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 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
 Exploration of the Role of Disinfection Timing, Duration, and Other Control Parameters on Bacterial Populations Using a Mathematical Model.
 Creator

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

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

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

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

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

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

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

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

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

In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the...
Show moreIn this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using geometric manipulations to fit the discrete domain to the curved domain given by the continuous problem, we use polynomial extensions to enforce that a suitably constructed extension of the numerical solution matches the boundary condition given by the continuous problem in the weak sense. This method is thus aptly named the Polynomial Extension Finite Element Method (PEFEM). Using this approach, we may approximate the solution of elliptic interface problems by enforcing that the extension of the solution on their respective subdomains matches weakly the continuity conditions prescribed by the continuous problem on a curved interface. This method is then called the Method of Virtual Interfaces (MVI), since, while the continuous interface exists in the context of the continuous problem, it is virtual in the context of its numerical approximation. The key benefits of this polynomial extension approach is that it is simple to implement and that it is optimally convergent with respect to the best approximation results given by interpolation. Theoretical analysis and computational results are presented.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Cheung_fsu_0071E_14328
 Format
 Thesis
 Title
 ReducedOrder Modeling of Reactive Solute Transport for AdvectionDominated Problems with Nonlinear Kinetic Reactions.
 Creator

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

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

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

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

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

Recently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due...
Show moreRecently nonlocal continuum models have gained interest as alternatives to traditional PDE models due to their capability of handling solutions with discontinuities and their ease of modeling anomalous diffusion. The typical approach used for approximating timedependent nonlocal integrodifferential models is to use finite element or discontinuous Galerkin methods; however, these approaches can be quite computationally intensive especially when solving problems in more than one dimension due to the approximation of the nonlocal integral. In this work, we propose a novel method based on using radial basis functions to generate accurate quadrature rules for the nonlocal integral appearing in the model and then coupling this with a finite difference approximation to the timedependent terms. The viability of our method is demonstrated through various numerical tests on time dependent nonlocal diffusion, nonlocal anomalous diffusion, and nonlocal advection problems in one and two dimensions. In addition to nonlocal problems with continuous solutions, we modify our approach to handle problems with discontinuous solutions. We compare some numerical results with analogous finite element results and demonstrate that for an equivalent amount of computational work we obtain much higher rates of convergence.
Show less  Date Issued
 2018
 Identifier
 2018_Fall_Lyngaas_fsu_0071E_14886
 Format
 Thesis