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 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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
 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