In our work we analyze and implement numerical schemes for the infinite Prandtl number model for convection. This model describes the convection that is a potential driving force behind the flow and temperature of the Earth's mantle. There are many schemes available, but most are given with no mention of their ability to adequately capture the long time statistical properties of the model. We investigate schemes with the potential to actually capture these statistics. We further show numerically that our schemes align with current knowledge of the model's characteristics at low Rayleigh numbers. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2015. / July 30, 2015. / infinite, long time, numerical, Prandtl, statistical / Includes bibliographical references. / Xiaoming Wang, Professor Directing Dissertation; Qing-Xiang Amy Sang, University Representative; Bettye Anne Case, Committee Member; Brian Ewald, Committee Member; Max Gunzburger, Committee Member.

Symmetric positive definite (SPD) matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. This dissertation investigates different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. In addition, we offer theoretical and empirical suggestions on how to choose between various methods and parameters. In the end, we evaluate the performance of different averaging techniques in applications. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / July 20, 2018. / Includes bibliographical references. / Kyle A. Gallivan, Professor Co-Directing Dissertation; Pierre-Antoine Absil, Professor Co-Directing Dissertation; Gordon Erlebacher, University Representative; Giray Okten, Committee Member; Martin Bauer, Committee Member.

The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. In this work we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method to further reduce the computational cost; in total, three algorithms are developed. Initially a first order accurate in time scheme for low Reynolds number flows is considered. Next a second order algorithm useful for applications that require long-term time integration, such as climate and ocean forecasting is developed. Lastly, in order to extend this approach to convection dominated flows a model incorporating a POD spatial filter is presented. For all these schemes stability and convergence results for the ensemble-based methods are extended to the ensemble-POD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / June 27, 2018. / ensemble computation, Navier-Stokes equations, Proper Orthogonal Decomposition / Includes bibliographical references. / Max Gunzburger, Professor Directing Thesis; Mark Sussman, University Representative; Janet Peterson, Committee Member; Gordon Erlebacher, Committee Member; Chen Huang, Committee Member.

Diseases such as tuberculosis, chronic pneumonia, and inner ear infections are caused by bacterial biofilms. Biofilms can form on any surface such as teeth, floors, or drains. Many studies show that it is much more difficult to kill the bacteria in a biofilm than planktonic bacteria because the structure of biofilms offers additional layered protection against diffusible antimicrobials. Among the bacteria in planktonic-biofilm populations, persisters is a subpopulation that is tolerant to antibiotics and that appears to play a crucial role in survival dynamics. Understanding the dynamics of persister cells is of fundamental importance for developing effective treatments. In this research, we developed a method to better describe the behavior of persistent bacteria through specific experiments and mathematical modeling. We derived an accurate mathematical model by tightly coupling experimental data and theoretical model development. By focusing on dynamic changes in antibiotic tolerance owing to phenotypic differences between bacteria, our experiments explored specific conditions that are relevant to specifying parameters in our model. We deliver deeper intuitions to experiments that address several current hypotheses regarding phenotypic expression. By comparing our theoretical model to experimental data, we determined a parameter regime where we obtain quantitative agreement with our model. This validation supports our modeling approach and our theoretical predictions. This model can be used to enhance the development of new antibiotic treatment protocols. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / March 1, 2018. / Antibiotic resistance, Biofilm, Mathematical Modeling, Persisters / Includes bibliographical references. / Nick Cogan, Professor Directing Dissertation; Peter Beerli, University Representative; Richard Bertram, Committee Member; Giray Okten, Committee Member; Theodore Vo, Committee Member.

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2018. / November 12, 2018. / diffusion approximation, Hawkes process, risk model / Includes bibliographical references. / Lingjiong Zhu, Professor Directing Dissertation; Xufeng Niu, University Representative; Arash Fahim, Committee Member; Sanghyun Lee, Committee Member.

This dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the low-rank projection of the neighborhood pattern similarity measure and our theoretical analysis of the relationship between the rank of the similarity measure and the number of roles in the graph motivates our proposal to use Riemannian optimization to compute a low-rank approximation of the similarity measure. We propose two indirect approaches to use to solve the role extraction problem. The first uses the standard two-phase process. For the first phase, we propose using Riemannian optimization to compute a low-rank approximation of the similarity of the graph, and for the second phase using k-means clustering on the low-rank factor of the similarity matrix to extract the role partition of the graph. This approach is designed to be efficient in time and space complexity while still being able to extract good quality role partitions. We use basic experiments and applications to illustrate the time, robustness, and quality of our two-phase indirect role extraction approach. The second indirect approach we propose combines the two phases of our first approach into a one-phase approach that iteratively approximates the low-rank similarity matrix, extracts the role partition of the graph, and updates the rank of the similarity matrix. We show that the use of Riemannian rank-adaptive techniques when computing the low-rank similarity matrix improves robustness of the clustering algorithm. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / September 21, 2017. / blockmodeling, graph partitioning, networks, Riemannian optimization, role extraction problem / Includes bibliographical references. / Kyle A. Gallivan, Professor Co-Directing Dissertation; Paul Van Dooren, Professor Co-Directing Dissertation; Gordon Erlebacher, University Representative; Giray Ökten, Committee Member; Mark Sussman, Committee Member.

abstract: Two urban flows are analyzed, one concerned with pollutant transport in a Phoenix, Arizona neighborhood and the other with windshear detection at the Hong Kong International Airport (HKIA). Lagrangian measures, identified with finite-time Lyapunov exponents, are first used to characterize transport patterns of inertial pollutant particles. Motivated by actual events the focus is on flows in realistic urban geometry. Both deterministic and stochastic transport patterns are identified, as inertial Lagrangian coherent structures. For the deterministic case, the organizing structures are well defined and are extracted at different hours of a day to reveal the variability of coherent patterns. For the stochastic case, a random displacement model for fluid particles is formulated, and used to derive the governing equations for inertial particles to examine the change in organizing structures due to ``zeroth-order'' random noise. It is found that, (1) the Langevin equation for inertial particles can be reduced to a random displacement model; (2) using random noise based on inhomogeneous turbulence, whose diffusivity is derived from $k$-$\epsilon$ models, major coherent structures survive to organize local flow patterns and weaker structures are smoothed out due to random motion. A study of three-dimensional Lagrangian coherent structures (LCS) near HKIA is then presented and related to previous developments of two-dimensional (2D) LCS analyses in detecting windshear experienced by landing aircraft. The LCS are contrasted among three independent models and against 2D coherent Doppler light detection and ranging (LIDAR) data. Addition of the velocity information perpendicular to the lidar scanning cone helps solidify flow structures inferred from previous studies; contrast among models reveals the intramodel variability; and comparison with flight data evaluates the performance among models in terms of Lagrangian analyses. It is found that, while the three models and the LIDAR do recover similar features of the windshear experienced by a landing aircraft (along the landing trajectory), their Lagrangian signatures over the entire domain are quite different - a portion of each numerical model captures certain features resembling those LCS extracted from independent 2D LIDAR analyses based on observations. Overall, it was found that the Weather Research and Forecast (WRF) model provides the best agreement with the LIDAR data. Finally, the three-dimensional variational (3DVAR) data assimilation scheme in WRF is used to incorporate the LIDAR line of sight velocity observations into the WRF model forecast at HKIA. Using two different days as test cases, it is found that the LIDAR data can be successfully and consistently assimilated into WRF. Using the updated model forecast LCS are extracted along the LIDAR scanning cone and compare to onboard flight data. It is found that the LCS generated from the updated WRF forecasts are generally better correlated with the windshear experienced by landing aircraft as compared to the LIDAR extracted LCS alone, which suggests that such a data assimilation scheme could be used for the prediction of windshear events. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2018

Spread of infectious diseases progresses as a result of contacts between the individuals in a population. Therefore, it is crucial to gain insight into the pattern of connections to better understand and possibly control the spread of infectious diseases. Moreover, people may respond to an epidemic by changing their social behaviors to prevent infection. as a result, the structure of the network of social contacts evolves adaptively as a function of the disease status of the nodes. Recently, the dynamic relationships between different network topologies and adaptation mechanisms have attracted great attention in modeling epidemic spread. However, in most of these models, the original network structure is not preserved due to the adaptation mechanisms involving random changes in the links. In this dissertation, we study more realistic models with network structure constraints to retain aspects of the original network structure.;We study a susceptible-infected-susceptible (SIS) disease model on an adaptive network with two communities. Different levels of heterogeneity in terms of average connectivity and connection strength are considered. We study the effects of a disease avoidance adaptation mechanism based on the rewiring of susceptible-infected links through which the disease could spread. We choose the rewiring rules so that the network structure with two communities would be preserved when the rewiring links occur uniformly. The high dimensional network system is approximated with a lower dimensional mean field description based on a moment closure approximation. Good agreement between the solutions of the mean field equations and the results of the simulations are obtained at the steady state. In contrast to the non-adaptive case, similar infection levels in both of the communities are observed even when they are weakly coupled. We show that the adaptation mechanism tends to bring both the infection level and the average degree of the communities closer to each other.;In this rewiring mechanism, the local neighborhood of a node changes and is never restored to its previous state. However, in real life people tend to preserve their neighborhood of friends. We propose a more realistic adaptation mechanism, where susceptible nodes temporarily deactivate their links to infected neighbors and reactivate the links to those neighbors after they recover. Although the original network is static, the subnetwork of active links is evolving.;We drive mean field equations that predict the behavior of the system at the steady state. Two different regimes are observed. In the slow network dynamics regime, the adaptation simply reduces the effective average degree of the network. However, in the fast network dynamics regime, the adaptation further suppresses the infection level by reducing the dangerous links. In addition, non-monotonic dependence of the active degree on the deactivation rate is observed.;We extend the temporary deactivation adaptation mechanism to a scale-free network, where the degree distribution shows heavy tails. It is observed that the tail of the degree distribution of the active subnetwork has a different exponent than that of the original network. We present a heuristic explanation supporting that observation. We derive improved mean field equations based on a new moment closure approximation which is derived by considering the active degree distribution conditioned on the total degree. These improved mean field equations show better agreement with the simulation results than standard mean field analysis based on homogeneity assumptions.

In addition to first passage times, many look-back stopping times play a significant role in modeling various risks in insurance and finance as well as in defining financial instruments. Motivated by many recently arisen problems in risk management and exotic options, we study some look-back stopping times including drawdown and drawup, Parisian time and inverse occupation time of some time-homogeneous Markov processes such as diffusion processes and jump-diffusion processes. Since the structures of these look-back stopping times are much more complex than fundamental stopping times such as first passage times, we aim to develop some general approaches to study these stopping times such as approximation approach and perturbation approach. These approaches can be transformed to a wide class of stochastic processes. Many interesting and explicit formulas for these stopping times are derived and based on which we gain quantitative understandings of these problems in insurance and finance. In our study, we mainly use the techniques of Laplace transforms and partial differential equations (PDEs). Due to the complex structures, the distributions of these look-back stopping times are usually not explicit even for the simplest linear Brownian motion. However, under Laplace transforms, many important formulas become explicit and it enables us to conduct further derivations and analysis. Besides, PDE methodology provides us an effective and efficient approach in both theoretical investigation and numerical study of these stopping times.

In Chapter 1, we study the effects of behavioral changes in a smallpox attack model. Response strategies to a smallpox bioterrorist attack have focused on interventions such as isolation, contact tracing, quarantine, ring vaccination, and mass vaccination. We formulate and analyze a mathematical model in which some individuals lower their daily contact activity rates once an epidemic has been identified in a community. We use computer simulations to analyze the effects of behavior change alone and in combination with other control measures. We demonstrate that the spread of the disease is highly sensitive to how rapidly people reduce their contact activity. In Chapter 2, we study mixing patterns between age groups using social networks. The course of an epidemic through a population is determined by the interactions among individuals. To capture these elements of reality, we use the contact network simulations for the city of Portland, Oregon that were developed as part of the TRANSIMS/EpiSims project to study and identify mixing patterns. We analyze contact patterns between different age groups and identify those groups who are at higher risk of infection. We describe a new method for estimating transmission matrices that describe the mixing and the probability of transmission between the age groups. We use this matrix in a simple differential equation model for the spread of smallpox. Our differential equation model shows that the epidemic size of a smallpox outbreak could be greatly affected by the level of residual immunity in the population. In Chapter 3, we study the effects of mixing patterns in the presence of population heterogeneity. We investigate the impact that different mixing assumptions have on the spread of a disease in an age-structured differential equation model. We use realistic, semi-bias and bias mixing matrices and investigate the impact that these mixing patterns have on epidemic outcomes when compared to random mixing. Furthermore, we investigate the impact of population heterogeneity such as differences in susceptibility and infectivity within the population for a smallpox epidemic outbreak. We find that different mixing assumptions lead to differences in disease prevalence and final epidemic size.