One of the basic questions of concern in mathematical biology is the long-term survival of each species in a set of populations. This question is particularly puzzling for a natural system with omnivory due to the fact that simple mathematical models of omnivory are prone to species extinction. Omnivory is defined as the consumption of resources from more than one trophic level. In this work, we investigate three omnivory models of increasing complexity. We use the notion of permanent coexistence, or permanence, to study the long-term survival of three interacting species governed by a mixture of competition and predation. We show the permanence of our models under certain parameter restrictions and include the biological interpretations of these parameter restrictions. Sensitivity analysis is used to obtain important information about meaningful parameter data collection. Examples are also given that demonstrate the ubiquity of omnivory in natural systems. / Ph. D.

A mathematical model for the determination of the aerodynamic forces acting on an aircraft is presented. The mathematical model is based on the generalization of the idea of aerodynamically steady motions. One important use of these results is the determination of steady (time-invariant) aerodynamic forces and moments. Such aerodynamic forces can be determined using computer simulation by determining numerically the associated steady flows around the aircraft when it is moving along such generalized steady trajectories. The method required the extension of standard (inertial) CFD formulations to general non-inertial reference frames. Generalized Navier-Stokes and Euler equations have been derived. The formulation is valid for all ranges of Mach numbers including transonic flow. The method was implemented numerically for the planar case using the generalized Euler equations. The developed computer codes can be used to obtain numerical flow solutions for airfoils moving in general steady motions (i.e. circular motions). From these numerical solutions it is possible to determine the variation of the lift, drag and pitching moment with respect to the pitch rate at different Mach numbers and angles of attack. One of the advantages of the mathematical model developed here is that the aerodynamic forces become well-defined functions of the motion variables (including angular rates). In particular, the stability derivatives are associated with partial derivatives of these functions. These stability derivatives can be computed using finite differences or the sensitivity equation method. / Ph. D.

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new "mostly linear" theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored. The main goal of this work is to begin to study the role of nonlinearity in transition. We use model problems to illustrate that small unmodeled disturbances can cause transition through movement or bifurcation of equilibria. We also demonstrate that small wall roughness can lead to transition by causing the linearized system to become unstable. Sensitivity methods are used to obtain important information about the disturbed problem and to illustrate that it is possible to have a precursor to predict transition. Finally, we apply linear feedback control to the model problems to illustrate the power of feedback to delay transition and even relaminarize fully developed chaotic flows. / Ph. D.

Simulating the complex physics and dynamics associated with unsteady aeroelastic systems is often attempted with high-fidelity numerical models. While these high-fidelity approaches are powerful in terms of capturing the main physical features, they may not discern the role of underlying phenomena that are interrelated in a complex manner. This often makes it difficult to characterize the relevant causal mechanisms of the observed features. Besides, the extensive computational resources and time associated with the use these tools could limit the capability of assessing different configurations for design purposes. These shortcomings present the need for the development of simplified and reduced-order models that embody relevant physical aspects and elucidate the underlying phenomena that help in characterizing these aspects. In this work, different fluid and aeroelastic systems are considered and reduced-order models governing their behavior are developed. In the first part of the dissertation, a methodology, based on the method of multiple scales, is implemented to show its usefulness and effectiveness in the characterization of the physics underlying the system, the implementation of control strategies, and the identification of high-impact system parameters. In the second part, the unsteady aerodynamic aspects of flapping micro air vehicles (MAVs) are modeled. This modeling is required for evaluation of performance requirements associated with flapping flight. The extensive computational resources and time associated with the implementation of high-fidelity simulations limit the ability to perform optimization and sensitivity analyses in the early stages of MAV design. To overcome this and enable rapid and reasonably accurate exploration of a large design space, a medium-fidelity aerodynamic tool (the unsteady vortex lattice method) is implemented to simulate flapping wing flight. This model is then combined with uncertainty quantification and optimization tools to test and analyze the performance of flapping wing MAVs under varying conditions. This analysis can be used to provide guidance and baseline for assessment of MAVs performance in the early stages of decision making on flapping kinematics, flight mechanics, and control strategies. / Ph. D.

A multidisciplinary approach for the modeling and analysis of the performance of Olympic rowing boats is presented. The goal is to establish methodologies and tools that would determine the effects of variations in applied forces and rowers motions and weights on mean surge speed and oscillatory boat motions. The coupling between the rowers motions with the hull and water forces is modeled with a system of equations. The water forces are computed using several fluid dynamic models that have different levels of accuracy and computational cost. These models include a solution of the Reynolds Averaged Navier--Stokes equations complemented by a Volume of Fluid method, a linearized 3D potential flow simulation and a 2D potential flow simulation that is based on the strip theory approximation. These results show that due to the elongated shape of the boat, the use of Sommerfeld truncation boundary condition does not yield the correct frequency dependence of the radiative coefficients. Thus, the radiative forces are not computed in the time-domain problem by means of a convolution integral, accounting for flow memory effects, but were computed assuming constant damping and added mass matrices. The results also show that accounting for memory effects significantly improves the agreement between the strip theory and the RANS predictions. Further improvements could be obtained by introducing corrections to account for longitudinal radiative forces, which are completely neglected in the strip theory. The coupled dynamical system and the multi-fidelity fluid models of the water forces were then used to perform a sensitivity analysis of boat motions to variations in rowers weights, exerted forces and cadence of motion. The sensitivity analysis is based on the polynomial chaos expansion. The coefficients of each random basis in the polynomial chaos expansion are computed using a non-intrusive strategy. Sampling, quadrature, and linear regression methods have been used to obtain the these coefficients from the outputs generated by the system at each sampling point. The results show that the linear regression method provides a very good approximation of the PCE coefficients. In addition, the number of samples needed for the expansion, does not grow exponentially with the number of varying input parameters. For this reason, this method has been selected for performing the sensitivity analysis. The sensitivity of output parameters to variations in selected input parameters of the system are obtained by taking the derivatives of the expansion with respect to each input parameter. Three test cases are considered: a light-weight female single scull, a male quad scull, and a male coxless four. For all of these cases, results that relate the effects of variations in rowers weights, amplitudes of exerted forces and cadence of rowing on mean boat speed and energy ratio, defined as the ratio of kinetic energy of the forward motion to that of the oscillatory motions, are presented. These results should be useful in the design of rowing boats as well as in the training of rowers. / Ph. D.

Modifications of large reflector antennas, such that their observing capabilities are enhanced in the range of about 10-500~MHz without affecting operation of the pre-existing higher-frequency systems, are addressed in this dissertation. The major contributions of this dissertation can be divided into two parts: 1) designing new low frequency feeds, and 2) developing new analysis methodologies which, as opposed to traditional techniques, are suitable for analyzing low frequency systems. First, we consider the performance of existing schemes that provide low frequency capability. Then, a new class of dipole-based low frequency feeds - namely, the ``distributed feed array'' - is designed to cover the frequency range of interest without affecting operation at higher frequencies. As an example, distributed feed arrays are designed for the Expanded Very Large Array (EVLA) to cover the range of 50-250~MHz. A method of moments (MoM) model of an EVLA antenna is developed for this purpose. The new design shows performance comparable to the existing 4 m system on the EVLA in the range of 50-88~MHz, and introduces observing capabilities in the range of 110-250~MHz (currently not covered by the EVLA). Moreover, the blockage presented to the existing EVLA L-band system is reduced significantly when the existing 4 m system is replaced by the proposed system. At low frequencies, external noise can be a significant or dominant contribution to the total noise of the system. This, combined with mutual coupling between the array elements of the distributed feed array, makes it difficult to predict the sensitivity of these systems. This dissertation describes a system model and procedure for estimating the system equivalent flux density (SEFD) - a useful and meaningful metric of the sensitivity of a radio telescope - that accounts for these issues. We consider the efficiency of methods other than MoM - in particular, Physical Optics (PO), Uniform Geometrical Theory of Diffraction (UTD), and hybrid methods - for accelerated computation at low frequencies. A method for estimating the blockage presented by low frequency systems to the pre-existing higher-frequency systems is also described. / Ph. D.

Quantitative phase imaging (QPI) is a category of imaging techniques that can retrieve the phase information of the sample quantitatively. QPI features label-free contrast and non-contact detection. It has thus gained rapidly growing attention in biomedical imaging. Capable of resolving biological specimens at tissue or cell level, QPI has become a powerful tool to reveal the structural, mechanical, physiological and spectroscopic properties. Over the past two decades, QPI has seen a broad spectrum of evolving implementations. However, only a few have seen successful commercialization. The challenges are manifold. A major problem for many QPI techniques is the necessity of a custom-made system which is hard to interface with existing commercial microscopes. For this type of QPI techniques, the cost is high and the integration of different imaging modes requires nontrivial hardware modifications. Another limiting factor is insufficient sensitivity. In QPI, sensitivity characterizes the system repeatability and determines the quantification resolution of the system. With more emerging applications in cell imaging, the requirement for sensitivity also becomes more stringent. In this work, a category of highly sensitive full-field QPI techniques based on wavelength shifting interferometry (WSI) is proposed. On one hand, the full-field implementations, compared to point-scanning, spectral domain QPI techniques, require no mechanical scanning to form a phase image. On the other, WSI has the advantage of preserving the integrity of the interferometer and compatibility with multi-modal imaging requirement. Therefore, the techniques proposed here have the potential to be readily integrated into the ubiquitous lab microscopes and equip them with quantitative imaging functionality. In WSI, the shifts in wavelength can be applied in fine steps, termed swept source digital holographic phase microscopy (SS-DHPM), or a multi-wavelength-band manner, termed low coherence wavelength shifting interferometry (LC-WSI). SS-DHPM brings in an additional capability to perform spectroscopy, whilst the LC-WSI achieves a faster imaging rate which has been demonstrated with live sperm cell imaging. In an attempt to integrate WSI with the existing commercial microscope, we also discuss the possibility of demodulation for low-cost sources and common path implementation. Besides experimentally demonstrating the high sensitivity (limited by only shot noise) with the proposed techniques, a novel sensitivity evaluation framework is also introduced for the first time in QPI. This framework examines the Cramér-Rao bound (CRB), algorithmic sensitivity and experimental sensitivity, and facilitates the diagnosis of algorithm efficiency and system efficiency. The framework can be applied not only to the WSI techniques we proposed, but also to a broad range of QPI techniques. Several popular phase shifting interferometry techniques as well as off-axis interferometry is studied. The comparisons between them are shown to provide insights into algorithm optimization and energy efficiency of sensitivity. / Doctor of Philosophy / The most common imaging systems nowadays capture the image of an object with the irradiance perceived by the camera. Based on the intensity contrast, morphological features, such as edges, humps, and grooves, can be inferred to qualitatively characterize the object. Nevertheless, in scientific measurements and research applications, a quantitative characterization of the object is desired. Quantitative phase imaging (QPI) is such a category of imaging techniques that can retrieve the phase information of the sample by properly design the irradiance capturing scheme and post-process the data, converting them to quantitative metrics such as surface height, material density and so on. The imaging process of QPI will neither harm the sample nor leave exogenous residuals. As a result, it has thus gained rapidly growing attention in biomedical imaging. Over the past two decades, QPI has seen a broad spectrum of evolving implementations, but only a few have seen successful commercialization. The challenges are manifold whilst one stands out - that they have expensive optical setups that are often incompatible with existing commercial microscope platforms. The setups are also very complicated such that without professionals having solid optics background, it is difficult to operate the system to perform imaging applications. Another limiting factor is the insufficient understanding of sensitivity. In QPI, sensitivity characterizes the system repeatability and determines its quantification resolution. With more emerging applications in cell imaging, the requirement for sensitivity also becomes more stringent. In this work, a category of highly sensitive full-field QPI techniques based on wavelength shifting interferometry (WSI) is proposed. WSI images the full-field of the sample simultaneously, unlike some other techniques requiring scanning one probe point across the sample. It also has the advantage of preserving the integrity of the interferometer, which is the key structure to enable highly sensitive measurement for QPI methods. Therefore, the techniques proposed here have the potential to be readily integrated into the ubiquitous lab microscopes and equip them with quantitative imaging functionality. Differed by implementations, two WSI techniques have been proposed, termed swept source digital holographic phase microscopy (SS-DHPM), and low coherence wavelength shifting interferometry (LC-WSI), respectively. SS-DHPM brings in an additional capability to perform spectroscopy, whilst the LC-WSI achieves a faster imaging rate which has been demonstrated with live sperm cell imaging. In an attempt to integrate WSI with the existing commercial microscope, we also discuss the possibility of demodulation for low-cost sources and common path implementation. Besides experimentally demonstrating the high sensitivity with the proposed techniques, a novel sensitivity evaluation framework is also introduced for the first time in QPI. This framework not only examines the realistic sensitivity obtained in experiments, but also compares it to the theoretical values. The framework can be widely applied to a broad range of QPI techniques, providing insights into algorithm optimization and energy efficiency of sensitivity.

Emulation of computer experiments via surrogate models can be difficult when the number of input parameters determining the simulation grows any greater than a few dozen. In this dissertation, we explore dimension reduction in the context of computer experiments. The active subspace method is a linear dimension reduction technique which uses the gradients of a function to determine important input directions. Unfortunately, we cannot expect to always have access to the gradients of our black-box functions. We thus begin by developing an estimator for the active subspace of a function using kernel methods to indirectly estimate the gradient. We then demonstrate how to deploy the learned input directions to improve the predictive performance of local regression models by ``undoing" the active subspace. Finally, we develop notions of sensitivities which are local to certain parts of the input space, which we then use to develop a Bayesian optimization algorithm which can exploit locally important directions. / Doctor of Philosophy / Increasingly, scientists and engineers developing new understanding or products rely on computers to simulate complex phenomena. Sometimes, these computer programs are so detailed that the amount of time they take to run becomes a serious issue. Surrogate modeling is the problem of trying to predict a computer experiment's result without having to actually run it, on the basis of having observed the behavior of similar simulations. Typically, computer experiments have different settings which induce different behavior. When there are many different settings to tweak, typical surrogate modeling approaches can struggle. In this dissertation, we develop a technique for deciding which input settings, or even which combinations of input settings, we should focus our attention on when trying to predict the output of the computer experiment. We then deploy this technique both to prediction of computer experiment outputs as well as to trying to find which of the input settings yields a particular desired result.

In recent years, several methods have been proposed for real-time modeling and forecasting of the epidemic curve. These methods range from simple compartmental models to complex agent-based models. In this dissertation, we present a model-based reasoning approach to forecasting the epidemic curve and estimating underlying disease model parameters. The method is based on building an epidemic library consisting of past and simulated influenza outbreaks. During an influenza epidemic, we use a combination of statistical, optimization and modeling techniques to either assign the epidemic to one of the cases in the library or propose parameters for modeling the epidemic. The method is presented in four steps. First, we discuss a sensitivity analysis study evaluating how minute changes in the disease model parameters influence the dynamics of simulated epidemics. Next, we present a supervised classification method for predicting the epidemic curve. The epidemic curve is forecasted by matching the partial surveillance curve to at least one of the epidemics in the library. We expand on the classification approach by presenting a method which identifies epidemics similar or different from those in the library. Lastly, we discuss a simulation optimization method for estimating model parameters to forecast the epidemic curve of an epidemic distinct from those in the library. / Ph. D.

The prevalence of obesity is high in the United States. One of the many concerns with the high prevalence of obesity is its association with an increased risk of falls and subsequent injury. Thus, it is important to understand factors affecting falls among individuals who are obese, to help develop effective intervention solutions to mitigate falls in this population. Obese individuals have been hypothesized to have an impaired plantar sensitivity, and this may influence their balance control, thus lead to more falls. Executive function deficits in individuals who are obese may affect their ability to allocate attentional resources to dual tasks (walking while performing other tasks), and may put them at higher risks of falls. Gait alterations and muscle strength deficits in individuals who are obese may also increase their fall risks. Therefore, three studies were carried out to provide better understanding into the factors affecting falls in individuals who are obese. The first study investigated the effects of obesity on plantar sensitivity, and explored the relationship between plantar sensitivity and postural sway during quiet standing. Plantar sensitivity was measured as the force threshold at which an increasing force applied to the plantar surface of the foot was first perceived, and the force threshold at which a decreasing force was last perceived. Measurements were obtained while standing, and at two locations on the plantar surface of the dominant foot. Postural sway during quiet standing was then measured under three different sensory conditions. Results indicated less sensitive plantar sensitivity and increased postural sway among individuals who are obese, and statistically significant correlations between plantar sensitivity and postural sway that were characterized as weak to moderate in strength. As such, impaired plantar sensitivity among individuals who are obese may be a mechanism by which obesity degrades standing balance among these individuals. The second study investigated the influence of obesity on executive function, and determined whether there is a relationship between executive function and fall risk (as estimated from selected gait parameters). Four major components of executive function were assessed, including selective attention, divided attention, semantic memory and working memory. Both single- and dual-task walking (walking-while-talking) were completed to evaluate fall risk during gait. Less effective selective attention, semantic memory, and working memory were found among young obese adults. Participants exhibited higher fall risks during dual-task walking, and executive function scores were associated with gait during dual-task walking. In conclusion, obese individuals exhibited less effective executive function, which may be associated with their increased fall risk. The third study explored differences in gait, plantar sensitivity, executive function, lower extremity muscle strength, and body size between fallers and non-fallers, and the strength of the association between the same factors and slip severity. Participants' gait, plantar sensitivity, executive function, lower extremity muscle strength, and body size measures were obtained. An unexpected slip was introduced in a laboratory setting to obtain slip severity related measures and slip outcome. Results indicated obese fallers exhibited better executive function (selective attention), stronger lower extremity muscle strength, lower BMI and smaller waist circumference. Results also indicated increased slip severity was associated with faster walking speed, longer step length, higher RCOF, worse executive function (working memory), and lower BMI. Slower reactive recovery response was also associated with lower BMI. As such, better selective attention and stronger muscle strength exhibited limited benefit in slip recovery among individuals who are obese. Altered gait pattern, and working memory may be factors by which obesity increased slip severity, and lower BMI among individuals who are obese may increase slip-induced fall risks. In conclusion, reduced plantar sensitivity, impairments in executive function, altered gait pattern were associated with deficits in standing and walking balance control, and increased slip severity among individuals who are obese. Therefore, appropriate fall prevention/intervention program targeting at some or all of these factors may be considered as solutions to decrease fall risks for obese individuals. / Ph. D.