<p> We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and exceeds that of perfectly matched layers for comparable effort. </p><p> We also describe the application of CRBC type boundary conditions for elastic waves in (an)isotropic media. The results show that we can optimize the CRBC problems for a subset of elastic media. Additionally, we propose a generalized CRBC type boundary conditions that may be more applicable to elastic wave equations and present some preliminary results.</p>

Many phenomena in ecology, evolution, and organismal biology relate to how a system changes through time. Unfortunately, most of the statistical methods that are common in these fields represent samples as static scalars or vectors. Since variables in temporally-dynamic systems do not have stable values this representation is unideal. Differential equation and basis function representations provide alternative systems for description, but they are also not without drawbacks of their own. Differential equations are typically outside the scope of statistical inference, and basis function representations rely on functions that solely relate to the original data in regards to qualitative appearance, not in regards to any property of the original system. In this dissertation, I propose that vector autoregressive-moving average (VARMA) and vector autoregressive (VAR) processes can represent temporally-dynamic systems. Under this strategy, each sample is a time series, instead of a scalar or vector. Unlike differential equations, these representations facilitate statistical description and inference, and, unlike basis function representations, these processes directly relate to an emergent property of dynamic systems, their cross-covariance structure. In the first chapter, I describe how VAR representations for biological systems lead to both a metric for the difference between systems, the Euclidean process distance, and to a statistical test to assess whether two time series may have originated from a single VAR process, the likelihood ratio test for a common process. Using simulated time series, I demonstrate that the likelihood ratio test for a common process has a true Type I error rate that is close to the pre-specified nominal error rate, regardless of the number of subseries in the system or of the order of the processes. Further, using the Euclidean process distance as a measure of difference, I establish power curves for the test using logistic regression. The test has a high probability of rejecting a false null hypothesis, even for modest differences between series. In addition, I illustrate that if two competitors follow the Lotka-Volterra equations for competition with some additional white noise, the system deviates from VAR assumptions. Yet, the test can still differentiate between a simulation based on these equations in which the constraints on the system change and a simulation where the constraints do not change. Although the Type I error rate is inflated in this scenario, the degree of inflation does not appear to be larger when the system deviates more noticeably from model assumptions. In the second chapter, I investigate the likelihood ratio test for a common process's performance with shape trajectory data. Shape trajectories are an extension of geometric morphometric data in which a sample is a set of temporally-ordered shapes as opposed to a single static shape. Like all geometric morphometric data, each shape in a trajectory is inherently high-dimensional. Since the number of parameters in a VAR representation grows quadratically with the number of subseries, shape trajectory data will often require dimension reduction before a VAR representation can be estimated, but the effects that this reduction will have on subsequent inferences remains unclear. In this study, I simulated shape trajectories based on the movements of roundworms. I then reduced the number of variables that described each shape using principle components analysis. Based on these lower dimensional representations, I estimated the likelihood ratio test's Type I error rate and power with the simulated trajectories. In addition, I also used the same workflow on an empirical dataset of women walking (originally from Morris13) but also tried varying amounts of preprocessing before applying the workflow as well. The likelihood ratio test's Type I error rate was mildly inflated with the simulated shape trajectories but had a high probability of rejecting false null hypotheses. Without preprocessing, the likelihood ratio test for a common process had a highly inflated Type I error rate with the empirical data, but when the sampling density is lowered and the number of cycles is standardized within a comparison the degree of inflation becomes comparable to that of the simulated shape trajectories. Yet, these preprocessing steps do not appear to negatively impact the test's power. Visualization is a crucial step in geometric morphometric studies, but there are currently few, if any, methods to visualize differences in shape trajectories. To address this absence, I propose an extension to the classic vector-displacement diagram. In this new procedure, the VAR representations for two trajectories' processes generate two simulated trajectories that share the same shocks. Then, a vector-displacement diagram compares the simulated shapes at each time step. The set of all diagrams then illustrates the difference between the trajectories' processes. I assessed the validity of this procedure using two simulated shape trajectories, one based on the movements of roundworms and the other on the movements of earthworms. The result provided mixed results. Some diagrams do show comparisons between shapes that are similar to those in the original trajectories but others do not. Of particular note, diagrams show a bias towards whichever trajectory's process was used to generate pseudo-random shocks. This implies that the shocks to the system are just as crucial a component to a trajectory's behavior as the VAR model itself. Finally, in the third chapter I discuss a new R library to study dynamic systems and represent them as VAR and VARMA processes, iPARMA. Since certain processes can have multiple VARMA representations, the routines in this library place an emphasis on the reverse echelon format. For every process, there is only one VARMA model in reverse echelon format. The routines in iPARMA cover a diverse set of topics, but they all generally fall into one of four categories: simulation and study, model estimation, hypothesis testing, and visualization methods for shape trajectories. Within the chapter, I discuss highlights and features of key routines' algorithms, as well as how they differ from analogous routines in the R package MTS \citep{mtsCite}. In many regards, this dissertation is foundational, so it provides a number of lines for future research. One major area for further work involves alternative ways to represent a system as a VAR or VARMA process. For example, the parameter estimates in a VAR or VARMA model could depict a process as a point in parameter space. Other potentially fruitful areas include the extension of representational applications to other families of time series models, such as co-integrated models, or altering the generalized Procrustes algorithm to better suit shape trajectories. Based on these extensions, it is my hope that statistical inference based on stochastic process representations will help to progress what systems biologists are able to study and what questions they are able to answer about them. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / May 3, 2017. / Function-valued Trait, Geometric morphometrics, Shape trajectory, Stochastic process, Time series analysis, Vector autoregressive-moving average (VARMA) model / Includes bibliographical references. / Dennis E. Slice, Professor Directing Dissertation; Paul M. Beaumont, University Representative; Peter Beerli, Committee Member; Anke Meyer-Baese, Committee Member; Sachin Shanbhag, Committee Member.

In many composite materials, rigid fibers are distributed throughout the material to tune the mechanical, thermal, and electric properties of the composite. The orientation and distribution of the fibers play a critical role in the properties of the composite. Many composites are processed as a liquid molten suspension of fibers and then solidified, holding the fibers in place. Once the fiber orientations are known, theoretical models exist that can predict properties of the composite.Modeling the suspended fibers in the liquid state is important because their ultimate configuration depends strongly on the flow history during the molten processing. Continuum models, such as the Folgar-Tucker model, predict the evolution of the fibers’ orientation in a fluid. These models are limited in several ways. First, they require empirical constants and closure relations that must be determined a priori, either by experiments or detailed computer simulations. Second, they assume that all the fibers are slender bodies of uniform length. Lastly, these methods break down for concentrated suspensions. For these reasons, it is desirable in certain situations to model the movement of individual fibers explicitly. This dissertation builds upon recent advances in boundary integral equations to develop a robust, accurate, and stable method that simulates fibers of arbitrary shape in a planar flow. In any method that explicitly models the individual fiber motion, care must be taken to ensure numerical errors do not cause the fibers to overlap. To maintain fiber separation, a repulsion force and torque are added when required. This repulsion force is free of tuning parameters and is determined by solving a sequence of linear complementarity problems to ensure that the configuration does not have any overlap between fibers. Numerical experiments demonstrate the stability of the method for concentrated suspensions. / 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. / July 16, 2018. / Boundary Integral Equations, Complementarity Problems, Rigid Particle Suspensions / Includes bibliographical references. / Bryan Quaife, Professor Co-Directing Dissertation; Sachin Shanbhag, Professor Co-Directing Dissertation; Nick Cogan, University Representative; Chen Huang, Committee Member; Nick Moore, Committee Member.

Unsteady fluid flows have complex dynamics due to the nonlinear interactions amongst vortical elements. In this thesis, a network-theoretic framework is developed to describe vortical and modal (coherent structure) interactions in unsteady fluid flows. A sparsified-dynamics model and a networked-oscillator model describe the complex dynamics in fluid flows in terms of vortical and modal networks, respectively. Based on the characterized network interactions, model-based feedback control laws are established, particularly for controlling the flow unsteadiness. Furthermore, to characterize model-free feedback control laws for suppressing flow separation in turbulent flows, a data-driven approach leveraging unsupervised clustering is developed. This approach alters the Markov transition dynamics of fluid flow trajectories in an optimal manner using a cluster-based control strategy. To describe vortical interactions, dense fluid flow graphs are constructed using discrete point vortices as nodes and induced velocity as edge weights. Sparsification techniques are then employed on these graph representations based on spectral graph theory to construct sparse graphs of the overall vortical interactions which maintain similar spectral properties as the original setup. Utilizing the sparse vortical graphs, a sparsified-dynamics model is developed which drastically reduces the computational cost to predict the dynamical behavior of vortices, sharing characteristics of reduced-order models. The model retains the nonlinearity of the interactions and also conserves the invariants of discrete vortex dynamics. The network structure of vortical interactions in two-dimensional incompressible homogeneous turbulence is then characterized. The strength distribution of the turbulence network reveals an underlying scale-free structure that describes how vortical structures are interconnected. Strong vortices serve as network hubs with smaller and weaker eddies predominantly influenced by the neighboring hubs. The time evolution of the fluid flow network informs us that the scale-free property is sustained until dissipation overtakes the flow physics. The types of perturbations that turbulence network is resilient against is also examined. To describe modal interactions in fluid flows, a networked-oscillator-based analysis is performed. The analysis examines and controls the transfer of kinetic energy for periodic bluff body flows. The dynamics of energy fluctuations in the flow field are described by a set of oscillators defined by conjugate pairs of spatial POD modes. To extract the network of interactions among oscillators, impulse responses of the oscillators to amplitude and phase perturbations are tracked. Using linear regression techniques, a networked oscillator model is constructed that reveals energy exchanges among the modes. In particular, a large collection of system responses are aggregated to capture the general network structure of oscillator interactions. The present networked oscillator model describes the modal perturbation dynamics more accurately than the empirical Galerkin reduced-order model. The linear network model for nonlinear dynamics is subsequently utilized to design a model-based feedback controller. The controller suppresses the modal fluctuations and amplitudes that result in wake unsteadiness leading to drag reduction. The strength of the approach is demonstrated for a canonical example of two-dimensional unsteady flow over a circular cylinder. The network-based formulation enables the characterization and control of modal interactions to control fundamental energy transfers in unsteady bluff body flows. Finally, unsupervised clustering and data-driven optimization of coarse-grained control laws is leveraged to manipulate post-stall separated flows. Optimized feedback control laws are deduced in high-fidelity simulations in an automated, model-free manner. The approach partitions the baseline flow trajectories into clusters, which corresponds to a characteristic coarse-grained phase in a low-dimensional feature space constituted by feature variables (sensor measurements). The feedback control law is then sought for each and every cluster state which is iteratively evaluated and optimized to minimize aerodynamic power and actuation power input. The control optimally transforms the Markov transition network associated with the baseline trajectories to achieve desired performance objectives. The approach is applied to two and three-dimensional separated flows over a NACA 0012 airfoil at an angle of attack of 9° Reynolds number Re = 23000 and free-stream Mach number M∞ = 0.3. The optimized control law minimizes power consumption for flight enabling flow to reach a low-drag state. The analysis provides insights for feedback flow control of complex systems characterizing global cluster-based control laws based on a data-driven, low-dimensional characterization of fluid flow trajectories. In summary, this thesis develops a novel network-theoretic and data-based framework for analyzing and controlling fluid flows. The framework incorporates advanced mathematical principles from network science, graph theory and dynamical systems to extract fundamental interactions in fluid flows. On manipulating these interactions, wake unsteadiness in bluff body flow is reduced leading to drag reduction. Finally, data-based methods are developed to deduce optimal feedback control laws for post-stall separated flows. The network-theoretic and data-based approaches provides insights on fundamental interactions in fluid flows which paves the way for design of novel flow control strategies. / A Dissertation submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2018. / July 23, 2018. / Cluster-based control, Modal interactions, Networked-oscillators, Scale-free networks, Sparsified vortex dynamics / Includes bibliographical references. / Kunihiko Taira, Professor Directing Dissertation; Mark Sussman, University Representative; Louis N. Cattafesta, Committee Member; William S. Oates, Committee Member; Farrukh S. Alvi, Committee Member; Steven L. Brunton, Committee Member.

In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the continuous problem in the weak sense. This method is thus aptly named the Polynomial Extension Finite Element Method (PE--FEM). Using this approach, we may approximate the solution of elliptic interface problems by enforcing that the extension of the solution on their respective subdomains matches weakly the continuity conditions prescribed by the continuous problem on a curved interface. This method is then called the Method of Virtual Interfaces (MVI), since, while the continuous interface exists in the context of the continuous problem, it is virtual in the context of its numerical approximation. The key benefits of this polynomial extension approach is that it is simple to implement and that it is optimally convergent with respect to the best approximation results given by interpolation. Theoretical analysis and computational results are presented. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / January 31, 2018. / Finite Element Method, High-Order Scheme, Polynomial Extension / Includes bibliographical references. / Max Gunzburger, Professor Directing Dissertation; Oliver Steinbock, University Representative; Pavel Bochev, Committee Member; Mauro Perego, Committee Member; Janet Peterson, Committee Member; Sachin Shanbhag, Committee Member; Peter Beerli, Committee Membe.

Neuron morphology plays a central role in characterizing cognitive health and functionality of brain structures. The problem of quantifying neuron shapes, and capturing statistical variability of shapes, is difficult because axons and dendrites have tree structures that differ in both geometry and topology. In this work, we restrict to the trees that consist of: (1) a main branch viewed as a parameterized curve in ℝ³, and (2) some number of secondary branches -- also parameterized curves in ℝ³ -- which emanate from the main branch at arbitrary points. We present two shape-analytic frameworks which each give a metric structure to the set of such tree shapes, Both frameworks are based on an elastic metric on the space of curves with certain shape-preserving nuisance variables modded out. In the first framework, the side branches are treated as a continuum of curve-valued annotations to the main branch. In the second framework, the side branches are treated as discrete entities and are matched to each other by permutation. We show geodesic deformations between tree shapes in both frameworks, and we show Fréchet means and modes of variability, as well as cross-validated classification between different experimental groups using the second framework. We conclude with a smaller project which extends some of these ideas to more general weighted attributed graphs. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2018. / April 18, 2018. / Includes bibliographical references. / Anuj Srivastava, Professor Directing Dissertation; Eric Klassen, University Representative; Wei Wu, Committee Member; Fred Huffer, Committee Member.

Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating time-dependent nonlocal integro-differential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due to the approximation of the nonlocal integral. In this work, we propose a novel method based on using radial basis functions to generate accurate quadrature rules for the nonlocal integral appearing in the model and then coupling this with a finite difference approximation to the time-dependent terms. The viability of our method is demonstrated through various numerical tests on time dependent nonlocal diffusion, nonlocal anomalous diffusion, and nonlocal advection problems in one and two dimensions. In addition to nonlocal problems with continuous solutions, we modify our approach to handle problems with discontinuous solutions. We compare some numerical results with analogous finite element results and demonstrate that for an equivalent amount of computational work we obtain much higher rates of convergence. / A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2018. / November 8, 2018. / Nonlocal Models, Radial Basis Functions / Includes bibliographical references. / Janet Peterson, Professor Directing Dissertation; Ziad Musslimani, University Representative; Max Gunzburger, Committee Member; Bryan Quaife, Committee Member; Sachin Shanbhag, Committee Member.

Dissertation focuses on exploring the capabilities of the SRSF statistical shape analysis framework through various applications. Each application gives rise to a specific mathematical shape analysis model. The theoretical investigation of the models, driven by real data problems, give rise to new tools and theorems necessary to conduct a sound inference in the space of shapes. From theoretical standpoint the robustness results are provided for the model parameters estimation and an ANOVA-like statistical testing procedure is discussed. The projects were a result of the collaboration between theoretical and application-focused research groups: the Shape Analysis Group at the Department of Statistics at Florida State University, the Center of Genomics and Personalized Medicine at FSU and the FSU's Department of Neuroscience. As a consequence each of the projects consists of two aspects—the theoretical investigation of the mathematical model and the application driven by a real life problem. The applications components, are similar from the data modeling standpoint. In each case the problem is set in an infinite dimensional space, elements of which are experimental data points that can be viewed as shapes. The three projects are: ``A new framework for Euclidean summary statistics in the neural spike train space''. The project provides a statistical framework for analyzing the spike train data and a new noise removal procedure for neural spike trains. The framework adapts the SRSF elastic metric in the space of point patterns to provides a new notion of the distance. ``SRSF shape analysis for sequencing data reveal new differentiating patterns''. This project uses the shape interpretation of the Next Generation Sequencing data to provide a new point of view of the exon level gene activity. The novel approach reveals a new differential gene behavior, that can't be captured by the state-of-the art techniques. Code is available online on github repository. ``How changes in shape of nucleosomal DNA near TSS influence changes of gene expression''. The result of this work is the novel shape analysis model explaining the relation between the change of the DNA arrangement on nucleosomes and the change in the differential gene expression. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2017. / October 30, 2017. / Functional Data Analysis, Genomics, Neuroscience, Next Generation Sequencing, Shape Analysis, Statistics / Includes bibliographical references. / Wei Wu, Professor Co-Directing Dissertation; Richard Bertram, Professor Co-Directing Dissertation; Anuj Srivastava, University Representative; Peter Beerli, Committee Member; Washington Mio, Committee Member; Giray Ökten, Committee Member.

<p> This dissertation considers a system consisting of self-interested users of a common multiple-input multiple-output (MIMO) channel and a jammer wishing to reduce the total capacity of the channel. In this setting, two games are constructed that model different system-level objectives. In the first&mdash;called &ldquo;utility games&rdquo;&mdash;the users maximize the mutual information between their transmitter and their receiver subject to a power constraint. In the other (termed &ldquo;cost games&rdquo;), the users minimize power subject to an information rate floor. A duality is established between the equilibrium strategies in these two games, and it is shown that Nash equilibria always exist in utility games. Via an exact penalty approach, a modified version of the cost game also possesses an equilibrium. Additionally, multiple equilibria may exist in utility games, but with mild assumptions on users&rsquo; own channels and the jammer-user channels, systems with no user-user interference, there can be at most one Nash equilibrium where a user transmits on all of its subchannels. A similar but weaker result is also found for channels with limited amounts of user-user interference. Two distributed update processes are proposed: gradient-play and best-response. The performance of these algorithms are compared via software simulation. Finally, previous results on network-level improvement via stream control are shown to carry over when a jammer is introduced. </p><p>

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