This thesis is concerned with an investigation and evaluation of collocation algorithms for solving two-point boundary value problems for systems of ordinary differential equations. An emphasis is on developing reliable and efficient adaptive mesh selection algorithms in piecewise collocation methods. General background materials including basic concepts and descriptions of the method as well as some functional analysis tools needed in developing some error estimates are given at the beginning. A brief review of some developments in the methods to be used is provided for later referencing. By utilising the special structure of the collocation matrices, a more compact block matrix structure is introduced and an algorithm for generating and solving the matrix is proposed. Some practical aspects and computational considerations of matrices involved in the collocation process such as analysis of arithmetic operations and amount of memory spaces needed are considered. An examination of scaling process to reduce the condition number is also presented. A numerical evaluation of some error estimates developed by considering the differential operator, the related matrices and the residual is carried out. These estimates are used to develop adaptive mesh selection algorithms, in particular as a cheap criterion for terminating the computation process. Following a discussion on mesh selection strategies, a criterion function for use in adaptive algorithms is introduced and a numerical scheme to equidistributing values of the criterion function is proposed. An adaptive algorithm based on this criterion is developed and the results of numerical experiments are compared with those using some well known criterion functions. The various examples are chosen in such a way that they include problems with interior or boundary layers. In addition, an algorithm has been developed to predict the necessary number of subintervals for a given tolerance, with the aim of improving the efficiency of the whole process. Using a good initial mesh in adaptive algorithms would be expected to provide some further improvement in the algorithms. This leads to the idea of locating the layer regions and determining suitable break points in such regions before the numerical process. Based on examining the eigenvalues of the coefficient matrix in the differential equation in the specified interval, using their magnitudes and rates of change, the algorithms for predicting possible layer regions and estimating the number of break points needed in such regions are constructed. The effectiveness of these algorithms is evaluated by carrying out a number of numerical experiments. The final chapter gives some concluding remarks of the work and comment on results of numerical experiments. Certain possible improvements and extensions for further research are also briefly given.

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The parameter estimation problem or the inverse problem of ordinary differential equations is prevalent in many process models in chemistry, molecular biology, control system design and many other engineering applications. It concerns the re-construction of auxillary parameters by fitting the solution from the system of ordinary differential equations( from a known mathematical model) to that of measured data obtained from observing the solution trajectory. Some of the traditional techniques (for example, initial value technques, multiple shooting, etc.) used to solve this class of problem have been discussed. A new algorithm, motivated by algorithms proposed by Childs and Osborne(1996) and Z.F.Li's dissertation(2000), has been proposed. The new algorithm inherited the advantages exhibited in the above-mentioned algorithms and, most importantly, the parameters can be transformed to a form that are convenient and suitable for computation. A statistical analysis has also been developed and applied to examples. The statistical analysis yields indications of the tolerance of the estimates and consistency of the observations used.

Under the assumption of mass-action kinetics, systems of chemical reactions can give rise to a wide variety of dynamical behaviour, including stability of a unique equilibrium concentration, multistability, periodic behaviour, chaotic behaviour, switching behaviour, and many others. In their canonical papers, M. Feinberg, F. Horn and R. Jackson developed so-called Chemical Reaction Network theory which drew a strong connection between the topological structure of the reaction graph and the dynamical behaviour of mass-action systems. A significant amount of work since that time has been conducted expanding upon this connection and fleshing out the theoretical underpinnings of the theory. In this thesis, I focus on three topics within the scope of Chemical Reaction Network theory. 1. Linearization: It is known that complex balanced systems possess within each invariant space of the system a unique positive equilibrium concentration and that that concentration is locally asymptotically stable. F. Horn and R. Jackson determined this through the use of an entropy-like Lyapunov function. In Chapter 4, I approach this problem through the alternative approach of linearizing the mass-action system about its equilibrium points. I show that this approach reproduces the results of F. Horn and R. Jackson and has the advantage of being able to give explicit exponential bounds on the convergence near equilibria. 2. Persistence: A well-known limitation of the theory is that the stabilities of the positive equilibrium concentrations guaranteed are locally limited. The conjecture that the equilibrium concentrations of complex balanced systems are global attractors of their respective invariant spaces has become known as the Global Attractor Conjecture and has received significant attention recently. This theory has been significantly aided by the realization that trajectories not tending toward the set of positive equilibria must tend toward the boundary of the positive orthant; consequently, persistence is a sufficient condition to affirm the conjecture. In Chapter 5, I present my contributions to this problem. 3. Linear Conjugacy: It is known that under the mass-action assumption two reaction networks with disparate topological structure may give rise to the same set of differential equations and therefore exhibit the same qualitative dynamical behaviour. In Chapter 6, I expand the scope of networks which exhibit the same behaviour to include ones which are related by a non-trivial linear mapping. I have called this theory Linear Conjugacy theory. I also show how networks exhibiting a linear conjugacy can be found using the mixed integer linear programming (MILP) framework introduced by G. Szederkenyi.

Under the assumption of mass-action kinetics, systems of chemical reactions can give rise to a wide variety of dynamical behaviour, including stability of a unique equilibrium concentration, multistability, periodic behaviour, chaotic behaviour, switching behaviour, and many others. In their canonical papers, M. Feinberg, F. Horn and R. Jackson developed so-called Chemical Reaction Network theory which drew a strong connection between the topological structure of the reaction graph and the dynamical behaviour of mass-action systems. A significant amount of work since that time has been conducted expanding upon this connection and fleshing out the theoretical underpinnings of the theory. In this thesis, I focus on three topics within the scope of Chemical Reaction Network theory. 1. Linearization: It is known that complex balanced systems possess within each invariant space of the system a unique positive equilibrium concentration and that that concentration is locally asymptotically stable. F. Horn and R. Jackson determined this through the use of an entropy-like Lyapunov function. In Chapter 4, I approach this problem through the alternative approach of linearizing the mass-action system about its equilibrium points. I show that this approach reproduces the results of F. Horn and R. Jackson and has the advantage of being able to give explicit exponential bounds on the convergence near equilibria. 2. Persistence: A well-known limitation of the theory is that the stabilities of the positive equilibrium concentrations guaranteed are locally limited. The conjecture that the equilibrium concentrations of complex balanced systems are global attractors of their respective invariant spaces has become known as the Global Attractor Conjecture and has received significant attention recently. This theory has been significantly aided by the realization that trajectories not tending toward the set of positive equilibria must tend toward the boundary of the positive orthant; consequently, persistence is a sufficient condition to affirm the conjecture. In Chapter 5, I present my contributions to this problem. 3. Linear Conjugacy: It is known that under the mass-action assumption two reaction networks with disparate topological structure may give rise to the same set of differential equations and therefore exhibit the same qualitative dynamical behaviour. In Chapter 6, I expand the scope of networks which exhibit the same behaviour to include ones which are related by a non-trivial linear mapping. I have called this theory Linear Conjugacy theory. I also show how networks exhibiting a linear conjugacy can be found using the mixed integer linear programming (MILP) framework introduced by G. Szederkenyi.

Philosophiae Doctor - PhD / The purpose of this study was to investigate the role of visualization in the conceptualisation and solution of problems in multivariate calculus and dynamical systems. The theoretical basis, and the visual and analytical aspects of evaluating multiple integrals, and the stability analysis of dynamical systems, were established. To address the research questions, a teaching experiment with activities to facilitate visualization of 3D objects and phase portraits of non-linear dynamic systems was conducted with an experimental class (n = 24) which received six activity sessions in the computer Laboratory in addition to traditional lectures. The control class (n = 26) received traditional lectures and tutorial instruction. Both groups were lectured by the researcher using the same set of class notes, assignments, worksheets and tutorials. Additional support materials were posted on the Blackboard on Web-City. The activities included tasks in the computer laboratory that reinforced visualization and spatial ability factors such as surface features, nets, projections, cross-sections and rotation of 3D objects as well as phase portraits of systems of differential equations. The students were tested at several time points, and over both the short and long term to assess the impact on their visual and analytical solutions to problems in the two study domains. The pre-test on prior knowledge indicated no significant differences between the means of the experimental and control groups. Results indicate that there were no significant differences between the achievement of the two groups in Test 1 and Test 2 while the activities were ongoing, but towards the end of the semester significant differences in favour of the experimental group were recorded. A multiple linear regression analysis confirmed that in addition to prior knowledge as measured by the pretest, two of the spatial factors were significant predictors of achievement for the domains under investigation. Students had difficulties in visualising 3D regions of integration and in switching the order of triple integrals. Very few (18%) recognised the need for split integrals to span the required area or volume. While students could find analytical solutions to systems of differential equations and describe the stability of individual equilibrium points using eigenvalues, they struggled with translating rates of change into slopes on the phase portraits, with the interpretation of the solutions and in describing the global behaviour of the system. Students had difficulties in visualizing the region of integration in R³, the stability of equilibrium points in the phase portraits, and in coordinating the treatments and conversions between the geometric, numerical, symbolic and algebraic registers. The tendency to work in the algebraic register to determine the limits of the integral was noted, and students opted to use analytic methods in conducting a stability analysis of the given dynamic system rather than the geometric method. This study adds to research on visualization in mathematics by examining how exposure to technologically enhanced representations complement and promote the conceptualisation of solutions to problems involving multiple integrals and systems of differential equations.

Global dynamic optimization arises in many engineering applications such as parameter estimation, global optimal control, and optimization-based worst-case uncertainty analysis. In branch-and-bound deterministic global optimization algorithms, a major computational bottleneck is generating appropriate lower bounds for the globally optimal objective value. These bounds are typically constructed using convex relaxations for the solutions of dynamic systems with respect to decision variables. Tighter convex relaxations thus translate into tighter lower bounds, which will typically reduce the number of iterations required by branch-and-bound. Subgradients, as useful local sensitivities of convex relaxations, are typically required by nonsmooth optimization solvers to effectively minimize these relaxations. This thesis develops novel techniques for efficiently computing tight convex relaxations with the corresponding subgradients for the solutions of ordinary differential equations (ODEs), to ultimately improve efficiency of deterministic global dynamic optimization. Firstly, new bounding and comparison results for dynamic process models are developed, which are more broadly applicable to engineering models than previous results. These new results show for the first time that in a state-of-the-art ODE relaxation framework, tighter enclosures of the original ODE system's right-hand side will necessarily translate into enclosures for the state variables that are at least as tight, which paves the way towards new advances for bounding in global dynamic optimization. Secondly, new convex relaxations are proposed for the solutions of ODE systems. These new relaxations are guaranteed to be at least as tight as state-of-the-art ODE relaxations. Unlike established ODE relaxation approaches, the new ODE relaxation approach can employ any valid convex and concave relaxations for the original right-hand side, and tighter such relaxations will necessarily yield ODE relaxations that are at least as tight. In a numerical case study, such tightness does indeed improve computational efficiency in deterministic global dynamic optimization. This new ODE relaxation approach is then extended in various ways to further tighten ODE relaxations. Thirdly, new subgradient evaluation approaches are proposed for ODE relaxations. Unlike established approaches that compute valid subgradients for nonsmooth dynamic systems, the new approaches are compatible with reverse automatic differentiation (AD). It is shown for the first time that subgradients of dynamic convex relaxations can be computed via a modified adjoint ODE sensitivity system, which could speed up lower bounding in global dynamic optimization. Lastly, in the situation where convex relaxations are known to be correct but subgradients are unavailable (such as for certain ODE relaxations), a new approach is proposed for tractably constructing useful correct affine underestimators and lower bounds of the convex relaxations just by black-box sampling. No additional assumptions are required, and no subgradients must be computed at any point. Under mild conditions, these new bounds are shown to converge rapidly to an original nonconvex function as the domain of interest shrinks. Variants of the new approach are presented to account for numerical error or noise in the sampling procedure. / Thesis / Doctor of Philosophy (PhD)

The overarching goal of this project is to develop intelligent, self-adapting numerical algorithms for the time discretization of complex real-world problems with Q-Learning methodologies. The specific application is ordinary differential equations which can resolve problems in mathematics, social and natural sciences, but which usually require approximations to solve because direct analytical solutions are rare. Using the traditional Brusellator and Lorenz differential equations as test beds, this research develops models to determine reward functions and dynamically tunes controller parameters that minimize both the error and number of steps required for approximate mathematical solutions. Our best reward function is based on an error that does not overly punish rejected states. The Alpha-Beta Adjustment and Safety Factor Adjustment Model is the most efficient and accurate method for solving these mathematical problems. Allowing the model to change the alpha/beta value and safety factor by small amounts provides better results than if the model chose values from discrete lists. This method shows potential for training dynamic controllers with Reinforcement Learning. / Master of Science / This research applies Q-Learning, a subset of Reinforcement Learning and Machine Learning, to solve complex mathematical problems that are unable to be solved analytically and therefore require approximate solutions. Specifically, this research applies mathematical modeling of ordinary differential equations which are used in many fields, from theoretical sciences such and physics and chemistry, to applied technical fields such as medicine and engineering, to social and consumer-oriented fields such as finance and consumer purchasing habits, and to the realms of national and international security and communications. Q-Learning develops mathematical models that make decisions, and depending on the outcome, learns if the decision is good or bad, and uses this information to make the next decision. The research develops approaches to determine reward functions and controller parameters that minimize the error and number of steps associated with approximate mathematical solutions to ordinary differential equations. Error is how far the model's answer is from the true answer, and the number of steps is related to how long it takes and how much computational time and cost is associated with the solution. The Alpha-Beta Adjustment and Safety Factor Adjustment Model is the most efficient and accurate method for solving these mathematical problems and has potential for solving complex mathematical and societal problems.

In this dissertation, we investigate the mechanisms behind diseases and the immune responses required for successful disease resolution in three projects: i) A study of HIV and HPV co-infection, ii) A germinal center dynamics model, iii) A study of monoclonal antibody therapy. We predict that the condition leading to HPV persistence during HIV/HPV co-infection is the permissive immune environment created by HIV, rather than the direct HIV/HPV interaction. In the second project, we develop a germinal center model to understand the mechanisms that lead to the formation of potent long-lived plasma. We predict that the T follicular helper cells are a limiting resource and present possible mechanisms that can revert this limitation in the presence of non-mutating and mutating antigen. Finally, we develop a pharmacokinetic model of 3BNC117 antibody dynamics and HIV viral dynamics following antibody therapy. We fit the models to clinical trial data and conclude that antibody binding is delayed and that the combined effects of initial CD4 T cell count, initial HIV levels, and virus production are strong indicators of a good response to antibody immunotherapy. / Ph. D. / In this dissertation, we investigate the mechanisms behind diseases and the immune responses required for successful disease resolution in three projects: i) A study of HIV and HPV co-infection, ii) A germinal center dynamics model, iii) A study of monoclonal antibody therapy. We predict that the condition leading to HPV persistence during HIV/HPV co-infection is the permissive immune environment created by HIV, rather than the direct HIV/HPV interaction. In the second project, we develop a germinal center model to understand the mechanisms that lead to the formation of potent long-lived plasma. We predict that the T follicular helper cells are a limiting resource and present possible mechanisms that can revert this limitation in the presence of non-mutating and mutating antigen. Finally, we develop a pharmacokinetic model of 3BNC117 antibody dynamics and HIV viral dynamics following antibody therapy. We fit the models to clinical trial data and conclude that antibody binding is delayed and that the combined effects of initial CD4 T cell count, initial HIV levels, and virus production are strong indicators of a good response to antibody immunotherapy.

The intent of this thesis is to develop ordinary differential equation models to better understand the mosquito population. We first develop a framework model, where we determine the condition under which a natural mosquito population can persist in the environment. Wolbachia is a bacterium which limits the replication of viruses inside the mosquito which it infects. As a result, infecting a mosquito population with Wolbachia can decrease the transmission of viral mosquito-borne diseases, such as dengue. We develop another ODE model to investigate the invasion of Wolbachia in a mosquito population. In a biologically feasible situation, we determine three coexisting equilibria: a stable Wolbachia-free equilibrium, an unstable coexistence equilibrium, and a complete invasion equilibrium. We establish the conditions under which a population of Wolbachia infected mosquitoes may persist in the environment via the next generation number and determine when a natural mosquito population may experience a complete invasion of Wolbachia.