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Page iii not included in the thesis and therefore, not included in the page count. / <p> In recent years, some efficient methods have been developed for calculating derivative price sensitivities, or the Greeks, using Monte Carlo simulation. However, the slow convergence, especially for discontinuous payoff functions, is well known for Monte Carlo simulation. In this project, we investigate the Malliavin calculus and its application in computation of the Greeks. Malliavin calculus and Wiener Chaos theory are introduced. The theoretical framework of the Malliavin weighted scheme of computation of the Greeks is explored in details, and the numerical implementation of the one-dimensional case and an example of the two-dimensional case are presented. Finally, the results are compared with those of finite difference scheme.</p> / Thesis / Master of Science (MSc)

Proximal risk factors including household chaos, parenting and maternal distress can have a broad impact on multiple domains of child development and functioning. Using multiple methodologies including a meta-analysis and structural equation modeling with an empirical, cross-sectional design from a larger longitudinal research study; in this dissertation, I examine the impact of household chaos on child executive functioning, socioemotional and physiological stress outcomes, the role that parenting plays in this association, and how maternal distress predicts chaos in the home. In study 1, I conduct a meta-analysis examining the direct association between household chaos and child executive functioning, as well as multiple potential moderators (e.g. child age, sex and race/ethnicity). It incorporates 26 studies, with 27 independent effect sizes with a total sample of 8,944 children. Overall, I found a significant effect of r = .22 between household chaos and child executive function. Among the moderators assessed, only measurement approach of executive functions (informant-completed questionnaire versus direct assessment) was significant, with informant-completed questionnaires yielding an effect of r = .27 compared to direct assessment, r = .16. I conducted a series of separate moderation analyses for questionnaire and direct assessment effects. No significant moderators emerged from the questionnaire analyses, despite heterogeneous effect sizes. Direct assessment analyses revealed that both household chaos dimensions (disorganization and instability) were significantly related to child executive functions, however instability was a stronger correlate (r = .21) than disorganization (r = .09). Composition of the sample was also a significant moderator with effects increased with the proportion of minorities, and with parents with lower levels of education. Building on this work, in studies 2 and 3, I used cross-sectional empirical data from a sample of 137 mothers and their school-aged (5-year old) children. During home visits, mothers completed questionnaires assessing their mood, stressful experiences, the home environment and their child’s socioemotional functioning. Mothers also completed a video tour of the home. Mother-child interactions were videotaped and later coded for parenting. Both mothers and children independently completed behavioural assessments of executive function. Also, hair samples were collected from mothers and children from which the stress hormone, cortisol, was extracted as a biomarker of chronic stress. In order to empirically test the findings from the meta-analysis, in my second study, I used structural equation modeling to examine the indirect effect of household chaos on child executive functioning via parenting. I found that household chaos was directly and indirectly (via maternal cognitive sensitivity and emotional availability) associated with a latent variable of child executive functioning. Furthermore, instability, but not disorganization, significantly predicted child executive functioning directly and indirectly via parenting. Finally, sex-based analyses indicated that the effect of chaos on child executive functioning was significant through indirect effects only for boys. In the third study, in order to elucidate potential contributing factors to household chaos, I used a structural equation model to examine the indirect effects of a linear regression-weighted composite variable of maternal distress (depression, negative affect and physiological stress) on child hair cortisol levels and externalizing and internalizing behaviour problems via household chaos. I found that maternal distress had both direct and indirect effects (via household chaos) on child hair cortisol levels; however, only indirect effects were significant for externalizing and internalizing behaviour problems. Also, the indirect effect was only significant for household disorganization, but not instability, for child hair cortisol and externalizing and internalizing behaviour problems. Taken together, the findings from my dissertation demonstrate that: 1) household chaos has a direct, negative effect on child executive functioning and an indirect effect via parenting; and 2) maternal distress plays an important role in predicting the levels of chaos within the home which has implications for child chronic stress levels and behavioural problems. Collectively, these findings highlight the need to take a multi-method approach to measuring executive functioning in children and further, to develop and evaluate interventions that aim to support mothers, improve parenting and promote order and stability within the home in order to foster healthy developmental trajectories for children. / Dissertation / Doctor of Philosophy (PhD) / Children exposed to household chaos may experience adverse outcomes across multiple domains. Parenting can also be negatively affected by household chaos which may impact the quality of parent-child interactions. Further, the physical and psychological health of the mother may regulate the levels of chaos in the home which has implications for child outcomes as well. This dissertation seeks to examine the influence of household chaos on child executive functioning, stress levels and socioemotional functioning, and the roles that parenting and maternal distress play. I address three primary objectives: 1) using meta-analytic techniques, I examine the magnitude of effect of household chaos on child executive functioning based on existing literature as well as potential factors that may modulate the strength of the linkage between household chaos and child executive functioning; and using cross-sectional data, I examine 2) how household chaos impacts parenting and subsequently, how parenting impacts child executive functioning; and 3) how maternal distress influences the level of chaos in the home and how this chaos impacts child stress levels and socioemotional functioning. Collectively, the results from this dissertation indicate that household chaos has a broad negative impact on child outcomes, and both parenting and maternal distress play important roles in understanding this impact. Further, it demonstrates the need for intervention research aimed at supporting the physical and psychological health of mothers, improving parenting and creating order and stability in homes for children.

Water is a scarce resource especially as the water demand is significantly increasing due to the rapid growth of population. Hydrological modelling has gained a lot of attention, as it is the key to predict water availability, optimize the use of water resources and develop risk mitigation schemes. There are still many challenges in hydrological modelling that researchers and designers are trying to solve. These challenges include, but not limited to: i) there is no single robust model that can perform well in all watersheds; ii) model parameters are often associated with uncertainty, which makes the results inconclusive; iii) the required computational power for uncertainty quantification increases with the increase in model complexity; iv) some modelling assumptions to simplify computational complexity, such as parameter independence are, are often not realistic. These challenges make it difficult to provide robust hydrological predictions and/or to quantify the uncertainties within hydrological models in an efficient and accurate way. This study aims to provide more robust hydrological predictions by developing a set of hybrid approaches. Firstly, a hybrid hydrological data-driven (HHDD) model based on the integration of a physically-based hydrological model (HYMOD) and a data-driven model (artificial neural network, ANN) is developed. The HHDD model is capable of improving prediction accuracy and generating interval flow prediction results. Secondly, a hybrid probabilistic forecasting approach is developed by linking the polynomial chaos expansion (PCE) method with ANN. The results indicate that PCE-ANN can be as reliable as but much more efficient than the traditional Monte-Carlo (MC) method for probabilistic flow forecasting. Finally, a hybrid uncertainty quantification approach that can address parameter dependence is developed through the integration of principal component analysis (PCA) with PCE. The results from this dissertation research can provide valuable technical and decision support for hydrological modeling and water resources management under uncertainty. / Thesis / Doctor of Engineering (DEng) / There is a water scarcity problem in the world, so it is vital to have reliable decision support tools for effective water resources management. Researchers and decision-makers rely on hydrological modelling to predict water availability. Hydrological model results are then used for water resources allocation and risk mitigation. Hydrological modelling is not a simple process, as there are different sources of uncertainty associated with it, such as model structure, model parameters, and data. In this study, data-driven techniques are used with process-driven models to develop hybrid uncertainty quantification approaches for hydrological modelling. The overall objectives are: i) to generate more robust probabilistic forecasts; ii) to improve the computational efficiency for uncertainty quantification without compromising accuracy; and, iii) to overcome the limitations of current uncertainty quantification methods, such as parameter interdependency. The developed hybrid approaches can be used by decision-makers in water resources management, as well as risk assessment and mitigation.

Fluid mixing is a challenging problem in laminar flow systems. Even in microfluidic systems, diffusion is often negligible compared to advection in the flow. The idea of chaotic advection can be applied in these systems to enhance mixing efficiency. Topological chaos can also lead to efficient and rapid mixing. In this dissertation, an approach to enhance fluid mixing in laminar flows without internal rods is demonstrated by using the idea of topological chaos. Periodic motion of three stirrers in a two-dimensional flow can lead to chaotic transport of the surrounding fluid. For certain stirrer motions, the generation of chaos is guaranteed solely by the topology of that motion and continuity of the fluid. This approach is in contrast to standard techniques. Appropriate stirrer motions are determined using the Thurston-Nielsen classification theorem, which also predicts a lower bound on the complexity of the dynamics in the flow. Work in this area has focused largely on using physical rods as stirrers, but the theorem also applies when the ``stirrers'' are passive fluid particles. In this thesis, topological chaos is theoretically and numerically investigated in lid-driven cavities and rectangular channels without internal rods. When a two-dimensional incompressible Newtonian flow is in the Stokes flow regime, the stream function satisfies the two-dimensional biharmonic equation. When the flow occurs in a lid-driven cavity with solid side walls, this equation can be solved using a method that is similar to the traditional Fourier expansion but uses an asymptotic approximation for the sum of high order terms. When the flow occurs between two infinite plates with spatially periodic boundary conditions, an exact solution in a rectangle with finite width, which represents the flow in this infinitely-wide cavity, can be obtained by using the principle of superposition. A fully developed, three-dimensional flow in a rectangular channel can be decomposed into an unperturbed Poiseuille flow in the axial direction and a lid-driven cavity secondary flow in the cross section. This model can be applied to numerically simulate either pressure-driven flow in a rectangular channel with surface grooves or electro-osmotic flow in a rectangular channel with variations in surface potential. In this dissertation, the occurrence of topological chaos in unsteady two-dimensional flows as well as steady three-dimensional flows without internal rods is demonstrated. For appropriate choices of boundary velocity on the top and/or bottom walls, there exist three periodic points in the flows that produce a chaos-generating motion. In steady flow through a three-dimensional rectangular channel, the axial direction plays the role of time and the periodic points lie on streamtubes that "braid" the surrounding fluid as it moves through the duct. When appropriate motion is applied on the boundary of the wide cavity or channel, topological chaos can also be generated in the flow. The stretching rate of non-trivial material lines in all these flows agrees with the prediction of the lower bound of topological entropy provided by the Thurston-Nielsen theorem. Ghost rod structures are found and analyzed in the lid-driven cavity and rectangular channel flows with solid side walls. The results suggest that the no-slip boundary condition on the stationary internal surfaces is one of the reasons for poor mixing in steady laminar three-dimensional flows considered previously with solid braided internal rods. / Ph. D.

A method utilizing H2 control concepts and the numerical method of Polynomial Chaos was developed in order to create a novel robust probabilistically optimal control approach. This method was created for the practical reason that uncertainty in parameters tends to be inherent in system models. As such, the development of new methods utilizing probability density functions (PDFs) was desired. From a more theoretical viewpoint, the utilization of Polynomial Chaos for studying and designing control systems has not been very thoroughly investigated. The current work looks at expanding the H2 and related Linear Quadratic Regulator (LQR) control problems for systems with parametric uncertainty. This allows solving deterministic linear equations that represent probabilistic linear differential equations. The application of common LTI (Linear Time Invariant) tools to these expanded systems are theoretically justified and investigated. Examples demonstrating the utilized optimization process for minimizing the H2 norm and parallels to LQR design are presented. The dissertation begins with a thorough background section that reviews necessary probability theory. Also, the connection between Polynomial Chaos and dynamic systems is explained. Next, an overview of related control methods, as well as an in-depth review of current Polynomial Chaos literature is given. Following, formal analysis, related to the use of Polynomial Chaos, is provided. This lays the ground for the general method of control design using Polynomial Chaos and H2. Then an experimental section is included that demonstrates controller synthesis for a constructed probabilistic system. The experimental results lend support to the method. / Ph. D.

An important phenomenon of systems driven far-from-equilibrium is spatiotemporal chaos where the dynamics are aperiodic in both time and space. We explored this numerically for three systems: the Lorenz-96 model, the Swift-Hohenberg equation, and Rayleigh-Bénard convection. The Lorenz-96 model is a continuous in time and discrete in space phenomenological model that captures important features of atmosphere dynamics. We computed the fractal dimension as a function of system size and external forcing to estimate characteristic length and time scales describing the chaotic dynamics. We found extensive chaos with significant deviations from extensivity for small changes in system size and also the power-law growth of the dimension with increasing forcing. The Swift-Hohenberg equation is a partial differential equation for a scalar field, which has been widely used as a model for the study of pattern formation. We found that the magnitude of the mean flow in this model must be sufficiently large for spiral defect chaos to occur. We also explored the spatiotemporal chaos in experimentally accessible Rayleigh-Bénard convection using large-scale numerical simulations of the Boussinesq equations and the corresponding tangent space equations. We performed a careful study analyzing the impact of variations in the domain size, Rayleigh number, and Prandtl number on the system dynamics and fractal dimension. In addition, we quantified the dynamics of the spectrum of Lyapunov exponents and the leading order Lyapunov vector in an effort to connect directly with the dynamics of the flow field patterns. Further, we numerically studied the synchronization of chaos in convective flows by imposing time-dependent boundary conditions from a principal domain onto an initially quiescent target domain. We identified a synchronization length scale to quantify the size of a chaotic element using only information from the pattern dynamics. We also explored the relationship of this length scale with the pattern wavelength. Finally, we analyzed bioconvection which occurs as the result of the collective behavior of a suspension of swimming microorganisms. We developed a series of simulations to capture the gyrotactic pattern formation of the swimming algae. The results can be compared with the corresponding trend of pattern instabilities observed in the experimental studies. / Ph. D.

Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models. A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined. We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes. / Master of Science / The world is an elaborate system of relationships between diverse processes. To accurately represent these relationships, increasingly complex models are defined to better match what is physically seen. These complex models can lead to issues when trying to use them to predict a realistic outcome, either requiring immensely powerful computers to run the simulations or long amounts of time to present a solution. To fix this, surrogates or approximations to these complex models are used. These surrogate models aim to reduce the resources needed to calculate a solution while remaining as accurate to the more complex model as possible. One way to make these surrogate models is through neural networks. Neural networks try to simulate a brain, making connections between some input and output given to the network. In the case of surrogate modeling, the input is some current state of the true process, and the output is what is seen later from the same system. But much like the human brain, the reasoning behind why choices are made when connecting the input and outputs is often largely unknown. Within this paper, we seek to add meaning to neural network surrogate models in two different ways. In the first, we change what each piece in a neural network represents to build large polynomials (e.g., $x^5 + 4x^2 + 2$) to approximate the larger complex system. We show that the building of these polynomials via neural networks performs much better than traditional ways to construct them. For the second, we guide the choices made by the neural network by enforcing restrictions in what connections it can make. We do this by using additional information from the larger system to ensure the connections made focus on the most important information first before trying to match the less important patterns. This guiding process leads to more information being captured when the surrogate model is compressed into only a few dimensions compared to traditional methods. Additionally, it allows for a faster learning time compared to similar surrogate models without the information.

<p dir="ltr">Complex physical systems provide a fertile ground for exploring various phenomena owing to the quantum nature inherent in their structure. Atoms and molecules not only serve as realistic systems for experimental investigation, but also exhibit a complexity stemming from their many-body interactions which is of significant theoretical interest. This thesis delves into the domain of ultracold collisions between different interacting species (where temperature T < 1mK), and introduces novel applications for probing such systems, particularly focusing on molecular formation via photoassociation. Molecular interactions, in comparison to their atomic counterparts, present heightened complexity. The interplay of electrostatic forces among electrons and nuclei intricately couples all degrees of freedom within a single molecule. Historically, the exploration of quantum dynamics between molecules was pioneered by Born and Oppenheimer. Their seminal work involved solving Schrödinger’s equation in two steps. First step is addressing a portion of the molecular Hamiltonian where the nuclei are clamped in space (adiabatic). This adiabatic solution yields effective potentials between nuclei, encapsulating the integrated influence of the surrounding electronic cloud. The second step is to solve for the nuclear degrees of freedom in the vicinity of the effective potentials. The validity of the Born-Oppenheimer approximation stems from the substantial mass disparity between electrons and nuclei, enabling a quasi-separation of the electronic and nuclear Hamiltonians. The first order Born-Oppenheimer approximation assumes a partial separation of the molecular wave function Ψmolecule ≈ ΞvibrationYrotationalΦelectronic.</p><p dir="ltr"> A comprehensive treatment is provided for systems with numerous degrees of freedom, elucidating how the Born-Oppenheimer approximation manifests when applied to molecules. This chapter also encapsulates the principal findings from collision theory and photoassociation spectroscopy, as well as foundational techniques underpinning this thesis. Spectroscopic investigations encompass four relevant transition types: boundbound (Rabi oscillations), bound-free (photoionization), free-free (elastic scattering), and free-bound (photoassociation) transitions. Photoassociation (PA) spectroscopy probes laserinduced processes where the reactants interact through a channel |i〉, and can absorb one or more photons causing a transition to a bound state in an excited channel |f〉. The excited complex usually decays with a high probability to the ground state of the formed molecule. The same process can be utilized experimentally to prepare a cold molecule in its vibrational ground state . Diatomic PA has been of great theoretical and experimental interest in recent years. Herein, we present a theoretical inquiry into photoassociation within triatomic systems, with a particular focus on alkali atom-dimer systems, and introduce a method for calculating PA rates.</p><p dir="ltr">Moreover, this thesis presents different methods for shielding polar molecules from their short-range interactions where inelastic collisions and chemical reactions can occur with high probability. Shielding polar molecules has been shown to suppress inelastic collisions substantially between two molecules. A technique to shield two polar molecules in their ground state is studied and applied to model collisions in a gas of ground state (NaCs) molecules at temperatures T ≈ 100nK. The results show a region of interactions between two polar molecules that has an extremely long-range nature and is well isolated from the short-range losses, allowing for long-range spectroscopic studies. A new long-range regime of molecular physics arises in the study of shielded molecules where long-range vibrational tetramer states form. Different tetramer formation pathways are studied within a range of different shielding parameters. In fact, microwave shielding provides a region to study collisions between polar molecules, and controls their dynamics without worrying about shortrange losses. It has also been applied in the observation of a Bose gas of polar molecules.</p>

A three-dimensional, nonlinear dynamic analysis is conducted on a fully submerged, rigid, solid cylinder to be used as a breakwater. The breakwater could potentially be used as a single cylinder to protect small structures. Alternatively, multiple cylinders could be positioned in series to protect shorelines, harbors, or moored vessels from destructive incident water waves. The cylinder is positioned with its axis horizontal and is moored to the seafloor with four symmetrically placed massless mooring lines connected at the ends of the cylinder. The mooring lines are modeled as both linearly elastic ("regular") springs and compressionless springs. All six degrees of freedom of the structure are considered. The breakwater is modeled in air with a net buoyant force acting through the cylinder's center of gravity. The six "dry" natural frequencies of the structure are computed. Both linear and nonlinear free vibrations of the structure are considered. Linear damping is used to model the fluid and mooring damping effects. Normal and oblique harmonic wave forces at various frequencies and amplitudes are applied to the cylinder. The effects of the forcing amplitude and frequency, and the coefficient of damping, on the motion of the breakwater are studied. The results show that more erratic behavior occurs for the breakwater with compressionless springs, mainly due to the development of snap loads in the mooring lines. / Master of Science