result in insufficient information to explain processes of metabolic expressions under environments of organisms. Consequently, mathematical modeling and computer simulations have been conducted to combine all possible cellular metabolic fluxes into single or multiple connected networks. Metabolic modeling based on linear programming (LP) subjected to constraints with an optimization approach is often applied metabolic reconstruction. The LP objective function is maximized to obtain an optimal value of biomass flux. Optimal solutions in LP problems can be used to explain how metabolites function in metabolic reactions. As an LP problem may have many optimal solutions, this study proposes a method for enumerating all alternate optimal solutions to evaluate important reactions of metabolic pathways in microorganisms. The algorithm for generating alternate optimal solutions is implemented in MetModelGUI, a Java-based software for creating and analyzing metabolic reconstructions. The algorithm is applied to models of five microorganisms: Trypanosoma cruzi, Thermobifida fusca, Helicobacter pylori, Cryptococcus neoformans and Clostridium thermocellum. The results are analyzed using principal component analysis, and insight into the essential and non-essential pathways for each organism is derived

Mixed integer programming (MIP) problems are highly parameterized, and finding parameter settings that achieve high performance for specific types of MIP instances is challenging. This paper presents a method to find the information about how CPLEX solver parameter settings perform for the different classes of mixed integer linear programs by using designed experiments and statistical models. Fitting a model through design of experiments helps in finding the optimal region across all combinations of parameter settings. The study involves recognizing the best parameter settings that results in the best performance for a specific class of instances. Choosing good setting has a large effect in minimizing the solution time and optimality gap.

Integer Programming problems are difficult to solve. The goal is to find an optimal solution that minimizes cost. With the help of Groebner based algorithms the optimal solution can be found if it exists. The application of the Groebner based algorithm and how it works is the topic of research. The Algorithms are The Conti-Traverso Algorithm and the Original Conti-Traverso Algorithm. Examples are given as well as proofs that correspond to the algorithms. The latter algorithm is more efficient as well as user friendly. The algorithms are not necessarily the best way to solve and integer programming problem, but they do find the optimal solution if it exists.

Cheng and Wu (2001) introduced a method for response surface exploration using only one design by using a 3-level design to first screen a large number of factors and then project onto the significant factors to perform response surface exploration. Previous work generally involved selecting designs based on projection properties first and aliasing structure second. However, having good projection properties is of little concern if the correct factors cannot be identified. We apply Jones and Nachtsheim’s (2009) method for finding optimal designs with minimal aliasing to find 18, 27, and 30-run designs to use for single-step screening and optimization. Our designs have better factor screening capabilities than the designs of Cheng and Wu (2001) and Xu et al. (2004), while maintaining similar D-efficiencies and allowing all projections to fit a full second order model.

We consider agent trading futures on a market with small transaction costs. Her capital is deposited on a money market account, where compounding is possible. Arithmetic Brownian motion with random coefficients is considered as a model for futures strike price. The coefficients are assumed to be bounded Itô processes with bounded coefficients. Under these assumptions, an almost optimal interval strategy is derived, which almost maximizes expected utility in certain stopping times under hyperbolic absolute risk aversion utility function. Furthermore, under logarithmic utility function the derived strategy almost maximizes expected utility in wide class of (integrable) stopping times.

Oceans cover two thirds of the Earth’s surface and the energy potential of ocean waves as a renewable energy source is huge. It would therefore be a tremendous achievement if the vast mechanical energy in waves was converted into a form of energy that could be used successfully by society. For years, scientists and engineers have endeavored to exploit this renewable energy by inventing various generators designed to transform wave energy into electrical energy. Generally, this sort of generator is called a Wave Energy Converter (WEC). In this thesis, the research is based on the WEC developed in the Lysekil Project. The Lysekil Project is led by a research group at Uppsala University and has a test site located on the west coast of Sweden. The project started in 2002. So far, more than ten prototypes of the WEC have been deployed and relevant experiments have been carried out at the test site. The WEC developed at Uppsala University can be categorized as a point absorber. It consists of a direct-drive linear generator connected to a floating buoy. The linear generator is deployed on the seabed and driven by a floating buoy to extract wave energy. The absorbed energy is converted to electricity and transmitted to a measuring station on land. The work presented in this thesis focuses on building a linear generator model which is able to predict the performance of the Lysekil WEC. Studies are also carried out on the damping behavior of the WEC under the impact of different sea climates. The purpose is to optimize the energy absorption with a specific optimal damping coefficient. The obtained results indicate an optimal damping for the Lysekil WEC which can be used for optimizing the damping control. Additionally, the impact two central engineering design features (the translator weight and the stroke length) are investigated. The aim is to find a reasonable structural design for the generator which balances the cost and the energy production.

Master of Science / Department of Electrical and Computer Engineering / Balasubramaniam Natarajan / Noel Schulz / With increasing costs for fuel there is a growing interest in improving fuel efficiency and performance of military vehicles by employing (1) hybrid drive train architecture; (2) reliable vehicle power system structure, and (3) effective power management strategies of multiple power sources (engine, battery and ultracapacitor) and vehicle electrical loads. However, current ruled-based power management strategies that focus primarily on traction fail to meet the rapidly increasing requirements of military vehicles, including: (1) better fuel economy; (2) the ability to support pulsed power weapon loads; (3) maintaining battery SOC for power offloading applications, and (4) the ability to perform load scheduling of vehicle non-traction electrical loads to save energy. In this thesis, we propose an optimal control based algorithm in conjunction with a rule-based control strategy to optimally manage three power sources (engine, battery and pulsed power supply module) and an effective power management solution for vehicle non-traction electrical loads such that: (1) all traction, non-traction and pulsed power needs are met; (2) power drawn from the engine for specific mission is minimized; (3) a certain desired battery SOC is guaranteed for offloading power, and (4) the ability to perform load scheduling based on different mission requirements. The proposed approach is validated using simulation of a mission specific profile and is compared with two other popular control strategies. The improvements in power efficiency, desired SOC level and ability to perform optimal load scheduling are demonstrated.