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 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
 An Overview of Homotopy Type Theory and the Univalent Foundations of Mathematics.
 Creator

Dunn, Lawrence, Department of Mathematics
 Abstract/Description

Homotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces''  not unstructured sets  are the most primitive objects. This perspective offers...
Show moreHomotopy type theory, the basis of ''univalent foundations'' of mathematics, is a formal system with intrinsic connections to computer science, homotopy theory, and higher category theory. Rooted in type theory, the theoretical basis of most modern proof assistants, the system admits an interpretation as a logical calculus for homotopy theory and suggests a foundational system for which abstract ''spaces''  not unstructured sets  are the most primitive objects. This perspective offers both a computational foundational for mathematics and a direct method for reasoning about homotopy theory. We present here a broad contextual overview of homotopy type theory, including a sufficiently thorough examination of the classical foundations which it replaces as to make clear the extent of its innovation. We will explain that homotopy type theory is, loosely speaking and among other things, a programming language for mathematics, especially one with native support for homotopy theory.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_uhm0304
 Format
 Thesis
 Title
 Dirichlet's Theorem and Analytic Number Theory.
 Creator

Frey, Thomas W., Department of Mathematics
 Abstract/Description

In 1837 Dirichlet proved the infinitude of primes in all arithmetic coprime sequences. This was done by look at Dirichlet Lfunctions, Dirichlet series, Dirichlet characters (modulo k), and Euler Products. In this thesis, the necessary facts, theorems, and properties are shown in order to prove Dirichlet's Theorem, concluding with a proof of Dirichlet's Theorem.
 Date Issued
 2015
 Identifier
 FSU_migr_uhm0560
 Format
 Thesis
 Title
 An Efficient And Longtime Accurate Thirdorder Algorithm For The Stokesdarcy System.
 Creator

Chen, Wenbin, Gunzburger, Max, Sun, Dong, Wang, Xiaoming
 Abstract/Description

A thirdorder in time numerical IMEXtype algorithm for the StokesDarcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel thirdorder AdamsMoulton scheme is used for the discretization of the dissipative term whereas a thirdorder explicit AdamsBashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. The scheme is efficient in the sense that one needs to solve, at each time step, decoupled...
Show moreA thirdorder in time numerical IMEXtype algorithm for the StokesDarcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel thirdorder AdamsMoulton scheme is used for the discretization of the dissipative term whereas a thirdorder explicit AdamsBashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. The scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. The scheme is also unconditionally stable and, with a mild timestep restriction, longtime accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. To the authors' knowledge, the novel algorithm is the first thirdorder accurate numerical scheme for the StokesDarcy system possessing its favorable efficiency, stability, and accuracy properties.
Show less  Date Issued
 201612
 Identifier
 FSU_libsubv1_wos_000386770800006, 10.1007/s0021101507893
 Format
 Citation
 Title
 Analytical Results on the Role of Flexibility in Flapping Propulsion.
 Creator

Moore, Nicholas
 Abstract/Description

Wing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop smallamplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with...
Show moreWing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop smallamplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with expressions for how thrust and efficiency are modified by compliance. The solutions recover a range of realistic behaviours and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple robust estimates afforded by our theory may prove valuable even in situations where details of the flapping motion and wing geometry differ.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_math_faculty_publications0002, 10.1017/jfm.2014.533
 Format
 Citation
 Title
 CAM Stochastic Volatility Model for Option Pricing.
 Creator

Huang, Wanwan, Ewald, Brian, Oekten, Giray
 Abstract/Description

The coupled additive and multiplicative (CAM) noises model is a stochastic volatility model for derivative pricing. Unlike the other stochastic volatility models in the literature, the CAM model uses two Brownian motions, one multiplicative and one additive, to model the volatility process. We provide empirical evidence that suggests a nontrivial relationship between the kurtosis and skewness of asset prices and that the CAM model is able to capture this relationship, whereas the traditional...
Show moreThe coupled additive and multiplicative (CAM) noises model is a stochastic volatility model for derivative pricing. Unlike the other stochastic volatility models in the literature, the CAM model uses two Brownian motions, one multiplicative and one additive, to model the volatility process. We provide empirical evidence that suggests a nontrivial relationship between the kurtosis and skewness of asset prices and that the CAM model is able to capture this relationship, whereas the traditional stochastic volatility models cannot. We introduce a control variate method and Monte Carlo estimators for some of the sensitivities (Greeks) of the model. We also derive an approximation for the characteristic function of the model.
Show less  Date Issued
 2016
 Identifier
 FSU_libsubv1_wos_000376329800001, 10.1155/2016/5496945
 Format
 Citation
 Title
 On The Evolution Of Partial Respect For Ownership.
 Creator

MestertonGibbons, Mike, Karabiyik, Tugba, Sherratt, Tom N.
 Abstract/Description

An early prediction of game theory was that respect for ownership"Bourgeois" or behaviorcan arise as an arbitrary convention to avoid costly disputes. However, its mirror imagethe disputeavoiding "antiBourgeois" or behavior through which owners concede their property to intrudersalso corresponds to an evolutionarily stable strategy (ESS) under the same conditions. It has since been found repeatedly that first finders of valuable resources are frequently left unchallenged in nature,...
Show moreAn early prediction of game theory was that respect for ownership"Bourgeois" or behaviorcan arise as an arbitrary convention to avoid costly disputes. However, its mirror imagethe disputeavoiding "antiBourgeois" or behavior through which owners concede their property to intrudersalso corresponds to an evolutionarily stable strategy (ESS) under the same conditions. It has since been found repeatedly that first finders of valuable resources are frequently left unchallenged in nature, while evidence for ceding property to intruders without a contest is rare at best. An early verbal rationale for the observed rarity of was that two individuals employing such behavior over repeated rounds would be interchanging roles repeatedly, a potentially inefficient outcome known as "infinite regress." This argument was formalized only recently, through a HawkDove model with ownership asymmetry and a fixed probability that two individuals meet again. The analysis showed that if and the cost of fighting exceed thresholds determined by the costs of assuming and relinquishing ownership, then becomes the only stable convention. However, contrary to expectation, and despite the inefficiency of the equilibrium, the analysis also showed that "infinite regress" does not invariably render unviable. Nevertheless, this model dealt only with ESSs at which respect for ownership is either absolute or entirely absent. Here, we extend the model to allow for polymorphic evolutionarily stable states, and we use it to explore the conditions that favor partial respect for ownership. In this way, we produce an analytic model that predicts a range of degrees of partial respect for ownership, dependent on model parameters. In particular, we identify a pathway through which any degree of respect for ownership can evolve from absolute disrespect under increasing with increasing costs of fighting.
Show less  Date Issued
 201609
 Identifier
 FSU_libsubv1_wos_000381210600006, 10.1007/s1323501501524
 Format
 Citation
 Title
 Investigation of drop impact on dry and wet surfaces with consideration of surrounding air.
 Creator

Guo, Yisen, Lian, Yongsheng, Sussman, Mark
 Abstract/Description

Numerical simulations were conducted to investigate drop impingement and splashing on both dry and wet surfaces at impact velocities greater than 50 m/s with the consideration of the effect of surrounding air. The NavierStokes equations were solved using the variable density pressure projection method on a dynamic block structured adaptive grid. The moment of fluid method was used to reconstruct interfaces separating different phases. A dynamic contact angle model was used to define the...
Show moreNumerical simulations were conducted to investigate drop impingement and splashing on both dry and wet surfaces at impact velocities greater than 50 m/s with the consideration of the effect of surrounding air. The NavierStokes equations were solved using the variable density pressure projection method on a dynamic block structured adaptive grid. The moment of fluid method was used to reconstruct interfaces separating different phases. A dynamic contact angle model was used to define the boundary condition at the moving contact line. Simulations showed that lowering the ambient gas density can suppress dry surface splashing, which is in agreement with the experiments. A recirculation zone was observed inside the drop after contact: a larger recirculation zone was formed earlier in the higher gas density case than in the lower gas density case. Increasing gas density also enhances the creation of secondary droplets from the lamella breakup. For high speed impact on a dry surface, lowering ambient gas density attenuates splashing. However, ambient air does not significantly affect splashing on a wet surface. Simulations showed that the splashed droplets are primarily from the exiting liquid film. Published by AIP Publishing.
Show less  Date Issued
 201607
 Identifier
 FSU_libsubv1_wos_000382446200015, 10.1063/1.4958694
 Format
 Citation
 Title
 A Confidence Building Exercise In Data And Identifiability: Modeling Cancer Chemotherapy As A Case Study.
 Creator

Eisenberg, Marisa C., Jain, Harsh V.
 Abstract/Description

Mathematical modeling has a long history in the field of cancer therapeutics, and there is increasing recognition that it can help uncover the mechanisms that underlie tumor response to treatment. However, making quantitative predictions with such models often requires parameter estimation from data, raising questions of parameter identifiability and estimability. Even in the case of structural (theoretical) identifiability, imperfect data and the resulting practical unidentifiability of...
Show moreMathematical modeling has a long history in the field of cancer therapeutics, and there is increasing recognition that it can help uncover the mechanisms that underlie tumor response to treatment. However, making quantitative predictions with such models often requires parameter estimation from data, raising questions of parameter identifiability and estimability. Even in the case of structural (theoretical) identifiability, imperfect data and the resulting practical unidentifiability of model parameters can make it difficult to infer the desired information, and in some cases, to yield biologically correct inferences and predictions. Here, we examine parameter identifiability and estimability using a case study of two compartmental, ordinary differential equation models of cancer treatment with drugs that are cell cyclespecific (taxol) as well as nonspecific (oxaliplatin). We proceed through model building, structural identifiability analysis, parameter estimation, practical identifiability analysis and its biological implications, as well as alternative data collection protocols and experimental designs that render the model identifiable. We use the differential algebra/inputoutput relationship approach for structural identifiability, and primarily the profile likelihood approach for practical identifiability. Despite the models being structurally identifiable, we show that without consideration of practical identifiability, incorrect cell cycle distributions can be inferred, that would result in suboptimal therapeutic choices. We illustrate the usefulness of estimating practically identifiable combinations (in addition to the more typically considered structurally identifiable combinations) in generating biologically meaningful insights. We also use simulated data to evaluate how the practical identifiability of the model would change under alternative experimental designs. These results highlight the importance of understanding the underlying mechanisms rather than purely using parsimony or information criteria/goodnessoffit to decide model selection questions. The overall roadmap for identifiability testing laid out here can be used to help provide mechanistic insight into complex biological phenomena, reduce experimental costs, and optimize modeldriven experimentation. (C) 2017 Published by Elsevier Ltd.
Show less  Date Issued
 10/27/2017
 Identifier
 FSU_libsubv1_wos_000410463600007, 10.1016/j.jtbi.2017.07.018
 Format
 Citation
 Title
 Twofold PT symmetry in doubly exponential optical lattices.
 Creator

Cole, J. T., Makris, K. G., Musslimani, Z. H., Christodoulides, D. N., Rotter, S.
 Abstract/Description

We introduce a family of nonHermitian optical potentials that are given in terms of doubleexponential periodic functions. The center of PT symmetry is not around zero and the potential satisfies a shifted PTsymmetry relation at two distinct locations. Motivated by wave transmission through thin phase screens and gratings, we examine these refractive index modulations from the perspective of optical lattices that are homogeneous along the propagation direction. The diffraction dynamics,...
Show moreWe introduce a family of nonHermitian optical potentials that are given in terms of doubleexponential periodic functions. The center of PT symmetry is not around zero and the potential satisfies a shifted PTsymmetry relation at two distinct locations. Motivated by wave transmission through thin phase screens and gratings, we examine these refractive index modulations 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 examined in detail. In addition, the nonlinear properties of wave propagation in Kerr nonlinearity media are studied. In particular, coherent structures such as lattice solitons are numerically identified by applying the spectral renormalization method. 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 stabilities are critically dependent on their spatial symmetry point.
Show less  Date Issued
 20160104
 Identifier
 FSU_libsubv1_wos_000367658200014, 10.1103/PhysRevA.93.013803
 Format
 Citation
 Title
 Upregulation Of An Inward Rectifying K+ Channel Can Rescue Slow Ca2+ Oscillations In K (atp) Channel Deficient Pancreatic Islets.
 Creator

Yildirim, Vehpi, Vadrevu, Suryakiran, Thompson, Benjamin, Satin, Leslie S., Bertram, Richard
 Abstract/Description

Plasma insulin oscillations are known to have physiological importance in the regulation of blood glucose. In insulinsecreting betacells of pancreatic islets, K(ATP) channels play a key role in regulating glucosedependent insulin secretion. In addition, they convey oscillations in cellular metabolism to the membrane by sensing adenine nucleotides, and are thus instrumental in mediating pulsatile insulin secretion. Blocking K(ATP) channels pharmacologically depolarizes the betacell plasma...
Show morePlasma insulin oscillations are known to have physiological importance in the regulation of blood glucose. In insulinsecreting betacells of pancreatic islets, K(ATP) channels play a key role in regulating glucosedependent insulin secretion. In addition, they convey oscillations in cellular metabolism to the membrane by sensing adenine nucleotides, and are thus instrumental in mediating pulsatile insulin secretion. Blocking K(ATP) channels pharmacologically depolarizes the betacell plasma membrane and terminates islet oscillations. Surprisingly, when K(ATP) channels are genetically knocked out, oscillations in islet activity persist, and relatively normal blood glucose levels are maintained. Compensation must therefore occur to overcome the loss of K(ATP) channels in K(ATP) knockout mice. In a companion study, we demonstrated a substantial increase in Kir2.1 protein occurs in betacells lacking K(ATP) because of SUR1 deletion. In this report, we demonstrate that betacells of SUR1 null islets have an upregulated inward rectifying K+ current that helps to compensate for the loss of K (ATP) channels. This current is likely due to the increased expression of Kir2.1 channels. We used mathematical modeling to determine whether an ionic current having the biophysical characteristics of Kir2.1 is capable of rescuing oscillations that are similar in period to those of wildtype islets. By experimentally testing a key model prediction we suggest that Kir2.1 current upregulation is a likely mechanism for rescuing the oscillations seen in islets from mice deficient in K(ATP) channels.
Show less  Date Issued
 201707
 Identifier
 FSU_libsubv1_wos_000406619800052, 10.1371/journal.pcbi.1005686
 Format
 Citation
 Title
 Automatic stage identification of Drosophila egg chamber based on DAPI images.
 Creator

Jia, Dongyu, Xu, Qiuping, Xie, Qian, Mio, Washington, Deng, WuMin
 Abstract/Description

The Drosophila egg chamber, whose development is divided into 14 stages, is a wellestablished model for developmental biology. However, visual stage determination can be a tedious, subjective and timeconsuming task prone to errors. Our study presents an objective, reliable and repeatable automated method for quantifying cell features and classifying egg chamber stages based on DAPI images. The proposed approach is composed of two steps: 1) a feature extraction step and 2) a statistical...
Show moreThe Drosophila egg chamber, whose development is divided into 14 stages, is a wellestablished model for developmental biology. However, visual stage determination can be a tedious, subjective and timeconsuming task prone to errors. Our study presents an objective, reliable and repeatable automated method for quantifying cell features and classifying egg chamber stages based on DAPI images. The proposed approach is composed of two steps: 1) a feature extraction step and 2) a statistical modeling step. The egg chamber features used are egg chamber size, oocyte size, egg chamber ratio and distribution of follicle cells. Methods for determining the onsite of the polytene stage and centripetal migration are also discussed. The statistical model uses linear and ordinal regression to explore the stagefeature relationships and classify egg chamber stages. Combined with machine learning, our method has great potential to enable discovery of hidden developmental mechanisms.
Show less  Date Issued
 20160106
 Identifier
 FSU_libsubv1_wos_000368658200001, 10.1038/srep18850
 Format
 Citation
 Title
 A Heuristic Method for a Rostering Problem with the Objective of Equal Accumulated Flying Time.
 Creator

Ye, Xugang, Blumsack, Steve, Bellenot, Steve, Braswell, Robert N., Department of Mathematics, Florida State University
 Abstract/Description

Crew costs are the second largest direct operating cost of airlines next to fuel costs. Therefore much research has been devoted to the planning and scheduling of crews over the last thirty years. The planning and scheduling of crews is a highly complex combinatorial problem that consists two independent phases. The first phase is the Crew Pairing Problem (CPP), which concerns finding a set of tasks with minimum cost while satisfying the service requirements. The second phase is the Crew...
Show moreCrew costs are the second largest direct operating cost of airlines next to fuel costs. Therefore much research has been devoted to the planning and scheduling of crews over the last thirty years. The planning and scheduling of crews is a highly complex combinatorial problem that consists two independent phases. The first phase is the Crew Pairing Problem (CPP), which concerns finding a set of tasks with minimum cost while satisfying the service requirements. The second phase is the Crew Rostering Problem (CRP), which concerns finding work assignment for crewmembers in a given period. In this thesis we focus on a Crew Rostering Problem, where a main pilot and a copilot perform a task. The model is a variance minimization problem with 01 variables and constraints associated with ensuring collective agreements, rules and guaranteeing the production of flights service. We choose a sequential constructive method (heuristic) to solve this difficult combinatorial problem since: (1), minimizing quadratic function of discrete variables makes linear methods difficult to use, a monthly schedule for one hundred pilots can generate tens of thousands variables and millions of constraints; (2), it is a NPhard problem, which means the CPU time of solution searching will grow exponentially as the instance dimension (the number of pilots and the number of tasks) increases. According to the characteristics of the model we propose, we do not find the global optimal solution; we find a satisfactory solution (or near optimal solution). The basic idea in our heuristic method is to decompose the assigning process into many subphases day by day. Then in dealing with minimizing the objective function, two heristic principals are employed. Meanwhile, in coping with the constraints, a weighted matching model and its algorithm will be used. In the numerical simulation, the comprehensive method is tested for its effectiveness. We show that our method can produce a solution whose objective value is below a satisfactory bound.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd0944
 Format
 Thesis
 Title
 A Comparison Study of Principal Component Analysis and Nonlinear Principal Component Analysis.
 Creator

Wu, Rui, Magnan, Jerry F., Bellenot, Steven, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

In the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various...
Show moreIn the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various artificial and natural datasets sampled from: "F(x) = sin(x) + x", the Lorenz Attractor, and sunspot data. The results from the experiments have been analyzed and compared. Generally speaking, NLPCA can explain more variance than a neural network PCA (NN PCA) in lower dimensions. However, as a result of increasing the dimension, the NLPCA approximation will eventually loss its advantage. Finally, we introduce a new combination of NN PCA and NLPCA, and analyze and compare its performance.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0704
 Format
 Thesis
 Title
 Steady Dynamics in Shearing Flows of Nematic Liquid Crystalline Polymers.
 Creator

Liu, Fangyu, Wang, Qi, Sussman, Mark, Song, Kaisheng, Department of Mathematics, Florida State University
 Abstract/Description

The biaxiality of the steady state solutions and their stability to inplane disturbances in shearing flows of nematic liquid crystalline polymers are studied by using simplified Wang (2002) model. We obtain all the steady states of Wang model exhibit biaxial symmetry in which two directors are confined to the shearing plane and analysis their stability with respect to inplane disturbances at isolated Debra numbers and polymer concentration values.
 Date Issued
 2004
 Identifier
 FSU_migr_etd1190
 Format
 Thesis
 Title
 QuasiMonte Carlo and Markov Chain QuasiMonte Carlo Methods in Estimation and Prediction of Time Series Models.
 Creator

Tzeng, YuYing, Ökten, Giray, Beaumont, Paul M., Srivastava, Anuj, Kercheval, Alec N., Kim, Kyounghee (Professor of Mathematics), Florida State University, College of Arts and...
Show moreTzeng, YuYing, Ökten, Giray, Beaumont, Paul M., Srivastava, Anuj, Kercheval, Alec N., Kim, Kyounghee (Professor of Mathematics), Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Randomized quasiMonte Carlo (RQMC) methods were first developed in mid 1990’s as a hybrid of Monte Carlo and quasiMonte Carlo (QMC) methods. They were designed to have the superior error reduction properties of lowdiscrepancy sequences, but also amenable to the statistical error analysis Monte Carlo methods enjoy. RQMC methods are used successfully in applications such as option pricing, high dimensional numerical integration, and uncertainty quantification. This dissertation discusses the...
Show moreRandomized quasiMonte Carlo (RQMC) methods were first developed in mid 1990’s as a hybrid of Monte Carlo and quasiMonte Carlo (QMC) methods. They were designed to have the superior error reduction properties of lowdiscrepancy sequences, but also amenable to the statistical error analysis Monte Carlo methods enjoy. RQMC methods are used successfully in applications such as option pricing, high dimensional numerical integration, and uncertainty quantification. This dissertation discusses the use of RQMC and QMC methods in econometric time series analysis. In time series simulation, the two main problems are parameter estimation and forecasting. The parameter estimation problem involves the use of Markov chain Monte Carlo (MCMC) algorithms such as MetropolisHastings and Gibbs sampling. In Chapter 3, we use an approximately completely uniform distributed sequence which was recently discussed by Owen et al. [2005], and an RQMC sequence introduced by O ̈kten [2009], in some MCMC algorithms to estimate the parameters of a Probit and SVlogAR(1) model. Numerical results are used to compare these sequences with standard Monte Carlo simulation. In the time series forecasting literature, there was an earlier attempt to use QMC by Li and Winker [2003], which did not provide a rigorous error analysis. Chapter 4 presents how RQMC can be used in time series forecasting with its proper error analysis. Numerical results are used to compare various sequences for a simple AR(1) model. We then apply RQMC to compute the valueatrisk and expected shortfall measures for a stock portfolio whose returns follow a highly nonlinear Markov switching stochastic volatility model which does not admit analytical solutions for the returns distribution. The proper use of QMC and RQMC methods in Monte Carlo and Markov chain Monte Carlo algorithms can greatly reduce the computational error in many applications from sciences, en gineering, economics and finance. This dissertation brings the proper (R)QMC methodology to time series simulation, and discusses the advantages as well as the limitations of the methodology compared the standard Monte Carlo methods.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Tzeng_fsu_0071E_13607
 Format
 Thesis
 Title
 On the Multidimensional Default Threshold Model for Credit Risk.
 Creator

Zhou, Chenchen, Kercheval, Alec N., Wu, Wei, Ökten, Giray, Fahim, Arash, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

This dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn of the firm's stock price jumps below a (possibly stochastic) "default threshold'' level. The stock price is assumed to follow an exponential L\'evy process and, in the multidimensional case, a multidimensional L\'evy process. This new structural model is...
Show moreThis dissertation is based on the structural model framework for default risk that was first introduced by garreau2016structural (henceforth: the "GK model"). In this approach, the time of default is defined as the first time the logreturn of the firm's stock price jumps below a (possibly stochastic) "default threshold'' level. The stock price is assumed to follow an exponential L\'evy process and, in the multidimensional case, a multidimensional L\'evy process. This new structural model is mathematically equivalent to an intensitybased model where the intensity is parameterized by a L\'evy measure. The dependence between the default times of firms within a basket is the result of the jump dependence of their respective stock prices and described by a L\'evy copula. To extend the previous work, we focus on generalizing the joint survival probability and related results to the ddimensional case. Using the link between L\'evy processes and multivariate exponential distributions, we derive the joint survival probability and characterize correlated default risk using L\'evy copulas. In addition, we extend our results to include stochastic interest rates. Moreover, we describe how to use the default threshold as the interface for incorporating additional exogenous economic factors, and still derive basket credit default swap (CDS) prices in terms of expectations. If we make some additional modeling assumptions such that the default intensities become affine processes, we obtain explicit formulas for the single name and firsttodefault (FtD) basket CDS prices, up to quadrature.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhou_fsu_0071E_14012
 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
 Monte Carlo Scheme for a Singular Control Problem: InvestmentConsumption under Proportional Transaction Costs.
 Creator

Tsai, WanYu, Fahim, Arash, Atkins, Jennifer, Zhu, Lingjiong, Florida State University, College of Arts and Sciences, Department of Mathematics
 Abstract/Description

Nowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative...
Show moreNowadays free boundary problems are considered as one of the most important directions in the mainstream of partial differential equations (PDEs) analysis, with an abundance of applications in various sciences and real world problems. Free boundary problems on finance have been extended in many areas, such as optimal portfolio selection, control credit risks, and different American style products etc. To modelling these financial problems in the real world, the qualitative and quantitative behaviors of the solution to a free boundary problem are still not well understood and also numerical solutions to free boundary problems remain a challenge. Stochastic control problems reduce to freeboundary problems in partial differential equations while there are no bounds on the rate of control. In a free boundary problem, the solution as well as the domain to the PDE need to be determined simultaneously. In this dissertation, we concern the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite time portfolio selection problem with proportional transaction costs. We consider optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a timedependent HamiltonJacobiBellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the highdimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_Tsai_fsu_0071E_14174
 Format
 Thesis
 Title
 Vortices In Boseeinstein Condensates With Ptsymmetric Gain And Loss.
 Creator

Schwarz, Lukas, Cartarius, Holger, Musslimani, Ziad H., Main, Joerg, Wunner, Guenter
 Abstract/Description

We investigate vortex excitations in dilute BoseEinstein condensates in the presence of complex PT symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to nonvortex states. In addition, we study...
Show moreWe investigate vortex excitations in dilute BoseEinstein condensates in the presence of complex PT symmetric potentials. These complex potentials are used to describe a balanced gain and loss of particles and allow an easier calculation of stationary states in open systems than in a full dynamical calculation including the whole environment. We examine the conditions under which stationary vortex states can exist and consider transitions from vortex to nonvortex states. In addition, we study the influences of PT symmetry on the dynamics of nonstationary vortex states placed at offcenter positions.
Show less  Date Issued
 20170512
 Identifier
 FSU_libsubv1_wos_000401189500012, 10.1103/PhysRevA.95.053613
 Format
 Citation
 Title
 Predictive Computer Models for Biofilm Detachment Properties in Pseudomonas aeruginosa.
 Creator

Cogan, Nick G., Harro, Janette M., Stoodley, Paul, Shirtliff, Mark E.
 Abstract/Description

Microbial biofilm communities are protected against environmental extremes or clearance by antimicrobial agents or the host immune response. They also serve as a site from which microbial populations search for new niches by dispersion via single planktonic cells or by detachment by protected biofilm aggregates that, until recently, were thought to become single cells ready for attachment. Mathematically modeling these events has provided investigators with testable hypotheses for further...
Show moreMicrobial biofilm communities are protected against environmental extremes or clearance by antimicrobial agents or the host immune response. They also serve as a site from which microbial populations search for new niches by dispersion via single planktonic cells or by detachment by protected biofilm aggregates that, until recently, were thought to become single cells ready for attachment. Mathematically modeling these events has provided investigators with testable hypotheses for further study. Such was the case in the recent article by Kragh et al. (K. N. Kragh, J. B. Hutchison, G. Melaugh, C. Rodesney, A. E. Roberts, Y. Irie, P. O. Jensen, S. P. Diggle, R. J. Allen, V. Gordon, and T. Bjarnsholt, mBio 7: e0023716, 2016, http://dx.doi.org/10.1128/mBio.0023716), in which investigators were able to identify the differential competitive advantage of biofilm aggregates to directly attach to surfaces compared to the singlecelled planktonic populations. Therefore, as we delve deeper into the properties of the biofilm mode of growth, not only do we need to understand the complexity of biofilms, but we must also account for the properties of the dispersed and detached populations and their effect on reseeding.
Show less  Date Issued
 201606
 Identifier
 FSU_libsubv1_wos_000383440300066, 10.1128/mBio.0081516
 Format
 Citation
 Title
 Quantifying the Relative Contributions of Divisive and Subtractive Feedback to Rhythm Generation.
 Creator

TabakSznajder, Joel, Rinzel, John, Bertram, R. (Richard)
 Abstract/Description

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes:...
Show moreBiological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high and lowactivity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? To answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_math_faculty_publications0003, 10.1371/journal.pcbi.1001124
 Format
 Citation
 Title
 Shape Spaces, Metrics and Their Applications to Brain Anatomy.
 Creator

Liu, Xinyang, Mio, Washington, Kumar, Piyush, Hurdal, Monica K., Quine, Jack, Cogan, Nick, Department of Mathematics, Florida State University
 Abstract/Description

We construct a framework for the analysis of shapes in Euclidean space of any dimension. In this framework, a shape is represented as a continuous map from a reference Riemannian manifold M. To quantify global shape differences, the framework employs a Sobolevtype metric considering information of both the position and the firstorder derivative at each point of the shape. Since firstorder derivatives are very sensitive to small variations, the derivative term in the metric is smoothed out...
Show moreWe construct a framework for the analysis of shapes in Euclidean space of any dimension. In this framework, a shape is represented as a continuous map from a reference Riemannian manifold M. To quantify global shape differences, the framework employs a Sobolevtype metric considering information of both the position and the firstorder derivative at each point of the shape. Since firstorder derivatives are very sensitive to small variations, the derivative term in the metric is smoothed out to reduce noise by a heat operator, which is constructed using the eigenvalues and eigenfunctions of the LaplaceBeltrami operator on M. A preshape space is constructed as an unit sphere of an inner product space, where each shape can be viewed as a point on it. A geodesic shape distance on the preshape space and an extrinsic distance are given based on the proposed metric. Besides global measures, we develop an energy function to quantify local shape divergence. Specifically, it is modified to reflect the magnitude of local shrinkage or expansion. In practical computations, the framework based on continuous representations is discretized using simplicial complex. To address issues in statistical shape analysis for a population of shapes, we present algorithms to calculate the mean shape and to perform principal component analysis on the tangent plane at the mean on the preshape space. The shape models and statistical tools are applied to three data sets of magnetic resonance (MR) scans of the hippocampus to study blindness and Alzheimer's disease. The shape of a hippocampus is represented using either a triangular mesh to represent its contour surface or a cubical mesh to represent its whole volume.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1148
 Format
 Thesis
 Title
 Closed Form Solutions of Linear Difference Equations.
 Creator

Cha, Yongjae, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this idea is how to find a solved equation Lg to which Linp can be reduced. To solve this problem, we use local data of a difference operator, that is invariant under the transformation.
 Date Issued
 2011
 Identifier
 FSU_migr_etd3960
 Format
 Thesis
 Title
 Computational Aeroacoustics Cascade Model of Fan Noise.
 Creator

Lepoudre, Philip P., Tam, Christopher, Shih, Chiang, Gallivan, Kyle, Hussaini, Yousuff, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

A Computational Aeroacoustics [CAA] cascade model has been built to study the generation and propagation mechanisms of noise resulting from the interaction of the fan and outlet guide vanes in a highbypass ratio turbofan engine. Also called rotorstator interaction noise, this noise source is a dominant contributor to the total tone and broadband noise levels produced by the engine, and therefore an improved understanding of the noise generation processes will assist in developing successful...
Show moreA Computational Aeroacoustics [CAA] cascade model has been built to study the generation and propagation mechanisms of noise resulting from the interaction of the fan and outlet guide vanes in a highbypass ratio turbofan engine. Also called rotorstator interaction noise, this noise source is a dominant contributor to the total tone and broadband noise levels produced by the engine, and therefore an improved understanding of the noise generation processes will assist in developing successful noise reduction strategies. The CAA cascade model directly solves the nonlinear compressible NavierStokes equations on a twodimensional linear cascade representation of the fan blade rows. The model incorporates real blade geometry and the rotor and stator blade rows are joined together with a sliding interface method. The fullycoupled aerodynamic flow and acoustic field are directly captured in one high resolution simulation, and therefore the noise production and propagation mechanisms can be visualized and measured in detail. The model includes the fullycoupled physics of the nonlinear sound generation and propagation in swirling wake flow, as well as the transmission and reflection of sound through the blade rows. Previous models of rotorstator interaction noise have typically involved some level of decoupling between the blade rows in order to simplify the noise problem. Stateoftheart CAA methodology is used to produce a high quality numerical solution with minimal dissipation and dispersion of supported waves. The multisizemesh multitimestep Dispersion Relation Preserving [DRP] scheme is used for efficient computation of the wide range of length and time scales in the problem. A conformal mapping technique is used to generate bodyfitted grids around the blade shapes, which are overset on a background grid to create the blade rows. An optimized interpolation scheme is employed for data transfer between the overset grids and also to create the sliding interface between the moving rotorfixed grid and stationary statorfixed grid. A completely new computer program was built for efficient implementation of the cascade model on parallel computers using Message Passing Interface [MPI], and the code was shown to have good parallel performance. The program is a general purpose solver for CAA calculations involving complex flow and geometry, and is a valuable resource for future research. A representative rotorstator cascade with three rotor blades and five stator blades was constructed using real fan and outlet guide vane crosssectional shapes from the NASA Glenn 22in. model fan. A fully developed flow was obtained through the blade rows at the approach condition of the model fan. The performance of the sliding interface method was analyzed by comparing the solution on the rotorfixed and statorfixed grids at the coincident sliding interface mesh line, and the error in grid transfer interpolation was found to be comparable to the low error levels of the underlying DRP scheme. The simulation was used to produce animations of pressure and Mach contour, which provided a wealth of visual information about the flow field and noise generation and propagation behaviour in the cascade. The ability of the CAA cascade model to produce a high fidelity picture of the interaction noise has been demonstrated. In addition, the velocity and pressure fields were measured at various axial locations in the domain to quantify the mean and fluctuating components of the swirling wake flow between the blade rows and after the stator. The tone noise results were compared with interaction tone linear theory. The theory predicted the existence of a small number of propagating spinning wave modes at harmonics of rotor blade passing frequency [BPF]. In particular, the dominant interaction tone at BPF, labelled , was predicted to have two wave fronts in the circumferential domain period and to spin counter to the direction of the rotor. This interaction tone was clearly visible in animations of the pressure contour as an intense shock wave moving at an oblique spiral angle between the blade rows and after the stator. The wave shape was measured using a moving average, and the high amplitude waveform showed characteristic nonlinear steepening, which calls into question the common assumption that the interaction tones can be adequately represented by single linear wave modes. The spinning modes in the solution were measured at various axial locations using a joint temporalspatial modal decomposition of the fluctuating pressure field, and very good agreement was observed with the modal content predicted by linear theory. The relationship of the mode spiral angle to blade stagger angle and the phase velocity of the spinning modes were shown to govern the transmission and reflection behaviour of the modes through the blade rows. The mode was reflected and frequency shifted by the rotor, and the reflected mode propagated through the stator blade row to the outlet. Only corotating modes were able to propagate through the rotor to the inlet, and hence the sound levels in the inlet were significantly lower than in the outlet. This behaviour is in good agreement with the trends observed in experimental studies of fan noise. The unsteady flow and surface pressure fluctuations around a stator blade were also measured. Spectral analysis of the surface pressure fluctuations revealed the highest sound pressure levels occurred near the blade leading edge and on the upper blade surface near the trailing edge. The sound source mechanisms on the stator blade are related to the fluctuating loading on the blade as it cuts through the rotor wake profile and experiences significant variation in the local angle of attack.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3115
 Format
 Thesis
 Title
 Envelopes, Duality, and Multipliers for Certain NonLocally Convex HardyLorentz Spaces.
 Creator

Lengfield, Marc, Oberlin, Daniel M., McKeague, Ian, Bellenot, Steven, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp...
Show moreThis dissertation is a study of the HardyLorentz spaces Hp,q for indices in the range 0 < p < 1, 0 < q ≤ ∞. More precisely, for p as indicated, we describe the dual spaces and Banach envelopes of the spaces Hp,q for finite values of q and do the same for H0p,∞, the closure of the polynomials in Hp,∞. In addition, we determine the sBanach envelopes for the spaces Hp,q in the cases 0 < q < p < s ≤ 1 and 0 < p < q ≤ s ≤ 1. As an application of our results we determine the multiplier spaces (Hp,q, ℓs) for 0 < p < 1, 0 < q, s ≤ ∞.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3131
 Format
 Thesis
 Title
 On Picard 2Stacks and Length 3 Complexes of Abelian Sheaves.
 Creator

Tatar, Ahmet Emin, Aldrovandi, Ettore, Capstick, Simon, Agashe, Amod, Aluﬃ, Paolo, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the...
Show moreIn Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the category of Picard stacks with morphisms being the isomorphism classes. In this dissertation, we generalize his work, following closely his steps in SGA4, to the case of Picard 2stacks. But this generalization requires first a clear description of a Picard 2category as well as of a 2functor between such 2categories that respects Picard structure. Once this has been done, we can talk about category of Picard 2stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2stacks.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1674
 Format
 Thesis
 Title
 An Asymptotically Preserving Method for Multiphase Flow.
 Creator

Jemison, Matthew, Sussman, Mark, Nof, Doron, Cogan, Nick, Gallivan, Kyle, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

A unified, asymptoticallypreserving method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The new algorithm uses a semiimplicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows,...
Show moreA unified, asymptoticallypreserving method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The new algorithm uses a semiimplicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows, including stiff materials, which enables large time steps characteristic of incompressible flow algorithms rather than the small time steps required by explicit methods. Shocks are captured and material discontinuities are tracked, without the aid of any approximate or exact Riemann solvers. The new method enables one to simulate the flow of multiple materials, each possessing a potentially exotic equation of state. Simulations of multiphase flow in one and two dimensions are presented which illustrate the effectiveness of the new algorithm at efficiently computing multiphase flows containing shock waves and material discontinuities with large ''impedance mismatch.'' Additionally, new techniques related to the MomentofFluid interface reconstruction are presented, including a novel, asymptoticallypreserving method for capturing ''filaments,'' and an improved method for initializing the MomentofFluid optimization problem on unstructured, triangular grids.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9012
 Format
 Thesis
 Title
 Constructing NonTrivial Elements of the ShafarevichTate Group of an Abelian Variety.
 Creator

Biswas, Saikat, Agashe, Amod, Aggarwal, Sudhir, Hironaka, Eriko, Van Hoeij, Mark, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non...
Show moreThe ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing nontrivial elements of the ShafarevichTate group of an elliptic curve by means of the MordellWeil group of an ambient curve. In this thesis, we extract a general theorem out of Cremona and Mazur's work and give precise conditions under which such a construction can be made. We then give an extension of our result which provides new theoretical evidence for the BSD conjecture. Finally, we prove a theorem that gives an alternative method to potentially construct nontrivial elements of the ShafarevichTate group of an elliptic curve by using the component groups of a second curve.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3717
 Format
 Thesis
 Title
 Uncertainty Quantification of Nonlinear Stochastic Phenomena.
 Creator

Jimenez, Edwin, Hussaini, M. Y., Srivastava, Anuj, Sussman, Mark, Kopriva, David, Department of Mathematics, Florida State University
 Abstract/Description

The present work quantifies uncertainty in two nonlinear problems using efficient sampling methods and polynomial chaos expansions. The first application is to the Rothermel wildland fire spread model. This model consists of a nonlinear system of algebraic and transcendental equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. The second application quantifies aeroacoustic uncertainty of a...
Show moreThe present work quantifies uncertainty in two nonlinear problems using efficient sampling methods and polynomial chaos expansions. The first application is to the Rothermel wildland fire spread model. This model consists of a nonlinear system of algebraic and transcendental equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. The second application quantifies aeroacoustic uncertainty of a Joukowski airfoil in stochastic vortical gusts. The stochastic gusts are described by random variables that model the gust amplitudes and frequency. The quantification of uncertainty is measured in terms of statistical moments. We construct moment estimates using a variance reduction procedure as well as an efficient stochastic collocation method.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3511
 Format
 Thesis
 Title
 Level Set and Conservative Level Set Methods on Dynamic Quadrilateral Grids.
 Creator

Simakhina, Svetlana, Sussman, Mark, Roper, Michael, Kopriva, David, Ewald, Brian, Peterson, Janet, Department of Mathematics, Florida State University
 Abstract/Description

The work in this thesis is motivated by the application of spray combustion. If one develops algorithms to simulate spray generation, for example the primary breakup of a liquid jet in a gas crossflow, then a bodyfitted or Lagrangian methods would require "surgery" in order to continue a simulation beyond the point at which a droplet is torn into multiple droplets. The liquid volume must also be conserved in simulating spray generation. In this thesis, an Eulerian front tracking method...
Show moreThe work in this thesis is motivated by the application of spray combustion. If one develops algorithms to simulate spray generation, for example the primary breakup of a liquid jet in a gas crossflow, then a bodyfitted or Lagrangian methods would require "surgery" in order to continue a simulation beyond the point at which a droplet is torn into multiple droplets. The liquid volume must also be conserved in simulating spray generation. In this thesis, an Eulerian front tracking method with conserved fluid volume is developed to represent and update an interface between two fluids. It's a level set (LS) method with global volume fix, and the underlying grid is a structured, dynamic, curvilinear grid. We compared our newly developed method to the coupled level set and volume of fluid method (CLSVOF) for two strategic test problems. The first problem, the rotation of a notched disk, tests for robustness. The second problem (proposed in this thesis), the deformation of a circular interface in an incompressible, deforming, velocity field, tests for order of accuracy. We found that for the notched disk problem, the CLSVOF method is superior to the new combined level set method/curvilinear grid method. For a given number of grid points, the CLSVOF method always outperforms the combined level set/curvilinear grid method. On the other hand, for the deformation of a circular interface problem, the combined level set/curvilinear grid method gives better accuracy than the CLSVOF method, for a given number of grid points. Unfortunately the new method is more expensive because a new mesh must be generated periodically. We note that the volume error of the new level set/curvilinear grid algorithm is comparable to that of the CLSVOF method for all test cases tried. We prove that the conservative level set (CLS) method has O(1) local truncation error in an advection scheme. The following developments of the conservative level set (CLS) method are presented in the thesis: new CLS function remapping algorithm and new CLS reinitialization algorithm. The new developments allow one to implement the CLS method on a dynamic quadrilateral grid but don't remedy the order of the method. A new algorithm for quasicubic interpolation is presented. Quasicubic interpolation has been used for local polynomial interpolation on an orthogonal mesh before, but never on a general, nonorthogonal curvilinear mesh. The new (tunnel quasicubic) algorithm enables one to find a global piecewise polynomial interpolation of degree three on an orthogonal mesh, and to find a local polynomial interpolation of degree three on a curvilinear mesh.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1724
 Format
 Thesis
 Title
 Finite Abelian Group Actions on Orientable Circle Bundles over Surfaces.
 Creator

Ibrahim, Caroline Maher Boulis, Heil, Wolfgang, Hollander, Myles, Hironaka, Eriko, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

A finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that...
Show moreA finite group G acts freely on an orientable manifold M if each element of G is a homeomorphism of M, without fixed points, and the multiplication in G is the composition of homeomorphisms. The map from M to M/G of M to the orbit space is a regular cover map. Algebraically, associated with the Gaction is a surjective homomorphism from the fundamental group of M into G. Two Gactions are equivalent if there exists an orientation preserving homeomorphism on M, inducing the identity on G, that preserves the group action. This topological definition is translated to an algebraic definition as two Gactions are equivalent if and only if the associated surjections into G are equivalent via an automorphism of the fundamental group of M. For the manifolds M considered in this dissertation every automorphism of the fundamental group of M can be realized by a homeomorphism of M. Hence there is a onetoone correspondence between the topological and algebraic equivalence. The problem of classifying fixedpoint free finite abelian group actions on surfaces had been investigated by, among others, Nielsen, Smith and Zimmermann. Nielsen classifies cyclic actions on surfaces. He gives a list of automorphisms which he uses in his classification. Smith does the classification for special abelian groups. His approach is different from Nielsen's in the algebraic methods he uses. Zimmermann gives an algebraic solution to the classification of any finite abelian group action on closed surfaces. His technique is to get every surjective homomorphism from the fundamental group of the surface into G in normal form and then differentiate between the normal forms. In this dissertation we classify fixedpoint free finite abelian group actions on circle bundles. By results of Waldhausen every homeomorphism of M is isotopic to a fiber preserving homeomorphism; that is, it preserves the S1 factor of the bundle. This corresponds to the algebraic condition that any automorphism on the fundamental group of M preserves the center of the group. We use the same approach as that of Nielsen on surfaces. We give algorithms to bring every surjective homomorphism from the fundamental group of the bundle into the group G to normal form. From there we differentiate between the normal forms based on Nielsen's results. The results obtained are for circle bundles over surfaces of genus g greater than or equal to 2. A complete classification is given for the case that the circle bundle is a product bundle and G is a finite abelian group. We also obtain a complete classification of cyclic group actions and finite abelian group actions on circle bundles that are not product bundles.
Show less  Date Issued
 2004
 Identifier
 FSU_migr_etd3887
 Format
 Thesis
 Title
 Analysis and Approximation of a TwoBand GinzburgLandau Model of Superconductivity.
 Creator

Chan, WanKan, Gunzburger, Max, Peterson, Janet, Manousakis, Efstratios, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

In 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field...
Show moreIn 2001, the discovery of the intermetallic compound superconductor MgB2 having a critical temperature of 39K stirred up great interest in using a generalization of the GinzburgLandau model, namely the twoband timedependent GinzburgLandau (2BTDGL) equations, to model the phenomena of twoband superconductivity. In this work, various mathematical and numerical aspects of the twodimensional, isothermal, isotropic 2BTDGL equations in the presence of a timedependent applied magnetic field and a timedependent applied current are investigated. A new gauge is proposed to facilitate the inclusion of a timedependent current into the model. There are three parts in this work. First, the 2BTDGL model which includes a timedependent applied current is derived. Then, assuming sufficient smoothness of the boundary of the domain, the applied magnetic field, and the applied current, the global existence, uniqueness and boundedness of weak solutions of the 2BTDGL equations are proved. Second, the existence, uniqueness, and stability of finite element approximations of the solutions are shown and error estimates are derived. Third, numerical experiments are presented and compared to some known results which are related to MgB2 or general twoband superconductivity. Some novel behaviors are also identified.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3923
 Format
 Thesis
 Title
 The Fractal Nature of Lightning: An Investigation of the Fractal Relationship of the Structure of Lightning to Terrain.
 Creator

GrahamJones, Brian Clay, Hunter, Christopher, Elsner, James B., Bellenot, Steve, Department of Mathematics, Florida State University
 Abstract/Description

This study focuses on the relationship between the structure of lightning and how it may or may not be related to the topography below it.
 Date Issued
 2006
 Identifier
 FSU_migr_etd4055
 Format
 Thesis
 Title
 All Speed MultiPhase Flow Solvers.
 Creator

Kadioglu, Samet Y., Sussman, Mark, Telotte, John, Hussaini, Yousuﬀ, Wang, Qi, Erlebacher, Gordon, Department of Mathematics, Florida State University
 Abstract/Description

A new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together...
Show moreA new second order primitive preconditioner technique (an all speed method) for solving all speed single/multiphase flow is presented. With this technique, one can compute both compressible and incompressible flows with Machuniform accuracy and efficiency (i.e., accuracy and efficiency of the method are independent of Mach numbers). The new primitive preconditioner (all speed/Mach uniform) technique can handle both strong and weak shocks, providing highly resolved shock solutions together with correct shock speeds. In addition, the new technique performs very well at the zero Mach limit. In the case of multiphase flow, the new primitive preconditioner technique enables one to accurately treat phase boundaries in which there is a large impedance mismatch. When solving multidimensional all speed multiphase flows, we introduce adaptive solution techniques which exploit the advantages of Machuniform methods. We compute a variety of problems from low (low speed) to high Mach number (high speed) flows including multiphase flow tests, i.e, computing the growth and collapse of adiabatic bubbles for study of underwater explosions
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3391
 Format
 Thesis
 Title
 Intersection Numbers of Divisors in Graph Varieties.
 Creator

Jones, Deborah, Aluffi, Paolo, Aldrovandi, Ettore, Hironaka., Eriko, Klassen, Eric, Reina, Laura, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations...
Show moreThis dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations explain a certain symmetry pattern we discovered.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd3426
 Format
 Thesis
 Title
 Biomedical Applications of Shape Descriptors.
 Creator

Celestino, Christian Edgar Laing, Sumners, De Witt, Greenbaum, Nancy, Mio, Washington, Hurdal, Monica, Department of Mathematics, Florida State University
 Abstract/Description

Given an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss...
Show moreGiven an edgeoriented polygonal graph in R3, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of pathconnected open regions on the unit sphere. Within each open region, the directional writhe is constant. We developed formulas for the writhe of polygons on Bravais lattices and a few crystallographic groups, and discuss applications to ring polymers. In addition, we obtained a closed formula for the writhe for graphs which extends the formula for the writhe of a polygon in R3, including the important special case of writhe of embedded open arcs. Additionally, we have developed shape descriptors based on a family of geometric measures for the purpose of classification and identification of shape differences for graphs. These shape descriptors involve combinations of writhe and average crossing numbers of curves, as well as total curvature, ropelength and thickness. We have applied these shape descriptors to RNA tertiary structures and families of sulcal curves from human brain surfaces. Preliminary results give an automatic method to distinguish RNA motifs. Clear differentiation among tRNA and/or ribozymes, and a distinction among mesophilic and thermophilic tRNA is shown. In addition, we notice a direct correlation between the length of an RNA backbone and its mean average crossing number which is described accurately by a power function. As a neuroscience application, human brain surfaces were extracted from MRI scans of human brains. In our preliminary results, an automatic differentiation between sulcal paths from the left or right hemispheres, an age differentiation and a malefemale classification were achieved.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd3314
 Format
 Thesis
 Title
 Calibration of Multivariate Generalized Hyperbolic Distributions Using the EM Algorithm, with Applications in Risk Management, Portfolio Optimization and Portfolio Credit Risk.
 Creator

Hu, Wenbo, Kercheval, Alec, Huﬀer, Fred, Case, Bettye, Nichols, Warren, Nolder, Craig, Department of Mathematics, Florida State University
 Abstract/Description

The distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical...
Show moreThe distributions of many financial quantities are wellknown to have heavy tails, exhibit skewness, and have other nonGaussian characteristics. In this dissertation we study an especially promising family: the multivariate generalized hyperbolic distributions (GH). This family includes and generalizes the familiar Gaussian and Student t distributions, and the socalled skewed t distributions, among many others. The primary obstacle to the applications of such distributions is the numerical difficulty of calibrating the distributional parameters to the data. In this dissertation we describe a way to stably calibrate GH distributions for a wider range of parameters than has previously been reported. In particular, we develop a version of the EM algorithm for calibrating GH distributions. This is a modification of methods proposed in McNeil, Frey, and Embrechts (2005), and generalizes the algorithm of Protassov (2004). Our algorithm extends the stability of the calibration procedure to a wide range of parameters, now including parameter values that maximize loglikelihood for our real market data sets. This allows for the first time certain GH distributions to be used in modeling contexts when previously they have been numerically intractable. Our algorithm enables us to make new uses of GH distributions in three financial applications. First, we forecast univariate ValueatRisk (VaR) for stock index returns, and we show in outofsample backtesting that the GH distributions outperform the Gaussian distribution. Second, we calculate an efficient frontier for equity portfolio optimization under the skewedt distribution and using Expected Shortfall as the risk measure. Here, we show that the Gaussian efficient frontier is actually unreachable if returns are skewed t distributed. Third, we build an intensitybased model to price Basket Credit Default Swaps by calibrating the skewed t distribution directly, without the need to separately calibrate xi the skewed t copula. To our knowledge this is the first use of the skewed t distribution in portfolio optimization and in portfolio credit risk.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3694
 Format
 Thesis
 Title
 Mathematical Models of Dengue Fever and Measures to Control It.
 Creator

Shen, Yingyun, MestertonGibbons, Mike, Schwartz, Daniel, Okten, Giray, Cogan, Nick, Ewald, Brian, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation, we build a compartment model to investigate the dynamics of spread of dengue fever in both human and mosquito populations. We study the demographic factors that influence equilibrium prevalence, and perform a sensitivity analysis on the basic reproduction number. Among several intervention measures, the effects of two potential control methods for dengue fever are estimated: introducing Wolbachia to the mosquito population and introducing vaccines to the human population...
Show moreIn this dissertation, we build a compartment model to investigate the dynamics of spread of dengue fever in both human and mosquito populations. We study the demographic factors that influence equilibrium prevalence, and perform a sensitivity analysis on the basic reproduction number. Among several intervention measures, the effects of two potential control methods for dengue fever are estimated: introducing Wolbachia to the mosquito population and introducing vaccines to the human population. A stochastic model for transmission of dengue fever is also built to explore the effect of some demographic factors.
Show less  Date Issued
 2014
 Identifier
 FSU_migr_etd9093
 Format
 Thesis
 Title
 Solutions of Second Order Recurrence Relations.
 Creator

Levy, Giles, Van Hoeij, Mark, Van Engelen, Robert A., Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such...
Show moreThis thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such a transformation exists. The second algorithm `Find Liouvillian' finds a gttransformation to a recurrence relation of the form u(n+2) + b(n)u(n) = 0 for some b(n) in C(n), if such a transformation exists. The third algorithm `Database Solver' takes advantage of a large database of sequences, `The OnLine Encyclopedia of Integer Sequences' maintained by Neil A. J. Sloane at AT&T Labs Research. It employs this database by using the recurrence relations that they satisfy as base equations from which to return a gttransformation, if such a transformation exists.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3099
 Format
 Thesis
 Title
 Singleand MultipleObjective Stochastic Programming Models with Applications to Aerodynamics.
 Creator

Croicu, AnaMaria, Hussaini, M. Yousuﬀ, Srivastava, Anuj, Kopriva, David, Wang, Qi, Department of Mathematics, Florida State University
 Abstract/Description

Deterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be...
Show moreDeterministic design assumes that there is no uncertainty in the modeling parameters, and as a consequence, there is no variability in the simulation outputs. Therefore, deterministic optimal designs that are obtained without taking into account uncertainty are usually unreliable. This is the case with transonic shape optimization, where the randomness in the cruise Mach number might have significant impact on the optimal geometric design. In this context, a stochastic search turns out to be more appropriate. Approaches to stochastic optimization have followed a variety of modeling philosophies, but little has been done to systematically compare different models. The goal of this thesis is to present a comparison between two stochastic optimization algorithms, with the emphasis on applications, especially on the airfoil shape optimization. Singleobjective and multiobjective optimization programs are analyzed as well. The relationship between the expected minimum value (EMV) criterion and the minimum expected value (MEV) criterion is explored, and it is shown that, under favorable conditions, a better optimal point could be obtained via the EMV approach. Unfortunately, the advantages of using the EMV approach are far outweighed by the prohibitive exorbitant computational cost.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd3027
 Format
 Thesis
 Title
 Massively Parallel Algorithms for CFD Simulation and Optimization on Heterogeneous ManyCore Architectures.
 Creator

Duffy, Austen C., Sussman, Mark, Hussaini, M. Yousuﬀ, Van Engelen, Robert, Cogan, Nick, Gallivan, Kyle, Department of Mathematics, Florida State University
 Abstract/Description

In this dissertation we provide new numerical algorithms for use in conjunction with simulation based design codes. These algorithms are designed and best suited to run on emerging heterogenous computing architectures which contain a combination of traditional multicore processors and new programmable manycore graphics processing units (GPUs). We have developed the following numerical algorithms (i) a new Multidirectional Search (MDS) method for PDE constrained optimization that utilizes a...
Show moreIn this dissertation we provide new numerical algorithms for use in conjunction with simulation based design codes. These algorithms are designed and best suited to run on emerging heterogenous computing architectures which contain a combination of traditional multicore processors and new programmable manycore graphics processing units (GPUs). We have developed the following numerical algorithms (i) a new Multidirectional Search (MDS) method for PDE constrained optimization that utilizes a Multigrid (MG) strategy to accelerate convergence, this algorithm is well suited for use on GPU clusters due to its parallel nature and is more scalable than adjoint methods (ii) a new GPU accelerated point implicit solver for the NASA FUN3D code (unstructured NavierStokes) that is written in the Compute Unified Device Architecture (CUDA) language, and which employs a novel GPU sharing model, (iii) novel GPU accelerated smoothers (developed using PGI Fortran with accelerator compiler directives) used to accelerate the multigrid preconditioned conjugate gradient method (MGPCG) on a single rectangular grid, and (iv) an improved pressure projection solver for adaptive meshes that is based on the MGPCG method which requires fewer grid point calculations and has potential for better scalability on hetergeneous clusters. It is shown that a multigrid  multidirectional search (MGMDS) method can run up to 5.5X faster than the MDS method when used on a one dimensional data assimilation problem. It is also shown that the new GPU accelerated point implicit solver of FUN3D is up to 5.5X times faster than the CPU version and that the solver can perform up to 40% faster on a single GPU being shared by four CPU cores. It is found that GPU accelerated smoothers for the MGPCG method on uniform grids can run over 2X faster than the nonaccelerated versions for 2D problems, and that the new MGPCG pressure projection solver for adaptive grids is up to 4X faster than the previous MG algorithm.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0651
 Format
 Thesis
 Title
 Openmath Library for Computing on Riemann Surfaces.
 Creator

Lebedev, Yuri, Seppälä, Mika, Van Engelen, Robert, Van Hoeij, Mark, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis carefully reviews computational methods that will act as a tool in the research of Riemann surfaces. We are interested in representing a Riemann surface from many equivalent points of view. The goal is to define a Riemann surface so it can be freely and unambiguously exchanged between mathematical servers by creating a set of suitable OpenMath CDs.
 Date Issued
 2008
 Identifier
 FSU_migr_etd3208
 Format
 Thesis
 Title
 Centroidal Voronoi Tessellations for Mesh Generation: from Uniform to Anisotropic Adaptive Triangulations.
 Creator

Nguyen, Hoa V., Gunzburger, Max D., ElAzab, Anter, Peterson, Janet, Wang, Xiaoming, Wang, Xiaoqiang, Department of Mathematics, Florida State University
 Abstract/Description

Mesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume methods. Mesh generation can be classified into several categories depending on the element sizes (uniform or nonuniform) and shapes (isotropic or anisotropic). Uniform meshes have been well studied and still find application in a wide variety of problems....
Show moreMesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume methods. Mesh generation can be classified into several categories depending on the element sizes (uniform or nonuniform) and shapes (isotropic or anisotropic). Uniform meshes have been well studied and still find application in a wide variety of problems. However, when solving certain types of partial differential equations for which the solution variations are large in some regions of the domain, nonuniform meshes result in more efficient calculations. If the solution changes more rapidly in one direction than in others, nonuniform anisotropic meshes are preferred. In this work, first we present an algorithm to construct uniform isotropic meshes and discuss several mesh quality measures. Secondly we construct an adaptive method which produces nonuniform anisotropic meshes that are well suited for numerically solving PDEs such as the convection diffusion equation. For the uniform Delaunay triangulation of planar regions, we focus on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVTbased grid generation. We also compare several methods, including CVTbased methods, for triangulating planar domains. Furthermore, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce highquality uniform isotropic grids and that the CVTbased grids are at least as good as any of the others. For more general grid generation settings, e.g., nonuniform and/or anistropic grids, such quantitative comparisons are much more difficult, if not impossible, to either make or interpret. This motivates us to develop CVTbased adaptive nonuniform anisotropic mesh refinement in the context of solving the convectiondiffusion equation with emphasis on convectiondominated problems. The challenge in the numerical approximation of this equation is due to large variations in the solution over small regions of the physical domain. Our method not only refines the underlying grid at these regions but also stretches the elements according to the solution variation. Three main ingredients are incorporated to improve the accuracy of numerical solutions and increase the algorithm's robustness and efficiency. First, a streamline upwind Petrov Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor. Our algorithm has been tested on a variety of 2dimensional examples. It is robust in detecting layers and efficient in resolving nonphysical oscillations in the numerical approximation.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2616
 Format
 Thesis
 Title
 Rheology and Mesoscale Morphology of Flows of Chlesteric and Nematic Liquid Crystal Polymers.
 Creator

Cui, Zhenlu, Wang, Qi, Liu, Guosheng, Magnan, Jerry F., Sussman, Mark, Tam, Christopher, Department of Mathematics, Florida State University
 Abstract/Description

Cholesteric liquid crystals(CLC) are mesophases, where the average direction of molecular orientation exhibits a chiral (twisted) pattern along its normal direction. In the past, the rheological and flow properties of CLC have been studied scarcely. This is due to the natural tendency of a cholesteric to favor its characteric, twisted configuration, which naturally leads to more complex arrangements of the optic axis than in pure nematics and complicated spatial structures. In this...
Show moreCholesteric liquid crystals(CLC) are mesophases, where the average direction of molecular orientation exhibits a chiral (twisted) pattern along its normal direction. In the past, the rheological and flow properties of CLC have been studied scarcely. This is due to the natural tendency of a cholesteric to favor its characteric, twisted configuration, which naturally leads to more complex arrangements of the optic axis than in pure nematics and complicated spatial structures. In this dissertation, we address the issues related to rheology and flow induced structures in CLC and nematic polymers, with emphasis on the role of the anisotropic elasticities. In the first part of this dissertation, we study the permeation flow problem using a mesoscopic theory obtained from the kinetic theory for Cholesteric liquid crystal polymers and resolve the inconsistency issue in the literature. Then we give a systematic study on steady structures and transient behavior in flows of nematic polymers. In the second part of this dissertation, we develop a hydrodynamic theory for flows of CLCPs following the continuum mechanics formulation of McMillan's second order tensor theory for liquid crystals and study phase transition in chiral nematic liquid crystals as well as the rheological hebaviors and the flow properties of CLCPs.
Show less  Date Issued
 2005
 Identifier
 FSU_migr_etd2952
 Format
 Thesis
 Title
 Adaptive Spectral Element Methods to Price American Options.
 Creator

Willyard, Matthew, Kopriva, David, Eugenio, Paul, Case, Bettye Anne, Gallivan, Kyle, Nolder, Craig, Okten, Giray, Department of Mathematics, Florida State University
 Abstract/Description

We develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or derefinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre...
Show moreWe develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or derefinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre coefficients and the global error estimator is based on the adjoint problem. One local error estimator uses the rate of decay of the Legendre coefficients to estimate the error. The other local error estimator compares the solution to an estimated solution using fewer Legendre coefficients found by the Tau method. The global error estimator solves the adjoint problem to weight local error estimates to approximate a terminal error functional. Both types of error estimators produce meshes that match expectations by being fine near the early exercise boundary and strike price and coarse elsewhere. The produced meshes also adapt as expected by derefining near the strike price as the solution smooths and staying fine near the moving early exercise boundary. Both types of error estimators also give solutions whose error is within prescribed tolerances. The adjointbased error estimator is more flexible, but costs up to three times as much as using the local error estimate alone. The global error estimator has the advantages of tracking the accumulation of error in time and being able to discount large local errors that do not affect the chosen terminal error functional. The local error estimator is cheaper to compute because the global error estimator has the added cost of solving the adjoint problem.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0892
 Format
 Thesis
 Title
 Combinatorial Type Problems for Triangulation Graphs.
 Creator

Wood, William E., Bowers, Philip, Hawkes, Lois, Bellenot, Steve, Klassen, Eric, Nolder, Craig, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situation, most significant in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results.
 Date Issued
 2006
 Identifier
 FSU_migr_etd0794
 Format
 Thesis
 Title
 ChernSchwartzMacpherson Classes of Graph Hypersurfaces and Schubert Varieties.
 Creator

Stryker, Judson P., Aluﬃ, Paolo, Van Engelen, Robert, Aldrovandi, Ettore, Hironaka, Eriko, Van Hoeij, Mark, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of...
Show moreThis dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of certain families of these varieties is proven. The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation. In the second chapter, a series of equivalences of graph hypersurfaces that are useful for reducing the number of cases that must be calculated are developed. A table of CSM classes of all but one graph with 6 or fewer edges are explicitly computed. This table also contains Fulton Chern classes and Milnor classes for the graph hypersurfaces. Using the equivalences and a series of formulas from a paper by Aluffi and Mihalcea, a new series of formulas for the CSM classes of certain families of graph hypersurfaces are deduced. I prove positivity for all graph hypersurfaces corresponding to graphs with first Betti number of 3 or less. Formulas for graphs equivalent to graphs with 6 or fewer edges are developed (as well as cones over graphs with 6 or fewer edges). In the third chapter, CSM classes of Schubert varieties are discussed. It is conjectured by Aluffi and Mihalcea that all Chern classes of Schubert varieties are represented by effective cycles. This is proven in special cases by B. Jones. I examine some positivity results by analyzing and applying combinatorial methods to a formula by Aluffi and Mihalcea. Positivity of what could be considered the ``typical' case for low codimensional coefficients is found. Some other general results for positivity of certain coefficients of Schubert varieties are found. This technique establishes positivity for some known cases very quickly, such as the codimension 1 case as described by Jones, as well as establishing positivity for codimension 2 and families of cases that were previously unknown. An unexpected connection between one family of cases and a second order PDE is also found. Positivity is shown for all cases of codimensions 14 and some higher codimensions are discussed. In both the graph hypersurfaces and Schubert varieties, all calculated ChernSchwartzMacPherson classes were found to be positive.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1531
 Format
 Thesis
 Title
 Geometric and Computational Generation, Correction, and Simplification of Cortical Surfaces of the Human Brain.
 Creator

Singleton, Lee William, Hurdal, Monica K., Kumar, Piyush, Mio, Washington, Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

The generation, correction, and simplification of brain surfaces from magnetic resonance imaging (MRI) data are important for studying brain characteristics, diseases, and functionality. Changes in cortical surfaces are used to compare healthy and diseased populations and they are used to understand how the brain changes as we age. We present several algorithms that use corrected MRI data to create a manifold surface, correct its topology, and simplify the resulting surface. We make...
Show moreThe generation, correction, and simplification of brain surfaces from magnetic resonance imaging (MRI) data are important for studying brain characteristics, diseases, and functionality. Changes in cortical surfaces are used to compare healthy and diseased populations and they are used to understand how the brain changes as we age. We present several algorithms that use corrected MRI data to create a manifold surface, correct its topology, and simplify the resulting surface. We make comparisons of several algorithmic choices and highlight the options that result in surfaces with the most desirable properties. In our discussion of surface generation, we present new approaches and analyze their features. We also provide a simple way to ensure that the created surface is a manifold. We compare our approaches to an existing method by examining the geometric and topological properties of the generated surfaces, including triangle count, surface area, Euler characteristic, and vertex degree. Our chapter on topology correction provides a description of our algorithm that can be used to correct the topology of a surface from the underlying volume data under a specific digital connectivity. We also present notation for new types of digital connectivities and show how our algorithm can be generalized to correct surfaces using these new connectivity schemes on the underlying volume. Our surface simplification algorithm is able to replace surface edges with new points in space rather than being restricted to the surface. We present new formulas for the fast and efficient computation of points for interior as well as boundary edges. We also provide results of several cost functions and report on their performances in surface simplification. Other algorithmic choices are also discussed and evaluated for effectiveness. We are able to produce high quality surfaces that reduce the number of surface triangles by 8586% on average while preserving surface topology, geometry, and anatomical features. On closed surfaces, our algorithm also preserves the volume inside the surface. This work provides an improvement to the general framework of surface processing. We are able to produce high quality surfaces with very few triangles and still maintain the general properties of the surface. These results have applicability to other downstream processes by reducing the processing time of applications such as flattening, inflation, and registration. Our surface results also produce much smaller files for use in future database systems. Furthermore, these algorithms can be applied to other areas of computational anatomy and scientific visualization. They have applicability to fields of medicine, computer graphics, and computational geometry.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1702
 Format
 Thesis
 Title
 Optimization Algorithms on Riemannian Manifolds with Applications.
 Creator

Huang, Wen, Gallivan, Kyle A., Absil, PierreAntoine, Duke, Dennis, Okten, Giray, Klassen, Eric P., Department of Mathematics, Florida State University
 Abstract/Description

This dissertation generalizes three wellknown unconstrained optimization approaches for Rn to solve optimization problems with constraints that can be viewed as a ddimensional Riemannian manifold to obtain the Riemannian Broyden family of methods, the Riemannian symmetric rankone trust region method, and Riemannian gradient sampling method. The generalization relies on basic differential geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as...
Show moreThis dissertation generalizes three wellknown unconstrained optimization approaches for Rn to solve optimization problems with constraints that can be viewed as a ddimensional Riemannian manifold to obtain the Riemannian Broyden family of methods, the Riemannian symmetric rankone trust region method, and Riemannian gradient sampling method. The generalization relies on basic differential geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on the more recent notions of (firstorder) retraction and vector transport. The effectiveness of the methods and techniques for their efficient implementation are derived and evaluated. Basic experiments and applications are used to illustrate the value of the proposed methods. Both the Riemannian symmetric rankone trust region method and the RBroyden family of methods are generalized from Euclidean quasiNewton optimization methods, in which a Hessian approximation exploits the wellknown secant condition. The generalization of the secant condition and the associated update formulas that define quasiNewton methods to the Riemannian setting is a key result of this dissertation. The dissertation also contains convergence theory for these methods. The Riemannian symmetric rankone trust region method is shown to converge globally to a stationary point and d+1step qsuperlinearly to a minimizer of the objective function. The RBroyden family of methods is shown to converge globally and qsuperlinearly to a minimizer of a retractionconvex objective function. A condition, called the locking condition, on vector transport and retraction that guarantees convergence for the RBroyden family method and facilitates efficient computation is derived and analyzed. The Dennis Mor\'e sufficient and necessary conditions for superlinear convergence, can be generalized to the Riemannian setting in multiple ways. This dissertation generalizes them in a novel manner that is applicable to both Riemannian optimization problems and root finding for a vector field on a Riemannian manifold. The convergence analyses of Riemannian symmetric rankone trust region method and RBroyden family methods assume a smooth objective function. For partly smooth Lipschitz continuous objective functions, a variation of one of the RBroyden family methods, RBFGS, is shown to be work well empirically. In addition, the Riemannian gradient sampling method is shown to work well empirically for both a Lipschitz continuous and a nonLipschitz continuous objective function associated with the important application nonlinear dimension reduction. Efficient and effective implementations for a manifold in Rn, a quotient manifold of total manifold in Rn and a product of manifolds, are presented. Results include efficient representations and operations of elements in a manifold, tangent vectors, linear operators, retractions and vector transports. Novel techniques for constructing and computing multiple kinds of vector transports are derived. In addition, the implementation details of all required objects for optimization on four manifolds, the Stiefel manifold, the sphere, the orthogonal group and the Grassmann manifold, are presented. Basic numerical experiments for the Brockett cost function on the Stiefel manifold, the Rayleigh quotient on the Grassmann manifold and the minmax problem on the sphere (Lipschitz and nonLipschitz forms), are used to illustrate the performance of the proposed methods and compare with existing optimization methods on manifolds. Three applications, Riemannian optimization for elastic shape analysis, a joint diagonalization problem for independent component analysis and a synchronization of rotation problem, that have smooth cost functions are used to show the advantages of the proposed methods. A secantbased nonlinear dimension reduction problem with a partly smooth function is used to show the advantages of the Riemannian gradient sampling method.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd8809
 Format
 Thesis