Current Search: Applied mathematics (x)
Search results
 Title
 LowRank Riemannian Optimization Approach to the Role Extraction Problem.
 Creator

Marchand, Melissa Sue, Gallivan, Kyle A., Dooren, Paul van, Erlebacher, Gordon, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the lowrank 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 lowrank approximation of the similarity measure. We propose two indirect...
Show moreThis dissertation uses Riemannian optimization theory to increase our understanding of the role extraction problem and algorithms. Recent ideas of using the lowrank 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 lowrank approximation of the similarity measure. We propose two indirect approaches to use to solve the role extraction problem. The first uses the standard twophase process. For the first phase, we propose using Riemannian optimization to compute a lowrank approximation of the similarity of the graph, and for the second phase using kmeans clustering on the lowrank 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 twophase indirect role extraction approach. The second indirect approach we propose combines the two phases of our first approach into a onephase approach that iteratively approximates the lowrank similarity matrix, extracts the role partition of the graph, and updates the rank of the similarity matrix. We show that the use of Riemannian rankadaptive techniques when computing the lowrank similarity matrix improves robustness of the clustering algorithm.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Marchand_fsu_0071E_14046
 Format
 Thesis
 Title
 Efficient and Accurate Numerical Schemes for Long Time Statistical Properties of the Infinite Prandtl Number Model for Convection.
 Creator

Woodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences,...
Show moreWoodruff, Celestine, Wang, Xiaoming, Sang, QingXiang Amy, Case, Bettye Anne, Ewald, Brian D., Gunzburger, Max D., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

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...
Show moreIn 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.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Woodruff_fsu_0071E_12813
 Format
 Thesis
 Title
 Riemannian Optimization Methods for Averaging Symmetric Positive Definite Matrices.
 Creator

Yuan, Xinru, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of...
Show moreYuan, Xinru, Gallivan, Kyle A., Absil, PierreAntoine, Erlebacher, Gordon, Ökten, Giray, Bauer, Martin, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

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...
Show moreSymmetric 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.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Yuan_fsu_0071E_14736
 Format
 Thesis
 Title
 Ensemble Proper Orthogonal Decomposition Algorithms for the Incompressible NavierStokes Equations.
 Creator

Schneier, Michael, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Erlebacher, Gordon, Huang, Chen, Florida State University, College of Arts and Sciences, Department of...
Show moreSchneier, Michael, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Erlebacher, Gordon, Huang, Chen, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

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 timedependent NavierStokes equations for which a recently developed ensemblebased method allows for...
Show moreThe 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 timedependent NavierStokes equations for which a recently developed ensemblebased 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) reducedorder model into the ensemblebased 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 longterm 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 ensemblebased methods are extended to the ensemblePOD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Schneier_fsu_0071E_14687
 Format
 Thesis
 Title
 Evolutionary Dynamics of Bacterial Persistence under Nutrient/Antibiotic Actions.
 Creator

Ebadi, Sepideh, Cogan, Nicholas G., Beerli, Peter, Bertram, R., Ökten, Giray, Vo, Theodore, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

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 planktonicbiofilm populations, persisters is a subpopulation that is tolerant to...
Show moreDiseases 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 planktonicbiofilm 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.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Ebadi_fsu_0071E_14324
 Format
 Thesis
 Title
 HighOrder, Efficient, Numerical Algorithms for Integration in Manifolds Implicitly Defined by Level Sets.
 Creator

Khanmohamadi, Omid, Sussman, Mark, Plewa, Tomasz, Moore, M. Nicholas J. (Matthew Nicholas J.), Ökten, Giray, Florida State University, College of Arts and Sciences, Department...
Show moreKhanmohamadi, Omid, Sussman, Mark, Plewa, Tomasz, Moore, M. Nicholas J. (Matthew Nicholas J.), Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

New numerical algorithms are devised for highorder, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension...
Show moreNew numerical algorithms are devised for highorder, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension reduction process are developed, including maps for identifying lowerdimensional hyperrectangle facets, algorithms for minimal coordinateflip vertex traversal, which, together with our multilinearformbased derivative approximation algorithms, are used for checking a proposed integration direction on a facet, as well as algorithms for detecting interfacefree subhyperrectangles. The multidimensional quadrature nodes generated by this method are inside their respective domains (hence, the method does not require any extension of the integrand) and the quadrature weights inherit any positivity of the underlying singledimensional quadrature method, if present. The accuracy and efficiency of the method are demonstrated through convergence and timing studies for test cases in spaces of up to seven dimensions. The strengths and weaknesses of the method in high dimensional spaces are discussed.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Khanmohamadi_fsu_0071E_14013
 Format
 Thesis
 Title
 Diffusion Approximation of a Risk Model.
 Creator

Cheng, Zailei, Zhu, Lingjiong, Niu, Xufeng, Fahim, Arash, Lee, Sanghyun, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

We consider a classical risk process with arrival of claims following a nonstationary 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 finitetime horizon. Numerical results will also be given.
 Date Issued
 2018
 Identifier
 2018_Fall_Cheng_fsu_0071E_14916
 Format
 Thesis
 Title
 Mass Conserving HamiltonianStructurePreserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme.
 Creator

Sockwell, K. Chad (Kenneth Chad), Gunzburger, Max D., Wahl, Horst, Peterson, Janet S., Quaife, Bryan, Huang, Chen, Florida State University, College of Arts and Sciences,...
Show moreSockwell, K. Chad (Kenneth Chad), Gunzburger, Max D., Wahl, Horst, Peterson, Janet S., Quaife, Bryan, Huang, Chen, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Ocean modeling, in a climatemodeling context, requires long timehorizons over global scales, which when combined with accurate resolution in time and space makes simulations very timeconsuming. While highresolution oceanmodeling simulations are still feasible on large HPC machines, performing uncertainty quantification or other many query applications at these resolutions is no longer feasible. Developing a more efficient model would allow for efficient uncertainty quantification, data...
Show moreOcean modeling, in a climatemodeling context, requires long timehorizons over global scales, which when combined with accurate resolution in time and space makes simulations very timeconsuming. While highresolution oceanmodeling simulations are still feasible on large HPC machines, performing uncertainty quantification or other many query applications at these resolutions is no longer feasible. Developing a more efficient model would allow for efficient uncertainty quantification, data assimilation, and spinup initializations. For these techniques to be feasible in practice, a faster model must be designed which can still attain sufficient accuracy. Techniques such as reduced order modeling produce an efficient reduced model based on existing highresolution simulation data. Models produced by these techniques provide a tremendous speedup at the cost of reduced accuracy. To offset this tradeoff, novel strategies are developed to retain as much accuracy as possible while still achieving tremendous speedups. Some of these methods improve accuracy by incorporating physical properties into the reduced model, leading to better solution quality. In this dissertation, a novel reduced order modeling method, the Hamiltonianstructurepreserving reduced order modeling method, will be derived and analyzed. The Hamiltonian structure is possessed by many physical systems and is directly related to energy conservation. This method produces a reduced model which retains the Hamiltonian structure of noncanonical Hamiltonian systems, which are the category of systems that many ocean models fall into. Error estimates are proven for the new method. The model is also be made to preserve linear invariants in the reduced model which are Casimirs. Casimirs are a class of special conserved quantities in the Hamiltonian Framework. For oceanmodeling, the Casimirs we consider are mass and potential vorticity. The new reduced model is proven to conserve both of these quantities. The model is also implemented in a special inner product derived from the Hamiltonian Framework, the approximate energy inner product. This special inner product not only improves the accuracy of the new method but also improves the accuracy of the traditional reduced order modeling method and leads to favorable analytical properties for problems with quadratic Hamiltonian functionals. The new method will be applied to the rotating shallow water equations, which act as a proxy to real ocean models, and compared to the traditional reduced order modeling method. Both energy conserving and forced testcases are considered where energy conservation, accuracy, and stability are investigated. Special techniques are also implemented to ensure that the new method is as efficient as possible.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_Sockwell_fsu_0071E_15277
 Format
 Thesis
 Title
 Distributed Neural Network Models for Birdsong Production.
 Creator

Galvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and...
Show moreGalvis, Daniel Eduardo, Bertram, R., Trombley, Paul Q., Cogan, Nicholas G., Hyson, Richard Lee, Johnson, Frank, Vo, Theodore, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Birdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are...
Show moreBirdsong is a model system for the production of learned, serially ordered motor movements, such as playing a musical instrument or riding a bicycle. To this end, the neural mechanisms underlying birdsong have been studied in great depth, and many tools have been developed for analyzing the spectral and temporal structure of song. In this dissertation, I develop mathematical neural network models to explain how the nuclei in the song system interact to produce song. These models are constrained by the structural connectivity of the song system, the signaling properties of individual neurons and circuits, and circuitbreaking behavioral studies. Chapter 1 provides an overview of songbirds as a model system for speech production, outlines the structure of song, and describes the structure and function of the song system. Chapter 2 describes the neurophysiology and mathematical models of a premotor nucleus, called HVC (proper name), that is essential for song learning and production. In Chapter 3, I develop the neural network model for song production and use it to explain the effects of partial lesions of HVC on song. Furthermore, I use the model to make predictions about the behavioral effects of these lesions and reanalyze the data, validating those predictions. Finally, in Chapter 4 I develop a simplified version of the model that sacrifices spiking dynamics of neurons while maintaining the essential higherlevel features of the model. I use this model to study interhemispheric synchronization and the effects of unilateral perturbations of HVC on song. The model captures the effects of these perturbations, particularly unilateral temperature manipulation and electrical stimulation of HVC, and makes predictions about the circuitlevel effects of these perturbations.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Galvis_fsu_0071E_14310
 Format
 Thesis
 Title
 Sensitivity Analysis of Options under Lévy Processes via Malliavin Calculus.
 Creator

Bayazit, Dervis, Nolder, Craig A., Huﬀer, Fred, Case, Bettye Anne, Kopriva, David, Okten, Giray, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via MonteCarlo...
Show moreThe sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via MonteCarlo simulations. We give explicit formulas that are used directly in MonteCarlo simulations. By using simulations, we show that a localized version of the Malliavin estimator outperforms others including the centered finite difference estimator for the call and digital options under Variance Gamma and Normal Inverse Gaussian processes driven option pricing models. In order to compare the performance of these methods we use an inverse Fourier transform method to calculate the exact values of the sensitivities of European call and digital options written on S&P 500 index. Our results show that a variation of localized Malliavin calculus approach gives a robust estimator while the convergence of centered finite difference method in MonteCarlo simulations varies with different Greeks and new sensitivities that we introduce. We also discuss an approximation method for the Variance Gamma process. We introduce new random number generators for the path wise simulations of the approximating process. We improve convergence results for a type of sensitivity by using a mixed Malliavin calculus on the increments of the approximating process.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1157
 Format
 Thesis
 Title
 Analysis of Regularity and Convergence of Discretization Methods for the Stochastic Heat Equation Forced by SpaceTime White Noise.
 Creator

Wills, Anthony Clinton, Wang, Xiaoming, Ewald, Brian D., Reina, Laura, Bowers, Philip L., Case, Bettye Anne, Ökten, Giray, Florida State University, College of Arts and Sciences...
Show moreWills, Anthony Clinton, Wang, Xiaoming, Ewald, Brian D., Reina, Laura, Bowers, Philip L., Case, Bettye Anne, Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

We consider the heat equation forced by a spacetime white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a spacetime white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as...
Show moreWe consider the heat equation forced by a spacetime white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a spacetime white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as the spatial discretization becomes arbitrarily fine and this is confirmed numerically. Dirichlet and Neumann boundary conditions are also considered. We then derive covariance structure formulas for the two dimensional stochastic heat equation using each of the different methods. In two dimensions the solution does not have a finite variance and the formulas for the covariance structure using different methods does not agree in the limit. This means we must analyze the convergence in a different way than the one dimensional problem. To understand this difference in the solution as the spatial dimension increases, we find the Sobolev space in which the approximate solution converges to the solution in one and two dimensions. This result is then generalized to n dimensions. This gives a precise statement about the regularity of the solution as the spatial dimension increases. Finally, we consider a generalization of the stochastic heat equation where the forcing term is the spatial derivative of a spacetime white noise. For this equation we derive formulas for the covariance structure of the discretized equation using the spectral truncation and finite difference method. Numerical simulation results are presented and some qualitative comparisons between these two methods are made.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9488
 Format
 Thesis
 Title
 Numerical Analysis of Nonlocal Problems.
 Creator

Guan, Qingguang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department...
Show moreGuan, Qingguang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the timedependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark...
Show moreIn this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the timedependent nonlocal diffusion and wave equations, formulated in the peridynamics setting. Initial and boundary data are given. For nonlocal diffusion equation, the time derivative is approximated using either an explicit Forward Euler, or implicit Backward Euler scheme. For nonlocal wave equation, we get the dispersion relations and use the Newmark method to discretize the equation. We have reformulated the standard timestep stability conditions, in light of the peridynamics formulation. Also we have obtained convergence results. Secondly, we consider the spacetime fractional diffusion equation which is used to model anomalous diffusion in physics. Finite difference, finite element and other methods are used to solve it. For finite difference method, the stability of the numerical schemes is well studied. However, for finite element method, we have not found the results for the stability of the θ schemes, especially for the explicit scheme. Here we get the stability and convergence results for all schemes with 0 ≤ θ ≤ 1. Thirdly, an obstacle problem for a nonlocal operator equation is considered; the operator is a nonlocal integral analogue of the Laplacian operator and, as a special case, reduces to the fractional Laplacian. In the analysis of classical obstacle problems for the Laplacian, the obstacle is taken to be a smooth function. For the nonlocal obstacle problem, obstacles are allowed to have jump discontinuities. We cast the nonlocal obstacle problem as a minimization problem wherein the solution is constrained to lie above the obstacle. We prove the existence and uniqueness of a solution in an appropriate function space. Then, the well posedness and convergence of finite element approximations are demonstrated. The results of numerical experiments are provided that illustrate the theoretical results and the differences between solutions of the nonlocal and local obstacle problems. Then we use sparse grid collocation, reduced basis and simplified reduced basis methods to solve nonlocal diffusion equation with random input data. Regularity of the solution and the convergence results for numerical methods are proved. The efficiency of these methods for solving the problem is investigated. As the radius of the spatial interaction zone changes, the computation cost varies due to the density of the stiffness matrix. This is quite different from local problems. Finally, the 1d nonlocal diffusion equation is solved by a continuous piecewiselinear collocation method using a uniform mesh. The time derivative is approximated using any of forward Euler, backward Euler, or CrankNicolson scheme. By developing a technique to deal with the singular integral, we are able to extend the method so that its validity is extended to include the case 1/2 ≤ s [less than] 1. We also derive stability conditions and convergence rates.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Guan_fsu_0071E_13425
 Format
 Thesis
 Title
 A Multiscale Implementation of Finite Element Methods for Nonlocal Models of Mechanics and Diffusion.
 Creator

Xu, Feifei, Gunzburger, Max D., Wang, Xiaoming, Burkardt, John V., Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

The nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon...
Show moreThe nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum extent of nonlocal interactions. A multiscale finite element implementation in one dimension and two dimensions of the nonlocal models is developed by taking advantage of the proven fact that, for smooth solutions, the nonlocal models reduce, as the horizon parameter tends to zero, to wellknown local partial differential equations models. The implementation features adaptive abrupt mesh refinement based on the detection of defects and resulting in an abrupt transition between refined elements that contain defects and unrefined elements that do not do so. Additional difficulties encountered in the implementation that are overcome are the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter. As a result, the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several numerical examples are provided to illustrate the efficacy of the multiscale methodology.
Show less  Date Issued
 2015
 Identifier
 FSU_2016SP_Xu_fsu_0071E_12974
 Format
 Thesis
 Title
 Exploration of the Role of Disinfection Timing, Duration, and Other Control Parameters on Bacterial Populations Using a Mathematical Model.
 Creator

Acar, Nihan, Cogan, Nicholas G., Keller, Thomas C. S., Bertram, R., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Tolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in...
Show moreTolerant bacteria enmeshed in a biofilm causes several difficult to treat illnesses like tuberculosis, chronic pneumonia, and chronic inner ear infections. These diseases typically respond poorly to antibiotics due to high tolerance. Bacterial tolerance can be genotypic (resistancee.g. MRSA), phenotypic (nonheritable) or environmental (e.g. nutrient gradients). Persister formation is phenotypic tolerance that is highly tolerant to disinfection. Constant dosing is typically ineffective in eliminating persister cells. To generate an effective treatment protocol, more research must examine the dynamics of persister cells. This study investigates how manipulating the application of antibiotics and the addition of nutrient may enhance the disinfection of a bacterial population in batch culture. Previous studies focused on the antimicrobial agent as a control variable to eliminate the bacterial population. In addition to antibiotic treatments, we consider the significance of the nutrient in eradicating the susceptible and persister cells since the disinfection of the susceptible population is dependent on nutrient intake. We present a mathematical model that captures the dynamics between susceptible and persister bacteria with antibiotic and nutrient as control variables. We investigate the optimal dosewithdrawal timing of antibiotic in several cases including: constant nutrient in time, dynamic nutrient in time, and piecewise constant nutrient in time. Also a global sensitivity analysis method, Partial Rank Correlation Coefficient (PRCC), is applied to determine the significance of model parameters for a quantity of interest. The highlights of this study are; 1.) Constant dosing is not an effective disinfection protocol. 2.) Nutrient plays a significant role such that in the presence of nutrient, bacterial population is eliminated much faster. 3.) Checking the eigenvalues of the established Poincaré map gives us information on how to choose withdrawdose timing for the nonlinear system. 4.) Periodic dosewithdraw offers a more efficient disinfection provided dose time is longer than withdrawal of antibiotic. 5.) As duration of dose decreases, the elimination of bacteria decreases and the death rate becomes insignificant.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Acar_fsu_0071E_14749
 Format
 Thesis
 Title
 Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to Second Order Elliptic Boundary Value and Interface Problems.
 Creator

Cheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and...
Show moreCheung, James, Gunzburger, Max D., Steinbock, Oliver, Bochev, Pavel B., Perego, Mauro, Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

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...
Show moreIn 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 (PEFEM). 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.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cheung_fsu_0071E_14328
 Format
 Thesis
 Title
 Using RBFGenerated Quadrature Rules to Solve Nonlocal Continuum Models.
 Creator

Lyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreLyngaas, Isaac R., Peterson, Janet S., Musslimani, Ziad H., Gunzburger, Max D., Quaife, Bryan, Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

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 timedependent nonlocal integrodifferential 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...
Show moreRecently 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 timedependent nonlocal integrodifferential 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 timedependent 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.
Show less  Date Issued
 2018
 Identifier
 2018_Fall_Lyngaas_fsu_0071E_14886
 Format
 Thesis
 Title
 Mathematical Modeling and Sensitivity Analysis for Biological Systems.
 Creator

Aggarwal, Manu, Cogan, Nicholas G., Hussaini, M. Yousuff, Chicken, Eric, Jain, Harsh Vardhan, Bertram, R. (Richard), Mio, Washington, Florida State University, College of Arts...
Show moreAggarwal, Manu, Cogan, Nicholas G., Hussaini, M. Yousuff, Chicken, Eric, Jain, Harsh Vardhan, Bertram, R. (Richard), Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

In this work, we propose a framework to develop testable hypotheses for the effects of changes in the experimental conditions on the dynamics of a biological system using mathematical models. We discuss the uncertainties present in this process and show how information from different experiment regimes can be used to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. To determine the significance of variation in the parameters due to varying...
Show moreIn this work, we propose a framework to develop testable hypotheses for the effects of changes in the experimental conditions on the dynamics of a biological system using mathematical models. We discuss the uncertainties present in this process and show how information from different experiment regimes can be used to identify a region in the parameter space over which subsequent mathematical analysis can be conducted. To determine the significance of variation in the parameters due to varying experimental conditions, we propose using sensitivity analysis. Using our framework, we hypothesize that the experimentally observed decrease in the survivability of bacterial populations of Xylella fastidiosa (causal agent of Pierce’s Disease) upon addition of zinc, might be because of starvation of the bacteria in the biofilm due to an inhibition of the diffusion of the nutrients through the extracellular matrix of the biofilm. We also show how sensitivity is related to uncertainty and identifiability; and how it can be used to drive analysis of dynamical systems, illustrating it by analyzing a model which simulates bursting oscillations in pancreatic βcells. For sensitivity analysis, we use Sobol’ indices for which we provide algorithmic improvements towards computational efficiency. We also provide insights into the interpretation of Sobol’ indices, and consequently, define a notion of the importance of parameters in the context of inherently flexible biological systems.
Show less  Date Issued
 2019
 Identifier
 2019_Spring_Aggarwal_fsu_0071E_15070
 Format
 Thesis
 Title
 Gliomas Diagnosis, Progress, and Treatment: A Mathematical Approach.
 Creator

Sorribes Rodriguez, Inmaculada Concepcion, Jain, Harsh Vardhan, Sang, QingXiang, Cogan, Nicholas G., Moore, Matthew Nicholas J., Florida State University, College of Arts and...
Show moreSorribes Rodriguez, Inmaculada Concepcion, Jain, Harsh Vardhan, Sang, QingXiang, Cogan, Nicholas G., Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The diagnosis and treatment of gliomas continues to pose a significant challenge for oncologists who not only have to contend with managing acute neurological symptoms, but also the almost inevitable development of resistance to treatment. Indeed, the last 25 years have produced minimal advancements in treatment efficacy, even though significant efforts and resources have been invested in the quest for breakthroughs. This effort has not been restricted only to clinicians or oncologists, with...
Show moreThe diagnosis and treatment of gliomas continues to pose a significant challenge for oncologists who not only have to contend with managing acute neurological symptoms, but also the almost inevitable development of resistance to treatment. Indeed, the last 25 years have produced minimal advancements in treatment efficacy, even though significant efforts and resources have been invested in the quest for breakthroughs. This effort has not been restricted only to clinicians or oncologists, with mathematical modeling also playing an increasingly important role. A variety of models aimed at providing new insights into glioma growth and response to treatment have been proposed. Initially designed to capture fundamental behavior of tumor cells, such as growth and motility, these models quickly became wellestablished and multiple extensions have since been introduced. However, as increasing biological details of how tumor cells respond to treatment at cellular and subcellular levels are revealed, mathematical models need to include this state of the art knowledge. The work presented in this thesis seeks to do this by refocusing our attention back to the most fundamental question: why are gliomas fatal? Biologically, it is known that glioma lethality is driven by a fast growth that increases intracranial pressure resulting in lethal neurological damage, which current treatments fail to prevent due to tumor cell resistance to treatments such as chemotherapy. By creating mathematical models inspired by these key elements of glioma malignancy, the work presented here seeks to elucidate what drives resistance to chemotherapy and how to overcome or mitigate it, as well as how malignancy correlated with intracranial pressure dynamics. Thus, the work comprises two main parts: (1) in silico optimization of treatment strategies using chemotherapy coupled with novel cellrepair inhibitors currently in various stages of the clinical trial; and (2) a study of tumorinduced intracranial pressure and edema in gliomas of grade IIV. A wide variety of mathematical modeling techniques are used, that incorporate biomechanical, biochemical, pharmacokinetics, and pharmacodynamics aspects, and include a level of detail hitherto unconsidered. The proposed models are validated and analyzed by employing a diverse set of mathematical tools that range from structural identifiability, parameter estimation, to global and local sensitivity analysis. As a result of this work, we propose a treatment strategy that showed a 30% improvement in patient survival time over conventional treatment when treating heterogeneous brain tumors in silico. Moreover, the second part of this work demonstrates how the spatiotemporal dynamics of tumorinduced intracranial pressure correlate with cancer grade, providing a better understanding of the mechanisms that underlie increased intracranial pressure onset. Both projects come together as a first step towards a better understanding of the poor survival rates of patients afflicted with gliomas. They raise new questions about what characterizes the malignancy of primary brain tumors and how clinicians can fight it. Continued modeling effort in these directions has the potential to make an impact in the field of brain cancer diagnostics and treatment.
Show less  Date Issued
 2019
 Identifier
 2019_Summer_SorribesRodriguez_fsu_0071E_15274
 Format
 Thesis
 Title
 ReducedOrder Modeling of Reactive Solute Transport for AdvectionDominated Problems with Nonlinear Kinetic Reactions.
 Creator

McLaughlin, Benjamin R. S., Peterson, Janet S., Ye, Ming, Duke, D. W. (Dennis W.), Gunzburger, Max D., Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreMcLaughlin, Benjamin R. S., Peterson, Janet S., Ye, Ming, Duke, D. W. (Dennis W.), Gunzburger, Max D., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Groundwater is a vital natural resource, and our ability to protect and manage this resource efficiently and effectively relies heavily on our ability to perform reliable and accurate computer modeling and simulation of subsurface systems. This frequently raises research questions involving parameter estimation and uncertainty quantification, which are often prohibitively expensive to answer using standard highdimensional computational models. We have previously demonstrated the ability to...
Show moreGroundwater is a vital natural resource, and our ability to protect and manage this resource efficiently and effectively relies heavily on our ability to perform reliable and accurate computer modeling and simulation of subsurface systems. This frequently raises research questions involving parameter estimation and uncertainty quantification, which are often prohibitively expensive to answer using standard highdimensional computational models. We have previously demonstrated the ability to replace the highdimensional models used to solve linear, uncoupled, diffusiondominated multispecies reactive transport systems with lowdimension approximations using reduced order modeling (ROM) based on proper orthogonal decomposition (POD). In this work, we seek to apply ROM to more general reactive transport systems, where the reaction terms may be nonlinear, mathematical models may be coupled, and the transport may be advectiondominated. We discuss the use of operator splitting, which is prevalent in the reactive transport field, to simplify the computation of complex systems of reactions in the transport model. We also discuss the use of some stabilization methods which have been developed in the computational science community to treat advectiondominated transport problems. We present a method by which we are able to incorporate stabilization and operator splitting together in the finite element setting. We go on to develop methods for implementing both operator splitting and stabilization in the ROM setting, as well as for incorporating both of them together within the ROM framework. We present numerical results which establish the ability of this new approach to produce accurate approximations with a significant reduction in computational cost, and we demonstrate the application of this method to a more realistic reactive transport problem involving bioremediation.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9649
 Format
 Thesis
 Title
 Unveiling Mechanisms for Electrical Activity Patterns in Neurons and Pituitary Cells Using Mathematical Modeling and Analysis.
 Creator

Sengul, Sevgi, Bertram, R. (Richard), TabakSznajder, Joel, Steinbock, Oliver, Quine, J. R. (John R.), Cogan, Nicholas G., Florida State University, College of Arts and Sciences...
Show moreSengul, Sevgi, Bertram, R. (Richard), TabakSznajder, Joel, Steinbock, Oliver, Quine, J. R. (John R.), Cogan, Nicholas G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Computational neuroscience is a relatively new area that utilizes the computational analyses of neural systems as well as development of mathematical models. Analyses of neural systems help us to gain a deeper understanding of how different dynamical variables contribute to generate a given electrical behavior and modelling helps to explain experimental results or make predictions that can be tested experimentally. Due to the complexity of nervous system behavior, mathematical models often...
Show moreComputational neuroscience is a relatively new area that utilizes the computational analyses of neural systems as well as development of mathematical models. Analyses of neural systems help us to gain a deeper understanding of how different dynamical variables contribute to generate a given electrical behavior and modelling helps to explain experimental results or make predictions that can be tested experimentally. Due to the complexity of nervous system behavior, mathematical models often have many variables, however simpler lowerdimensional models are also important for understanding complex behavior. The work described herein utilizes both approaches in two separate, but related, studies in computational neuroscience. In the first study, we determined the contributions of two negative feedback mechanisms in the HodgkinHuxley model. Hodgkin and Huxley pioneered the use of mathematics in the description of an electrical impulse in a squid axon, developing a differential equation model that has provided a template for the behavior of many other neurons and other excitable cells. The HodgkinHuxley model has two negative feedback variables. The activation of a current (n), subtracts from the positive feedback responsible for the upstroke of an impulse. We call this subtractive negative feedback. Divisive feedback is provided by the inactivation of the positive feedback current (h), which divides the current. Why are there two negative feedback variables when only one type of negative feedback can produce rhythmic spiking? We detect if there is any advantage to having both subtractive and divisive negative feedback in the system and the respective contributions of each to rhythmic spiking by using three different metrics. The first measures the width of a parameter regime within which tonic spiking is a unique and stable limit cycle oscillation. The second metric, contribution analysis, measures how changes in the time scale parameters of the feedback variables affect the durations of the "active phase" during the action potential and the interspike interval "silent phase" of a tonically spiking model. The third metric, dominant scale analysis, measures a sensitivity of the voltage dynamics to each of the ionic currents and ranks their influence. xi In the second study, we used electrophysiology data provided from the collaborating lab of Mike Shipston combined with mathematical modelling to show how two different neurohormones regulate patterns of electrical activity in corticotrophs. Corticotroph cells of the anterior pituitary are electrically excitable cells and are an integral component of the stress the neuroendocrine response to stress. Stress activates neurons in the hypothalamus to release corticotrophinreleasing hormone (CRH) and arginine vasopressin (AVP). These neurohormones act on corticotrophs in the anterior pituitary gland, which secrete another hormone, adrenocorticotropic hormone (ACTH). ACTH enters the general circulation and stimulates the adrenal cortex to secrete corticosteroid (cortisol in humans). Corticotrophs display single spike activity under basal conditions which can be converted to complex bursting behavior after stimulation by the combination of CRH and AVP. Bursting is much more effective at releasing ACTH than is spiking, so this transition is physiologically important. We investigated the underlying mechanisms controlling this transition to bursting by mathematical modelling combined with the experimental data. The significance of the work in this dissertation is that it provides a very good example of how experiments and modelling can complement each other and how the right mathematical tools can increase our understanding of even a very old and much studied model.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9245
 Format
 Thesis
 Title
 Approximating Nonlocal Diffusion Problems Using Quadrature Rules Generated by Radial Basis Functions.
 Creator

Lyngaas, Isaac Ron, Peterson, Janet S., Gunzburger, Max D., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits so...
Show moreNonlocal models differ from traditional partial differential equation (PDE) models because they contain no spatial derivatives; instead an appropriate integral is used. Nonlocal models are especially useful in the case where there are issues calculating the spatial derivatives of a PDE model. In many applications (e.g., biological systems, flow through porous media) the observed rate of diffusion is not accurately modeled by the standard diffusion differential operator but rather exhibits socalled anomalous diffusion. Anomalous diffusion can be represented in a PDE model by using a fractional Laplacian operator in space whereas the nonlocal approach only needs to slightly modify its integral formulation to model anomalous diffusion. Anomalous diffusion is one such case where approximating the spatial derivative operator is a difficult problem. In this work, an approach for approximating standard and anomalous nonlocal diffusion problems using a new technique that utilizes radial basis functions (RBFs) is introduced and numerically tested. The typical approach for approximating nonlocal diffusion problems is to use a Galerkin formulation. However, the Galerkin formulation for nonlocal diffusion problems can often be difficult to compute efficiently and accurately especially for problems in multiple dimensions. Thus, we investigate the alternate approach of using quadrature rules generated by RBFs to approximate the nonlocal diffusion problem. This work will be split into three major parts. The first will introduce RBFs and give some examples of how they are used. This part will motivate our approach for using RBFs on the nonlocal diffusion problem. In the second part, we will derive RBFgenerated quadrature rules in one dimension and show they can be used to approximate nonlocal diffusion problems. The final part will address how the RBF quadrature approach can be extended to higher dimensional problems. Numerical test cases are shown for both the standard and anomalous nonlocal diffusion problems and compared with standard finite element approximations. Preliminary results show that the method introduced is viable for approximating nonlocal diffusion problems and that highly accurate approximations are possible using this approach.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Lyngaas_fsu_0071N_13512
 Format
 Thesis
 Title
 Modeling Credit Risk in the Default Threshold Framework.
 Creator

Chiu, ChunYuan, Kercheval, Alec N., Chicken, Eric, Ökten, Giray, Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642673, 2016] enjoys the advantages of both the structural form models and the reduced form models, including excellent analytical tractability. In their paper, the closed form default time distribution of a company is derived when the default threshold is a constant or a deterministic function. As for stochastic default threshold, it is shown that the...
Show moreThe default threshold framework for credit risk modeling developed by Garreau and Kercheval [SIAM Journal on Financial Mathematics, 7:642673, 2016] enjoys the advantages of both the structural form models and the reduced form models, including excellent analytical tractability. In their paper, the closed form default time distribution of a company is derived when the default threshold is a constant or a deterministic function. As for stochastic default threshold, it is shown that the survival probability can be derived as an expectation. How to specify the stochastic default threshold so that this expectation can be obtained in closed form is however left unanswered. The purpose of this thesis is to fulfill this gap. In this thesis, three credit risk models with stochastic default thresholds are proposed, under each of which the closed form default time distribution is derived. Unlike Garreau and Kercheval's work where the logreturn of a company's stock price is assumed to be independent and identically distributed and the interest rate is assumed constant, in our new proposed models the random interest rate and the stochastic volatility of a company's stock price are taken into consideration. While in some cases the defaultable bond price, the credit spread and the CDS premium are derived in closed form under the new proposed models, in others it seems not so easy. The difficulty that stops us from getting closed form formulas is also discussed in this thesis. Our new models involve the Heston model, which has a closed form characteristic function. We found the common characteristic function formula used in the literature not always applicable for all input variables. In this thesis the safe region of the formula is analyzed completely. A new formula is also derived that can be used to find the characteristic function value in some cases when the common formula is not applicable. An example is given where the common formula fails and one should use the new formula.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Chiu_fsu_0071E_13584
 Format
 Thesis
 Title
 Nonlinear SchrödingerType Systems: Complex Lattices and NonParaxiality.
 Creator

Cole, Justin, Musslimani, Ziad H., Höflich, Peter A., Wang, Xiaoming, Moore, M. Nicholas J. (Matthew Nicholas J.), Florida State University, College of Arts and Sciences,...
Show moreCole, Justin, Musslimani, Ziad H., Höflich, Peter A., Wang, Xiaoming, Moore, M. Nicholas J. (Matthew Nicholas J.), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis investigates nonlinear systems that are dispersive and conservative in nature and wellapproximated by the nonlinear Schrödinger (NLS) equation. The NLS equation is the prototypical equation for describing such phenomena and it has been utilized in a large number of physical systems. This work considers novel applications and exotic parameter regimes that fall inside the class of solutions well described by nonlinear Schrödingertype systems. A brief historical, physical, and...
Show moreThis thesis investigates nonlinear systems that are dispersive and conservative in nature and wellapproximated by the nonlinear Schrödinger (NLS) equation. The NLS equation is the prototypical equation for describing such phenomena and it has been utilized in a large number of physical systems. This work considers novel applications and exotic parameter regimes that fall inside the class of solutions well described by nonlinear Schrödingertype systems. A brief historical, physical, and mathematical introduction to deriving the NLS equation and its variants is presented. The topics considered in detail cover optical systems in various media and are naturally divided into two parts: nonparaxiality through the inclusion of higherorder dispersion/diffraction and beam propagation in the presence of complex lattices. The higherorder dispersion/diffraction effects on soliton solutions are considered in detail. The propagation of a short soliton pulse as it travels down a fiber optic in the presence of a linear timeperiodic potential is considered. Due to the short duration of the pulse fourthorder dispersive effects are relevant. The band gap structure is determined using FloquetBloch theory and the shape of its dispersion curves as a function of the fourthorder dispersion coupling constant β is discussed. Several features not observed in the absence of highorder dispersion (β=0) are highlighted, such as a nonzero threshold value of potential strength below which there is no band gap and the formation of novel localized modes at large potential amplitudes. A higher order two band tight binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semiinfinite and first band gaps are constructed. Stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation. Next the spectral transverse instabilities of onedimensional solitary wave solutions to the twodimensional NLS equation with biharmoinc diffraction and subject to higherdimensional perturbations are studied. Physically, the inclusion of the biharmonic term corresponds to spatial beams with a narrow width in comparison to their wavelength. A linear boundary value problem governing the evolution of the transverse perturbations is derived. The eigenvalues of the perturbations are numerically computed and a finite band of unstable transverse modes is found to exist. In the long wavelength limit an asymptotic formula for the perturbation growth rate that agrees well with the numerical findings. Using a variational formulation based on Lagrangian model reduction, an approximate expression for the perturbation eigenvalues is obtained and its validity is compared with both the asymptotic and numerical results. The dynamics of a onedimensional soliton stripe in the presence of a transverse perturbation is studied using direct numerical simulations. The second half of the dissertation is concerned with beam propagation in the presence of complex lattices, in particular lattices that possess paritytime (PT) symmetries. A new family of nonhermitian optical potentials given in terms of double exponential periodic functions is introduced. The center of PTsymmetry is not around zero and the potential satisfies a shifted PTsymmetry relation at two distinct locations. These novel refractive index modulations are examined from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics, abrupt phase transitions in the eigenvalue spectrum and exceptional points in the band structure are studied in detail. In addition, the nonlinear properties of coherent structures in Kerr media is probed. The spatial symmetries of such lattice solitons follow the shifted PTsymmetric relations. Furthermore, such lattice solitons have a power threshold and their linear and nonlinear stability is critically dependent on their spatial symmetry point. In the final chapter a class of exact multicomponent constant energy solutions to a Manakov system in the presence of an external PTsymmetric complex potential is constructed. This type of uniform wave pattern displays a nontrivial phase whose spatial dependence is induced from the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogenous gain and loss. The constantintensity continuous waves are then used to perform a modulational instability analysis in the presence of both nonhermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using FourierFloquetBloch theory. The stability of the selffocusing and defocusing cases is considered and compared to the zeropotential results. Our linear stability results are supplemented with direct (nonlinear) numerical simulations.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_Cole_fsu_0071E_13102
 Format
 Thesis
 Title
 Modelling Limit Order Book Dynamics Using Hawkes Processes.
 Creator

Chen, Yuanda, Kercheval, Alec N., Beaumont, Paul M., Ewald, Brian D., Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

The Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in...
Show moreThe Hawkes process serves as a natural choice for modeling selfexciting dynamics, such as the behavior of an electronic exchangehosted limit order book (LOB). However, due to the lack of analytical solutions, probability estimates of future events often must rely on Monte Carlo simulation. Although Monte Carlo simulation is known to be good at solving pathdependent problems, it has the limitation that a high computation time is often required to get good accuracy. This is a concern in fields like algorithmic trading where fast calculation is essential. In this dissertation we propose the use of a 4dimensional Hawkes process to model the LOB and to forecast midprice movement probabilities using Monte Carlo simulation. We study the feasibility of making this prediction quickly enough to be applicable in practice. We show that fast predictions are feasible, and show in tests on real data that the model has some trading value in forecasting midprice movements. This dissertation also compares the performance of several popular computer languages, Python, MATLAB, Cython and C, in singlecore experiments, and examines the scalability for parallel computing using Cython and C.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Chen_fsu_0071E_13187
 Format
 Thesis
 Title
 Confined supersonic mixing layers: A computational investigation of instability and mixing enhancement.
 Creator

Hu, Fang Qiang., Florida State University
 Abstract/Description

Hydrodynamic instability of a compressible shear mixing layer plays a very important role in controlling and promoting the mixing processes in supersonic combustion problems. At supersonic convective Mach numbers the natural mixing rates of the shear layer are observed to be very small. The current research interest is in the mixing enhancement. In the first part of the present study, the linear spatial instability problem of a supersonic shear mixing layer inside a rectangular channel is...
Show moreHydrodynamic instability of a compressible shear mixing layer plays a very important role in controlling and promoting the mixing processes in supersonic combustion problems. At supersonic convective Mach numbers the natural mixing rates of the shear layer are observed to be very small. The current research interest is in the mixing enhancement. In the first part of the present study, the linear spatial instability problem of a supersonic shear mixing layer inside a rectangular channel is solved. A systematic way of calculating and classifying all the normal modes is developed. It is demonstrated that a vortex sheet at high supersonic convective Mach number, neutrally stable when unconfined, becomes unstable when confined. Extensive numerical computations indicate that two classes of unstable waves, named as class A and class B supersonic instability waves, can be identified. In addition to unstable waves, two families of neutral waves, named as class C and class D neutral acoustic waves, can also be identified. The characteristics of the unstable waves as well as the neutral waves are determined. It is concluded that the new supersonic instabilities found in this study are the dominant instabilities of a confined mixing layer at high supersonic convective Mach numbers. As such they are very relevant to the supersonic combustion problem. The second part of the study explores the possibility of enhancing the rate of supersonic mixing by means of a periodic Mach wave system. The Mach wave system may be generated by small amplitude waviness of the confining channel walls. The stability of the periodic basic flow is analyzed numerically by the Floquet theory and the spectralcollocation method. The convergence of the numerical solutions is discussed. New secondary instabilities of the shear layer induced by the periodic Mach waves are found computationally. The, growth rate of the new instabilities is found to vary nearly linearly with the ratio of the wavy wall amplitude to the wave length. Therefore it is a potentially promising scheme for the enhancement of supersonic mixing.
Show less  Date Issued
 1990, 1990
 Identifier
 AAI9027799, 3162059, FSDT3162059, fsu:78257
 Format
 Document (PDF)
 Title
 Developing SRSF Shape Analysis Techniques for Applications in Neuroscience and Genomics.
 Creator

Wesolowski, Sergiusz, Wu, Wei, Bertram, R. (Richard), Srivastava, Anuj, Beerli, Peter, Mio, Washington, Florida State University, College of Arts and Sciences, Department of...
Show moreWesolowski, Sergiusz, Wu, Wei, Bertram, R. (Richard), Srivastava, Anuj, Beerli, Peter, Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

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...
Show moreDissertation 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 ANOVAlike statistical testing procedure is discussed. The projects were a result of the collaboration between theoretical and applicationfocused 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 stateofthe 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.
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Wesolowski_fsu_0071E_14177
 Format
 Thesis
 Title
 Parma: Applications of VectorAutoregressive Models to Biological Inference with an Emphasis on ProcrustesBased Data.
 Creator

Soda, K. James (Kenneth James), Slice, Dennis E., Beaumont, Paul M., Beerli, Peter, MeyerBaese, Anke, Shanbhag, Sachin, Florida State University, College of Arts and Sciences,...
Show moreSoda, K. James (Kenneth James), Slice, Dennis E., Beaumont, Paul M., Beerli, Peter, MeyerBaese, Anke, Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

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 temporallydynamic 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....
Show moreMany 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 temporallydynamic 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 autoregressivemoving average (VARMA) and vector autoregressive (VAR) processes can represent temporallydynamic 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 crosscovariance 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 prespecified 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 LotkaVolterra 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 temporallyordered shapes as opposed to a single static shape. Like all geometric morphometric data, each shape in a trajectory is inherently highdimensional. 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 vectordisplacement diagram. In this new procedure, the VAR representations for two trajectories' processes generate two simulated trajectories that share the same shocks. Then, a vectordisplacement 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 pseudorandom 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 cointegrated 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.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Soda_fsu_0071E_13917_P
 Format
 Set of related objects
 Title
 NetworkTheoretic and DataBased Analysis and Control of Unsteady Fluid Flows.
 Creator

Nair, Aditya Gopimohan, Taira, Kunihiko, Sussman, Mark, Cattafesta, Louis N., Oates, William, Alvi, Farrukh S., Brunton, Steven L. (Steven Lee), Florida State University,...
Show moreNair, Aditya Gopimohan, Taira, Kunihiko, Sussman, Mark, Cattafesta, Louis N., Oates, William, Alvi, Farrukh S., Brunton, Steven L. (Steven Lee), Florida State University, College of Engineering, Department of Mechanical Engineering
Show less  Abstract/Description

Unsteady fluid flows have complex dynamics due to the nonlinear interactions amongst vortical elements. In this thesis, a networktheoretic framework is developed to describe vortical and modal (coherent structure) interactions in unsteady fluid flows. A sparsifieddynamics model and a networkedoscillator model describe the complex dynamics in fluid flows in terms of vortical and modal networks, respectively. Based on the characterized network interactions, modelbased feedback control laws...
Show moreUnsteady fluid flows have complex dynamics due to the nonlinear interactions amongst vortical elements. In this thesis, a networktheoretic framework is developed to describe vortical and modal (coherent structure) interactions in unsteady fluid flows. A sparsifieddynamics model and a networkedoscillator model describe the complex dynamics in fluid flows in terms of vortical and modal networks, respectively. Based on the characterized network interactions, modelbased feedback control laws are established, particularly for controlling the flow unsteadiness. Furthermore, to characterize modelfree feedback control laws for suppressing flow separation in turbulent flows, a datadriven approach leveraging unsupervised clustering is developed. This approach alters the Markov transition dynamics of fluid flow trajectories in an optimal manner using a clusterbased 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 sparsifieddynamics model is developed which drastically reduces the computational cost to predict the dynamical behavior of vortices, sharing characteristics of reducedorder 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 twodimensional incompressible homogeneous turbulence is then characterized. The strength distribution of the turbulence network reveals an underlying scalefree 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 scalefree 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 networkedoscillatorbased 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 reducedorder model. The linear network model for nonlinear dynamics is subsequently utilized to design a modelbased 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 twodimensional unsteady flow over a circular cylinder. The networkbased formulation enables the characterization and control of modal interactions to control fundamental energy transfers in unsteady bluff body flows. Finally, unsupervised clustering and datadriven optimization of coarsegrained control laws is leveraged to manipulate poststall separated flows. Optimized feedback control laws are deduced in highfidelity simulations in an automated, modelfree manner. The approach partitions the baseline flow trajectories into clusters, which corresponds to a characteristic coarsegrained phase in a lowdimensional 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 threedimensional separated flows over a NACA 0012 airfoil at an angle of attack of 9° Reynolds number Re = 23000 and freestream Mach number M∞ = 0.3. The optimized control law minimizes power consumption for flight enabling flow to reach a lowdrag state. The analysis provides insights for feedback flow control of complex systems characterizing global clusterbased control laws based on a datadriven, lowdimensional characterization of fluid flow trajectories. In summary, this thesis develops a novel networktheoretic and databased 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, databased methods are developed to deduce optimal feedback control laws for poststall separated flows. The networktheoretic and databased approaches provides insights on fundamental interactions in fluid flows which paves the way for design of novel flow control strategies.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Nair_fsu_0071E_14745
 Format
 Thesis
 Title
 ContactFree Simulations of Rigid Particle Suspensions Using Boundary Integral Equations.
 Creator

Bystricky, Lukas, Quaife, Bryan, Shanbhag, Sachin, Cogan, Nicholas G., Huang, Chen, Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences,...
Show moreBystricky, Lukas, Quaife, Bryan, Shanbhag, Sachin, Cogan, Nicholas G., Huang, Chen, Moore, Matthew Nicholas J., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

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...
Show moreIn 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 FolgarTucker 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.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Bystricky_fsu_0071E_14725
 Format
 Thesis
 Title
 Statistical Shape Analysis of Neuronal Tree Structures.
 Creator

Duncan, Adam, Srivastava, Anuj, Klassen, E., Wu, Wei, Huffer, Fred W., Florida State University, College of Arts and Sciences, Department of Statistics
 Abstract/Description

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...
Show moreNeuron 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 shapeanalytic 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 shapepreserving nuisance variables modded out. In the first framework, the side branches are treated as a continuum of curvevalued 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 crossvalidated 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.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Duncan_fsu_0071E_14500
 Format
 Thesis
 Title
 Characterization of Coherent Structures in QuasiSteady State Astrophysical Fluid Flows.
 Creator

Learn, Ryan Joseph, Plewa, Tomasz, Ye, Ming, Erlebacher, Gordon, Navon, Ionel Michael, Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

In astrophysical hydrodynamical objects, multiple physical processes take place on a wide variety of spatial and temporal scales simultaneously, making direct numerical simulation of such objects dicult computationally. Our work focuses on developing and testing reducedorder models of such physical processes and objects in order to mitigate this diculty. We use the singular value decomposition on snapshot data the systems generated by a highdelity model in order to generate a singular...
Show moreIn astrophysical hydrodynamical objects, multiple physical processes take place on a wide variety of spatial and temporal scales simultaneously, making direct numerical simulation of such objects dicult computationally. Our work focuses on developing and testing reducedorder models of such physical processes and objects in order to mitigate this diculty. We use the singular value decomposition on snapshot data the systems generated by a highdelity model in order to generate a singular eigenvalue spectrum as well as a orthogonal eigenfunction basis. The original equations of the system are then projected onto this basis via a Galerkin or discontinuous Galerkin projection, giving rise to a system of ordinary dierential equations that serve as the reduced order model. These models are then propagated forward in time, and their accuracy and computational cost are compared with our highdelity models. We nd that for the systems of interest (quasisteady systems), high accuracy reduced order models can be created with only a small number of basis functions at a cost of an order of magnitude less computational time. We further propose methods to increase the savings for these systems even further.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9637
 Format
 Thesis
 Title
 Asset Pricing Equilibria for Heterogeneous, LimitedInformation Agents.
 Creator

Jones, Dawna Candice, Kercheval, Alec N., Beaumont, Paul M, Van Winkle, David H., Nichols, Warren, Ökten, Giray, Florida State University, College of Arts and Sciences,...
Show moreJones, Dawna Candice, Kercheval, Alec N., Beaumont, Paul M, Van Winkle, David H., Nichols, Warren, Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

The standard general equilibrium asset pricing models typically make two simplifying assumptions: homogeneous agents and the existence of a rational expectations equilibrium. This context sometimes yields outcomes that are inconsistent with the empirical findings. We hypothesize that allowing agent heterogeneity could assist in replicating the empirical results. However, the inclusion of heterogeneity in models where agents are fully rational proves impossible to solve without severe...
Show moreThe standard general equilibrium asset pricing models typically make two simplifying assumptions: homogeneous agents and the existence of a rational expectations equilibrium. This context sometimes yields outcomes that are inconsistent with the empirical findings. We hypothesize that allowing agent heterogeneity could assist in replicating the empirical results. However, the inclusion of heterogeneity in models where agents are fully rational proves impossible to solve without severe simplifying assumptions. The reason for this difficulty is that heterogeneous agent models generate an endogenously complicated distribution of wealth across the agents. The state space for each agent's optimization problem includes the complex dynamics of the wealth distribution. There is no general way to characterize the interaction between the distribution of wealth and the macroeconomic aggregates. To address this issue, we implement an agentbased model where the agents have bounded rationality. In our model, we have a complete markets economy with two agents and two assets. The agents are heterogeneous and utility maximizing with constant coefficient of relative risk aversion [CRRA] preferences. How the agents address the stochastic behaviour of the evolution of the wealth distribution is central to our task since aggregate prices depend on this behaviour. An important component of this dissertation involves dealing with the computational difficulty of dynamic heterogeneousagent models. That is, in order to predict prices, agents need a way to keep track of the evolution of the wealth distribution. We do this by allowing each agent to assume that a priceequivalent representative agent exists and that the representative agent has a constant coefficient of relative risk aversion. In so doing, the agents are able to formulate predictive pricing and demand functions which allow them to predict aggregate prices and make consumption and investment decisions each period. However, the agents' predictions are only approximately correct. Therefore, we introduce a learning mechanism to maintain the required level of accuracy in the agents' price predictions. From this setup, we find that the model, with learning, will converge over time to an approximate expectations equilibrium, provided that the the initial conditions are close enough to the rational expectations equilibrium prices. Two main contributions in our work are: 1) to formulate a new concept of approximate equilibria, and 2) to show how equilibria can be approximated numerically, despite the fact that the true state space at any point in time is mathematically complex. These contributions offer the possibility of characterizing a new class of asset pricing models where agents are heterogeneous and only just slightly limited in their rationality. That is, the partially informed agents in our model are able to forecast and utilitymaximize only just as well as economists who face problems of estimating aggregate variables. By using an exogenously assigned adaptive learning rule, we analyse this implementation in a Lucastype heterogeneous agent model. We focus on the sensitivity of the risk parameter and the convergence of the model to an approximate expectations equilibrium. Also, we study the extent to which adaptive learning is able to explain the empirical findings in an asset pricing model with heterogeneous agents.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9624
 Format
 Thesis
 Title
 Stochastic Modeling of Epidemic Diseases Considering Dynamic Contact Networks and Genealogy Information.
 Creator

Ashki, Haleh, Beerli, Peter, Coutts, Christopher, Shanbhag, Sachin, Slice, Dennis E., Lemmon, Alan R., Florida State University, College of Arts and Sciences, Department of...
Show moreAshki, Haleh, Beerli, Peter, Coutts, Christopher, Shanbhag, Sachin, Slice, Dennis E., Lemmon, Alan R., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Human life and diseases are inseparable. For millions of years, humans and their ancestors suffered from diseases, caused by infectious pathogens (e.g., bacteria, viruses, parasites) and caused by our own bodies as they age and degenerate. Within the last century, with the advent of public health measures, improved nutrition and medicine, such as antibiotics, some of the infectious diseases have been controlled. However, infectious diseases still lead to most of the nonage related deaths in...
Show moreHuman life and diseases are inseparable. For millions of years, humans and their ancestors suffered from diseases, caused by infectious pathogens (e.g., bacteria, viruses, parasites) and caused by our own bodies as they age and degenerate. Within the last century, with the advent of public health measures, improved nutrition and medicine, such as antibiotics, some of the infectious diseases have been controlled. However, infectious diseases still lead to most of the nonage related deaths in the world, especially in nations with insufficient health support. My research has taken the complex and dynamic contact networks as well as heterogeneity in disease transmission and recovery into account. Real social networks among individuals were used to generate an adjacency matrix in my formulas. Both, transition and recovery rates have been used as unique variables for each individual. I have used the forward Kolmogorov equation to solve the system. To control and prevent the infectious diseases such as influenza, sexually transmitted diseases, we have to model the dynamics of a particular disease, estimate the parameters, and forecast the behavior of the disease over time. The estimated parameters help us to design and implement interventions, such as vaccination, closure of public places, to limit the spread of diseases. R0, the reproduction number is an important parameter in epidemiology. R0 is the average number of secondary infections produced by a primary infection. If R0 is larger than one an epidemic will most likely happen, an R0 smaller than one suggests that the disease outbreak is local and will die out. In this study I have shown that R0 estimators that only use the the number of contacts and some network features such as covariance of coefficient are not enough to estimate the epidemic threshold. I have formulated R0 to consider both node degree distribution as well as the spectral gap in the eigenvalue of a weighted adjacency matrix of contact network. Only recently, researchers have developed theoretical approaches that can take into account dynamic networks and, independently, that can use genomic data of the pathogen, sampled from infected persons, to reconstruct the path of an epidemic. By considering the location and time of the sampled pathogen sequence data we can combine the sampled infection network and the mutational history of the pathogen to reconstruct a more accurate contact network. We can reconstruct this dynamic contact network using genetic data and epidemic parameters via a Hidden Markov Model. Sampled genome sequenced data of the pathogen are the observation and a set of dynamic networks are the hidden states in our HMM framework. The system switches between the set of dynamic contact networks to fit the best pattern to observation data. The outcome of such an analysis is the accurate dynamic network among samples of the pathogen. These set of dynamic networks capture the dynamics of the social contact network of the infected people. My model will most likely enable earlier detection of infectious disease spread in dynamic social networks than currently available methods.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9542
 Format
 Thesis
 Title
 Structural Health Monitoring with LambWave Sensors: Problems in Damage Monitoring, Prognostics and Multisensory Decision Fusion.
 Creator

Mishra, Spandan, Vanli, Omer Arda, Okoli, Okenwa, Jung, Sungmoon, Park, Chiwoo, Florida State University, FAMUFSU College of Engineering, Department of Industrial and...
Show moreMishra, Spandan, Vanli, Omer Arda, Okoli, Okenwa, Jung, Sungmoon, Park, Chiwoo, Florida State University, FAMUFSU College of Engineering, Department of Industrial and Manufacturing Engineering
Show less  Abstract/Description

Carbon ﬁber reinforced composites (CFRC) have several desirable traits that can be exploited in the design of advanced structures and systems. The applications requiring high strength toweight ratio and high stiﬀnesstoweight ratio such as, fuselage of airplanes, wind turbine blades, waterboats etc. have found profound use of CFRC. Furthermore, low density, good vibration damping ability, easy manufacturability, carbon ﬁber’s electrical conductivity, as well as high thermal conductivity...
Show moreCarbon ﬁber reinforced composites (CFRC) have several desirable traits that can be exploited in the design of advanced structures and systems. The applications requiring high strength toweight ratio and high stiﬀnesstoweight ratio such as, fuselage of airplanes, wind turbine blades, waterboats etc. have found profound use of CFRC. Furthermore, low density, good vibration damping ability, easy manufacturability, carbon ﬁber’s electrical conductivity, as well as high thermal conductivity and smooth surface ﬁnish provide additional beneﬁts to the users. Various applications of CFRC can be relevant for aerospace, military, windturbines, robotics, sports equipment etc. However, among many advantages of CFRC there are a few disadvantages; CFRC undergo completely diﬀerent failure patterns compared to metals. Once the yield strength is exceeded, CFRC will fail suddenly and catastrophically. The inherent anisotropic nature of CFRC makes it very diﬃcult for traditional condition monitoring methods to assess the condition of the structure. The complex failure patterns, including delamination, microcracks, and matrixcracks require specialized sensing and monitoring schemes for composite structure. This Ph.D. research is focuses on developing an integrated structural health monitoring methodology for damage monitoring, remaining useful life estimation (RUL), and decision fusion using Lambwave data. The main objective of this research is to develop an integrated damage detection method that utilizes Lambwave sensor data to infer the state of the damage condition and make an accurate prognosis of the structure. Slow fatigue loading results in very unique failure patterns in the CFRC structures, fatigue damage ﬁrst manifests itself as ﬁberbreakage and then slowly progresses to matrixcracks and that ultimately leads to delamination damage. This type of failure process is very diﬃcult to monitor using the traditionally used damage monitoring methods such as Xray evaluation, ultrasonic evaluation, infrared evaluation etc. For this research, we have used principal component (PC) based multivariate cumulative sum (MCUSUM) to monitor the structure. MCUSUM chart is very useful when monitoring structures undergoing slow and gradual change. For remainingusefullife (RUL) estimation, we have proposed to use the Wiener process model coupled with principal component regression (PCR). For damage detection/classiﬁcation we studied discriminant analysis, inspite of the popular use in image analysis and in the gene data classiﬁcation problem, has not been widely used for damage classiﬁcation. In this research, we showed that discriminant analysis is a useful detecting known damage modes, while dealing with the high dimensionality of Lambwave data. We modiﬁed the standard Gaussian discriminant analysis by introducing regularization parameters to directly process raw Lambwave data without requiring an intermediate feature extraction step.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SU_Mishra_fsu_0071E_13346
 Format
 Thesis
 Title
 Improvement of a Tracer Correlation Problem with a NonIterative Limiter.
 Creator

Lopez, Nicolas A., Gunzburger, Max D., Burkardt, John V., Peterson, Janet C., Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

A functional relation between two chemical species puts observational constraints on attempts to model the atmosphere. For example, adequate representation of these relations is important when modeling the depletion of stratospheric ozone by nitrous oxide. Previous work has shown a case where a linear functional relation is not preserved in the tracer transport scheme of the Higher Order Methods Modeling Environment (HOMME), which is the spectral element dynamics core used by the Community...
Show moreA functional relation between two chemical species puts observational constraints on attempts to model the atmosphere. For example, adequate representation of these relations is important when modeling the depletion of stratospheric ozone by nitrous oxide. Previous work has shown a case where a linear functional relation is not preserved in the tracer transport scheme of the Higher Order Methods Modeling Environment (HOMME), which is the spectral element dynamics core used by the Community Atmosphere Model (CAM). Application of a certain simple tracer chemistry reaction before each model time step can test whether the scheme actually preserves linear tracer correlations (LCs) to machine precision. Using this method, we confirm previous results that, the implementation of the default shapepreserving filter of HOMME used in the transport scheme does not preserve LCs. However, since we prove that this limiter along with a few other limiter algorithms do in fact preserve LCs in exact arithmetic, we suggest that these limiter algorithms exacerbate the growth of roundoff error in elements where tracers have very different magnitudes. Nevertheless, we manage to put forth a limiting scheme that improves the tracer correlation problem. We also derive another new limiter that relies on multiplicative rescaling of nodal values within a given element. This algorithm does not rely on iterations for convergence and thus has the advantage of being more computationally efficient than the current default CAMSE limiter. Results also show that the default limiter does not always introduce the lowest amount of L₂ error, which contradicts its purpose, since it was derived to minimize error in the L₂ norm.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SU_Lopez_fsu_0071N_13470
 Format
 Thesis
 Title
 Insulin Secretion Rhythms: Calcium Regulation of BetaCell Metabolism and Rescue of Islet Oscillations.
 Creator

McKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University,...
Show moreMcKenna, Joseph P., Bertram, R. (Richard), Roper, Michael Gabriel, Muslimani, Ziad H., Moore, M. Nicholas J. (Matthew Nicholas J.), Miller, Brian G., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Pancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect...
Show morePancreatic islet betacells play a vital role in regulating blood glucose levels by releasing insulin into the bloodstream. Insulin is released in pulses that parallel interacting betacell rhythms, including oscillatory glucose metabolism and periodic calcium influx. We present concurrent time series records of metabolic variables and intracellular calcium levels in glucosestimulated betacells that support regulation of mitochondrial dehydrogenases is the dominant calcium feedback effect onto metabolism in the insulin secretory pathway. We include this effect into the betacell Dual Oscillator Model to reconcile model simulations with experimental data, then we determine the oscillation mechanism in the modified model. Islets lose the rhythms that govern insulin pulses when glucose is elevated to hyperglycemic levels. We demonstrate with modeling and experiments that oscillations lost to elevated glucose can be recovered by converting the elevated glucose stimulus to a sinusoidal wave. We predict with modeling which periodic glucose stimuli can recover islet oscillations.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_McKenna_fsu_0071E_13864
 Format
 Thesis
 Title
 Mathematical Modeling and Analysis of Gene Knockout Compensation in Pancreatic BCells.
 Creator

Yildirim, Vehpi, Bertram, R. (Richard), Fadool, Debra Ann, Mio, Washington, Ökten, Giray, Roper, Michael Gabriel, Florida State University, College of Arts and Sciences,...
Show moreYildirim, Vehpi, Bertram, R. (Richard), Fadool, Debra Ann, Mio, Washington, Ökten, Giray, Roper, Michael Gabriel, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Living systems consist of several complex interacting components. Depending on the complexity of the organism, these components can span from molecules to tissues and organs. Systems biology is the interdisciplinary field of study that uses mathematical and computational tools to describe and investigate the roles these components play in biological systems and the way their interactions result in functionality. The collaborative work between biological and mathematical sciences brings deeper...
Show moreLiving systems consist of several complex interacting components. Depending on the complexity of the organism, these components can span from molecules to tissues and organs. Systems biology is the interdisciplinary field of study that uses mathematical and computational tools to describe and investigate the roles these components play in biological systems and the way their interactions result in functionality. The collaborative work between biological and mathematical sciences brings deeper insights into understanding living systems because, even with the recent advancements in technology, it is impossible to acquire all types of empirical data on many living systems. Technical restrictions together with the complexity of the system components usually give rise to this limitation. Hence, this interdisciplinary field of study makes great contributions to both clinical and basic research by solving these complexities and helping to better interpret the acquired data. Besides, biological experiments can be expansive and time consuming. Therefore, testing biological hypotheses with mathematical models can be significantly beneficial. In this regard, mathematical models can be thought as microscopes developed for specific living systems and inexpensive and fast ways of simulating experiments. Insulin secreting pancreatic betacells are very good examples of such complex systems. Activity of these cells is controlled by extremely complex metabolic and electrophysiological pathways. Therefore, mathematical modeling approaches are proven to be very effective in the study of pancreatic betacells. Impairments in the activity of these cells lead to impaired insulin secretion, which can have life threatening complications in the body. Thus, understanding the mechanisms underlying betacell activity and insulin secretion is crucial. Pancreatic betacells are excitable cells and they produce electrical activity with the ion channels they express in their plasma membranes. In pancreatic betacells, insulin secretion is regulated through pathways that link cellular metabolism to the membrane potential through ion channels they express in their plasma membranes. In the initiation and modulation of the insulin secretion ATPsensitive K+ channels (K(ATP) channels) play a significant role by coupling cell metabolism to the membrane potential. Defects in the expression of K(ATP) channels lead to hypoglycemia associated with excessive insulin secretion in humans. However, mice seem to be able to overcome these defects by employing alternative mechanisms. In this dissertation, we investigate the pathological conditions associated with ATPsensitive K+ channel deficiency in betacells and, with a systems biology approach, we propose mechanisms through which mice can compensate for these defects. Using mathematical modeling we explain the dynamics of these compensatory mechanisms and make predictions to test their plausibility. We also demonstrate the results of the in vitro experiments performed in accordance with our model predictions. One of the longterm goals of this study is helping to identify possible therapeutic targets for the treatment of the congenital hypoglycemia that results from K(ATP) channel deficiency. The overall aim of this dissertation is using mathematical modeling and analysis techniques to better understand the experimental data on pancreatic betacells and guide future research by making testable predictions.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Yildirim_fsu_0071E_13685
 Format
 Thesis
 Title
 ErrorInformed Likelihood Calculations for More Realistic Genetic Analyses.
 Creator

Bricker, Justin, Beerli, Peter, MeyerBaese, Anke, Lemmon, Alan R., Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

Next generation sequencing can rapidly analyze entire genomes in just hours. However, due to the nature of the sequencing process, errors may arise which limit the accuracy of the reads obtained. Luckily, modern sequencing technologies associate with their reads, a quality score, derived from the sequencing procedures, which represents our confidence in each nucleotide in the sequence. Currently, these quality scores are used as a criteria for the removal or modification of reads in the data...
Show moreNext generation sequencing can rapidly analyze entire genomes in just hours. However, due to the nature of the sequencing process, errors may arise which limit the accuracy of the reads obtained. Luckily, modern sequencing technologies associate with their reads, a quality score, derived from the sequencing procedures, which represents our confidence in each nucleotide in the sequence. Currently, these quality scores are used as a criteria for the removal or modification of reads in the data set. These methods result in the loss of information contained in those sequences and rely on parameters that are somewhat arbitrary; this may lead to a biased sample and inaccurate analyses. I propose an alternative method for incorporating the error of the sequences without discarding poor quality reads by including the error probabilities of the reads in the likelihood calculations used for sequence analysis. It was found that, despite introducing variability, using the errorinformed likelihood method improved analyses compared with those which ignored the error altogether. While this method will likely result in analyses with less definite results compared with those in which the data was treated with a preprocessing technique, these results will utilize all of the provided data and will be more grounded in reality as we take into account the uncertainty that we have in our sequenced samples.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Bricker_fsu_0071N_12977
 Format
 Thesis
 Title
 Neural Rule Ensembles: Encoding Feature Interactions into Neural Networks.
 Creator

Dawer, Gitesh, Barbu, Adrian G., Gallivan, Kyle A., Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of...
Show moreDawer, Gitesh, Barbu, Adrian G., Gallivan, Kyle A., Erlebacher, Gordon, Ökten, Giray, Sussman, Mark, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Artificial Neural Networks form the basis of very powerful learning methods. It has been observed that a naive application of fully connected neural networks often leads to overfitting. In an attempt to circumvent this issue, a prior knowledge pertaining to feature interactions can be encoded into these networks. This defines a taskspecific structure on an underlying representation and helps in reducing the number of learnable parameters. Convolutional Neural Network is such an adaptation of...
Show moreArtificial Neural Networks form the basis of very powerful learning methods. It has been observed that a naive application of fully connected neural networks often leads to overfitting. In an attempt to circumvent this issue, a prior knowledge pertaining to feature interactions can be encoded into these networks. This defines a taskspecific structure on an underlying representation and helps in reducing the number of learnable parameters. Convolutional Neural Network is such an adaptation of artificial neural networks for image datasets which exploits the spatial relationship among the features and explicitly encodes the translational equivariance. Similarly, Recurrent Neural Networks are designed to exploit the temporal relationship inherent in sequential data. However, for tabular datasets, any prior structure on feature relationships is not apparent. In this work, we use decision trees to capture such feature interactions for this kind of datasets and define a mapping to encode extracted relationships into a neural network. This addresses the initialization related concerns of fully connected neural networks and enables learning of compact representations compared to state of the art treebased approaches. Empirical evaluations and simulation studies show the superiority of such an approach over fully connected neural networks and treebased approaches.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Dawer_fsu_0071E_14670
 Format
 Thesis
 Title
 Use of Multiple Satellite Total Ozone Observations within and Around Tropical Cyclones.
 Creator

Wang, Hui, Ellingson, R. G., Wang, Xiaoming, Hart, Robert E. (Robert Edward), Misra, Vasubandhu, Liu, Guosheng, Florida State University, College of Arts and Sciences,...
Show moreWang, Hui, Ellingson, R. G., Wang, Xiaoming, Hart, Robert E. (Robert Edward), Misra, Vasubandhu, Liu, Guosheng, Florida State University, College of Arts and Sciences, Department of Earth, Ocean, and Atmospheric Science
Show less  Abstract/Description

This study explores whether or not tropical cyclone (TC) structure information may be retrieved from satellite total ozone observations and how to link total ozone with analysis fields for potential application to TC vortex initialization schemes. Satellite total ozone retrievals from the Atmospheric Infrared Sounder (AIRS), the second Global Ozone Monitoring Experiment (GOME2), the Ozone Monitoring Instrument (OMI), and the Ozone Mapping and Profiler Suite (OMPS) are used in this study. The...
Show moreThis study explores whether or not tropical cyclone (TC) structure information may be retrieved from satellite total ozone observations and how to link total ozone with analysis fields for potential application to TC vortex initialization schemes. Satellite total ozone retrievals from the Atmospheric Infrared Sounder (AIRS), the second Global Ozone Monitoring Experiment (GOME2), the Ozone Monitoring Instrument (OMI), and the Ozone Mapping and Profiler Suite (OMPS) are used in this study. The study is divided into two portions: updating the quality control (QC) scheme for the AIRS total ozone data and examining applications of total ozone data in TC analyses. A modified QC scheme for AIRS total ozone is proposed to identify erroneous data while avoiding removal of potential useful data, as well as to keep the data consistent with a numerical prediction model. The modified QC scheme produces smaller bias and standard deviation of total ozone relative to the original AIRS QC scheme with less data being removed. Since ultraviolet (UV) retrieved total ozone generally suffers less cloud contamination, GOME2 total ozone is used to examine possible TC structures captured by the data. All the TCs in the 20102012 Atlantic Hurricane seasons are used. Detailed comparisons of total ozone from GOME2 and geopotential heights on isentropic surfaces from the European Centre for MediumRange Weather Forecasts (ECMWF) Interim Reanalysis are carried out for the representative case of Hurricane Earl (2010). It is shown that GOME2 total ozone can capture the characteristics of the storm center including the eye region and spiral bands. Changes in total ozone can also reveal the variations of the storm intensity. Alternatively, locally elevated total ozone in the hurricane eye region can be used to identify the upper tropospheric storm center, depending on the stage and the radius of the maximum winds of the storm. Further, the distribution of total ozone shows that the isentropic geopotential heights may have misplaced the vortex center. Total ozone and the 'aligned' geopotential heights at isentropic levels tend to form a linear relationship. Total ozone and geopotential heights correlate better in isentropic level range of 330K to 345K than at other isentropic levels, indicating better inference of the geopotential heights from total ozone at these isentropic levels. Stronger storms tend to have larger correlations of total ozone and geopotential heights at nearly all isentropic levels examined in this study. Intersatellite calibration is shown to be necessary for better use of ozone observations in TC applications. A general intersatellite total ozone calibration is carried out during August of 2012 with total ozone data from the four instruments. Total ozone from different instruments shows a dependence on cloud fraction, solar zenith angle (SZA), geolocation, and possibly aerosols. Overall, GOME2, OMI and OMPS ozone observations agree well globally for both clearsky and cloudy conditions, whereas AIRS shows large relative differences in both the southern Polar Regions and in the 30S30N tropical regions. The large relative errors in the southern Polar Regions are associated with large AIRS SZAs, while the large relative errors in the 30S30N zone may be caused by aerosol contamination and high cirrus clouds. Latitudinally varying coefficients are derived to remove large, zonally varying biases. Hurricane Michael (2012) is chosen to illustrate the resulting impacts of the intersatellite total ozone calibrations on an individual TC's structure information. It is shown that the transformed observations are more comparable spatially and radially than the original ones. The impacts of the calibrated total ozone on the regression models linking isentropic geopotential heights and total ozone formed from all the TCs are examined. The calibration improves the performance of AIRS regression models at nearly all isentropic levels by increasing the explained variance and decreasing the rootmeansquareerror (RMSE), while it does not change much with the regression models for OMI and GOME2. Overall, stronger storms tend to have larger inference of geopotential heights at all the isentropic levels from these total ozone observations before and after the intersatellite calibration. The intersatellite total ozone calibration improves the comparability among different regression models. The resulting more consistent regression models based on TC's intensities can provide more consistent initial analysis fields from different total ozone observations for potential hurricane vortex initialization application.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9481
 Format
 Thesis
 Title
 DTIBased Connectivity in Isolated Neural Ganglia: A Default Structural Graph in a Small World Framework.
 Creator

Ould Ismail, Abdol Aziz O., Grant, Samuel C., Guan, Jingjiao, Li, Yan, Florida State University, College of Engineering, Department of Chemical and Biomedical Engineering
 Abstract/Description

Diffusion Tensor Imaging (DTI) provides a unique contrast based on the restricted directionality of water movement in an anisotropic environment. As such, DTIbased tractography can be used to characterize and quantify the structural connectivity within neural tissue. Here, DTIbased connectivity within isolated abdominal ganglia (ABG) of aplysia Californica is analyzed using network theory. In addition to quantifying the regional physical proprieties of the fractional anisotropy (FA) and...
Show moreDiffusion Tensor Imaging (DTI) provides a unique contrast based on the restricted directionality of water movement in an anisotropic environment. As such, DTIbased tractography can be used to characterize and quantify the structural connectivity within neural tissue. Here, DTIbased connectivity within isolated abdominal ganglia (ABG) of aplysia Californica is analyzed using network theory. In addition to quantifying the regional physical proprieties of the fractional anisotropy (FA) and apparent diffusion coefficient (ADC), DTI tractography was used to probe innerconnections of local communities, yielding unweighted, undirected graphs that represent community structures. Local and global efficiency, characteristic path lengths and clustering analysis are performed on both experimental and simulated data. The relevant intensity and velocity by which these specific nodes communicate is probed through weighted clustering coefficient measurements for the descriptive weighted matrices. Both smallworldness and novel small world metrics were used as tools to verify the smallworld properties for the experimental results. The aim of this manuscript is to categorize and quantify the properties exhibited by structural networks in a model neural tissue to derive unique mean field information that quantitatively describe macroscopic connectivity. For ABG, findings demonstrate a default structural network with preferential specific smallworld properties when compared to simulated lattice and random networks that are equivalent in order and degree.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_OuldIsmail_fsu_0071N_13048
 Format
 Thesis
 Title
 On the motion of a rigid cylinder parallel to its axis in a rotating electrically conducting fluid.
 Creator

Ruan, Kezhi., Florida State University
 Abstract/Description

In an effort to understand better the flow in the core of the Earth, we investigate the steady rise of an infinitely long vertical rigid cylinder parallel to its axis in a rotating electrically conducting fluid in the presence of uniform prescribed transverse magnetic field. The rotation and magneticfield vectors have arbitrary orientation. We suppose the circular cylinder is forced to rise with a constant speed and investigate the structure of the flow and calculate the drag on the cylinder...
Show moreIn an effort to understand better the flow in the core of the Earth, we investigate the steady rise of an infinitely long vertical rigid cylinder parallel to its axis in a rotating electrically conducting fluid in the presence of uniform prescribed transverse magnetic field. The rotation and magneticfield vectors have arbitrary orientation. We suppose the circular cylinder is forced to rise with a constant speed and investigate the structure of the flow and calculate the drag on the cylinder. The flow structure is found by solving a twodimensional (independent of the axial coordinate) mixed boundary value problem. Approximate analytic solutions for velocity field and perturbed magnetic field are obtained. The buoyancy driven rise speed of the cylinder is calculated. The results are consistent with the those derived from Moore and Saffman (1969) and given by Hasimoto (1960) as limiting cases. The numerical value of dimensional rise speed obtained is in good agreement with the typically quoted rise speed in geophysics.
Show less  Date Issued
 1995, 1995
 Identifier
 AAI9525925, 3088642, FSDT3088642, fsu:77444
 Format
 Document (PDF)
 Title
 An analysis of mushchimney structure.
 Creator

Yang, YoungKyun., Florida State University
 Abstract/Description

When a multicomponent liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its...
Show moreWhen a multicomponent liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its permeability to the liquid flow. In particular, if a binary alloy (for example, NH$\sb4$ClH$\sb2$O solution) is cooled at bottom and a dense component (for example, NH$\sb4$Cl) is solidified, buoyant material released during freezing in the pores returns to the melt only through thin, vertical, but widely separated, 'chimneys', the flow through the matrix between them being organized to supply these chimneys., We presented photos of a mushchimney system obtained from the ammonium chloride experiment, and we studied how convection with horizontal divergence affects the structure and flow of the mushchimney system. We use a simple ODE system in the mush derived by assuming that the temperature depends on vertical coordinate only. We find that the mass fraction of solid increases and the depth of a mush decreases when the strength of convection increases., We present an axisymmetric model containing only one chimney to analyze the structure of the mushchimney system. We find solutions of the temperature, the solid fraction, and the pressure in the chimney wall. In particular, the pressure expression shows that the fluid flow needs a huge pressure in order to pass through the chimney wall if its permeability is very small., We assume that a ratio of composition is large, which allows us to neglect the pressure contribution of the chimney wall. We use the knowledge of the variables in the mush, evaluated on the chimney wall, to find the fluid flow in the chimney and the radius of chimney. Our procedure employs the von KarmanPohlhausen technique for determining chimney flow (Roberts & Loper, 1983) and makes use of the fact that the radius of the chimney is much less than the thickness of the mush. We find a relation between a parameter measuring the ratio of viscous and buoyancy forces in the chimney and the vertical velocity component on the top of the mush, and estimate numerically the value of this velocity measuring the strength of convection. The results obtained show reasonably good agreement with theoretical and experimental works (Roberts & Loper (1983), Chen & Chen (1991), Tait & Jaupart (1992), Hellawell etc. (1993), Worster (1991)).
Show less  Date Issued
 1995, 1995
 Identifier
 AAI9540067, 3088707, FSDT3088707, fsu:77509
 Format
 Document (PDF)
 Title
 Computational Methods for AgeatDeath Estimation Based on the Pubic Symphysis.
 Creator

Stoyanova, Detelina, Slice, Dennis E., Creswell, Michael H., AlgeeHewitt, Bridget, Beerli, Peter, Wang, Xiaoqiang, Florida State University, College of Arts and Sciences,...
Show moreStoyanova, Detelina, Slice, Dennis E., Creswell, Michael H., AlgeeHewitt, Bridget, Beerli, Peter, Wang, Xiaoqiang, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

The identification of forensic cases often includes the use of skeletal elements to assess the ageatdeath of an individual. The pubic symphysis is the preferred and most often used skeletal age indicator. Standard techniques, such as the SucheyBrooks system, require that the morphology of the pubic symphysis is visually compared to shape characteristics typical for phases with associated age intervals. As individual factors accumulate during the aging process, estimating the ageatdeath...
Show moreThe identification of forensic cases often includes the use of skeletal elements to assess the ageatdeath of an individual. The pubic symphysis is the preferred and most often used skeletal age indicator. Standard techniques, such as the SucheyBrooks system, require that the morphology of the pubic symphysis is visually compared to shape characteristics typical for phases with associated age intervals. As individual factors accumulate during the aging process, estimating the ageatdeath for older individuals becomes increasingly more difficult. In addition, methods based on visual inspection of the bones introduce some level of subjectivity and observerrelated error. This research makes use of about 100 3D laser scans of the pubic symphysis of white male skeletons with known agesatdeath, and proposes several objective, quantitative methods for shape analysis that aim to provide a surface or outline measure of the shape of the scans that minimizes the ageestimation error. The proposed methods include the use of thin plate splines, twodimensional Fourier, wavelet and elliptic Fourier analysis, and a technique that uses the radius of a best fitting circle (in 2D) or sphere (in 3D) as a measure of the curvature of a shape. In addition some refinement and partitioning techniques were implemented. The project investigates the relationship between the exact ageatdeath and the different measures produced by each method. Also included are results of applying a recently proposed computational method, the SAHScore, to new scan data and scan partitions. As a final result, the project proposes multivariate regression models that combine the measures with highest statistical significance to minimize the age estimation error (about 12 years) and maximize the adjusted Rsquared value (over 55%). Furthermore, the results are subjected to two crossvalidation analysis to test for the accuracy of the models when used in practice.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Stoyanova_fsu_0071E_12868
 Format
 Thesis
 Title
 An Oblate Spheroid Model of Cortical Folding.
 Creator

Grazzini, Courtney, Department of Mathematics
 Abstract/Description

In previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reactiondiffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is...
Show moreIn previous work, Striegel and Hurdal have developed a mathematical model for cortical folding pattern formation during development (Striegel). A Turing reactiondiffusion system and a prolate spheroid domain were used to model the shape of the ventricle during development. They assumed a chemical hypothesis for cortical folding development. The chemical hypothesis suggests that a radial glial cell duplicates into an intermediate progenitor (IP) cell and a new radial glial cell only if it is activated. In turn, the IP cell duplicates into one or two neuroblasts. These cells form the cortical layer. The amplifications due to activated radial glial cells create gyral walls, and the inhibited cells create sulcal valleys. In this research, we modify Striegel and Hurdal's model to use an oblate spheroid domain. We develop mathematical equations using this new domain and investigate the role of various parameters through numerical stimulations. We suggest how these results can be applied to diseases, such as ventriculomegaly and holoprosencephaly, which alter the shape and size of the brain.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_uhm0487
 Format
 Thesis
 Title
 Algorithmic Lung Nodule Analysis in Chest Tomography Images: Lung Nodule Malignancy Likelihood Prediction and a Statistical Extension of the Level Set Image Segmentation Method.
 Creator

Hancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences,...
Show moreHancock, Matthew C. (Matthew Charles), Magnan, Jeronimo Francisco, Duke, D. W., Hurdal, Monica K., Mio, Washington, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Lung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an...
Show moreLung cancer has the highest mortality rate of all cancers in both men and women in the United States. The algorithmic detection, characterization, and diagnosis of abnormalities found in chest CT scan images can aid radiologists by providing additional medicallyrelevant information to consider in their assessment of medical images. Such algorithms, if robustly validated in clinical settings, carry the potential to improve the health of the general population. In this thesis, we first give an analysis of publicly available chest CT scan annotation data, in which we determine upper bounds on expected classification accuracy when certain radiological features are used as inputs to statistical learning algorithms for the purpose of inferring the likelihood of a lung nodule as being either malignant or benign. Second, a statistical extension of the level set method for image segmentation is introduced and applied to both syntheticallygenerated and real threedimensional image volumes of lung nodules in chest CT scans, obtaining results comparable to the current stateoftheart on the latter.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Hancock_fsu_0071E_14427
 Format
 Thesis
 Title
 A Study of Shock Formation and Propagation in the ColdIon Model.
 Creator

Cheung, James, Gunzburger, Max D., Peterson, Janet S., Shanbhag, Sachin, Florida State University, College of Arts and Sciences, Department of Scientific Computing
 Abstract/Description

The central purpose of this thesis is to explore the behavior of the numerical solution of the Cold Ion model with shock forming conditions in one and two dimensions. In the one dimensional case, a comparison between the numerical solution of the Vlasov equation is made. It is observed that the ColdIon model is no longer representative of the coldion limit of the VlasovPoisson equation when a spike forms in the solution. It was found that the lack of a spike in the solution of the Cold...
Show moreThe central purpose of this thesis is to explore the behavior of the numerical solution of the Cold Ion model with shock forming conditions in one and two dimensions. In the one dimensional case, a comparison between the numerical solution of the Vlasov equation is made. It is observed that the ColdIon model is no longer representative of the coldion limit of the VlasovPoisson equation when a spike forms in the solution. It was found that the lack of a spike in the solution of the ColdIon model does not necessarily mean that a bifurcation has not formed in the solution of the VlasovPoisson equation. It was also determined that the spike present in the solution of the one dimensional problem appears again in the two dimensional simulation. The findings presented in this thesis opens up the question of determining which initial and boundary conditions of the ColdIon model causes a shock to form in the solution.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9158
 Format
 Thesis
 Title
 Extending and Simplifying Existing PiecewiseLinear Homotopy Methods for Solving Nonlinear Systems of Equations.
 Creator

Wheaton, Ira Monroe, Awoniyi, Samuel A. (Samuel Ayodele), Foo, Simon Y., Park, Chiwoo, Vanli, Omer Arda, Florida State University, FAMUFSU College of Engineering, Department of...
Show moreWheaton, Ira Monroe, Awoniyi, Samuel A. (Samuel Ayodele), Foo, Simon Y., Park, Chiwoo, Vanli, Omer Arda, Florida State University, FAMUFSU College of Engineering, Department of Industrial and Manufacturing Engineering
Show less  Abstract/Description

This dissertation research extends and simplfiies existing piecewiselinear homotopy (PL) methods to solve G(x) = 0, with G : ℝⁿ → ℝ[superscript m]. Existing PL methods are designed to solve F(x) = 0, with F : ℝⁿ → ℝⁿ and some related pointtoset mappings. PL methods are a component of what is also known as numerical continuation methods, and they are known for being globally convergent methods. First, we present a new PL method for computing zeros of functions of the form ƒ : ℝⁿ → ℝ by...
Show moreThis dissertation research extends and simplfiies existing piecewiselinear homotopy (PL) methods to solve G(x) = 0, with G : ℝⁿ → ℝ[superscript m]. Existing PL methods are designed to solve F(x) = 0, with F : ℝⁿ → ℝⁿ and some related pointtoset mappings. PL methods are a component of what is also known as numerical continuation methods, and they are known for being globally convergent methods. First, we present a new PL method for computing zeros of functions of the form ƒ : ℝⁿ → ℝ by mimicking classical PL methods for computing zeros of functions of the form ƒ : ℝ → ℝ. Our PL method avoids traversing subdivisions of ℝⁿ x [0, 1] and instead uses an object that we refer to as triangulationgraph, which is essentially a triangulation of ℝ x [0, 1] with hypercubes of ℝⁿ as its vertices. The hypercubes are generated randomly, and a sojourn time of an associated discretetime Markov chain is used to show that not too many cubes are generated. Thereafter, our PL method is applied to solving G(x) = 0 for G : ℝⁿ → ℝ[superscript m] under inequality constraints. The resultant method for solving G(x) = 0 translates into a new type of iterative method for solving systems of linear equations. Some computational illustrations are reported. A possible application to optimization problems is also indicated as a direction for further work.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Wheaton_fsu_0071E_13419
 Format
 Thesis
 Title
 Autonomous RC Aircraft with Collision Avoidance Capabilities.
 Creator

Timmons, Cristopher, Department of Electrical and Computer Engineering
 Abstract/Description

This thesis directly deals with setting up all of the electrical components to create an inexpensive automatic dependent surveillance broadcast (ADSB) receiver that can transmit the ADSB data received to a ground control station (GCS) using a Raspberry Pi (single chip computer), operating an autopilot system from the same GCS, and sending the telemetry data to and from the autopilot system and the GCS also using the Raspberry Pi. Furthermore, it will discuss how this setup allows the system...
Show moreThis thesis directly deals with setting up all of the electrical components to create an inexpensive automatic dependent surveillance broadcast (ADSB) receiver that can transmit the ADSB data received to a ground control station (GCS) using a Raspberry Pi (single chip computer), operating an autopilot system from the same GCS, and sending the telemetry data to and from the autopilot system and the GCS also using the Raspberry Pi. Furthermore, it will discuss how this setup allows the system to be capable to performing autonomous collision avoidance maneuvers. Finally, it will show how these capabilities can be demonstrated in a HardwareintheLoop Simulation (HILSim). As the member in charge of performing autonomous flight for the Autonomous Aerial Vehicle senior design team last year, it was ideal to continue research in the field of autonomous flight. The original goal was to compete in the Unmanned Air Systems (UAS) Airspace Operations Challenge (AOC) NASA Centennial Challenge, however, not enough funds were acquired to enter this competition. Therefore, the focus of the thesis shifted to setting up the electrical components of an RC aircraft that would be able to meet the general requirements of the competition.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_uhm0338
 Format
 Thesis