Current Search: Peterson, Janet S. (x)
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 Title
 Reduced Order Modeling of Reactive Transport in a Column Using Proper Orthogonal Decomposition.
 Creator

McLaughlin, Benjamin R. S., Peterson, Janet, Ye, Ming, Shanbhag, Sachin, Department of Scientific Computing, Florida State University
 Abstract/Description

Estimating parameters for reactive contaminant transport models can be a very computationally intensive. Typically this involves solving a forward problem many times, with many degrees of freedom that must be computed each time. We show that reduced order modeling (ROM) by proper orthogonal decomposition (POD) can be used to approximate the solution to the forward model using many fewer degrees of freedom. We provide background on the finite element method and reduced order modeling in one...
Show moreEstimating parameters for reactive contaminant transport models can be a very computationally intensive. Typically this involves solving a forward problem many times, with many degrees of freedom that must be computed each time. We show that reduced order modeling (ROM) by proper orthogonal decomposition (POD) can be used to approximate the solution to the forward model using many fewer degrees of freedom. We provide background on the finite element method and reduced order modeling in one spatial dimension, and apply both methods to a system of linear uncoupled timedependent equations simulating reactive transport in a column. By comparing the reduced order and finite element approximations, we demonstrate that the reduced model, while having many fewer degrees of freedom to compute, gives a good approximation of the highdimensional (finite element) model. Our results indicate that one may substitute a reduced model in place of a highdimensional model to solve the forward problem in parameter estimation with many fewer degrees of freedom.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd5030
 Format
 Thesis
 Title
 A Study of Shock Formation and Propagation in the ColdIon Model.
 Creator

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

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

Gu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific...
Show moreGu, Rui, Wang, Xiaoqiang, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet S., Ye, Ming, Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

We construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions...
Show moreWe construct a phase field model for simulating the adhesion of a cell membrane to a substrate. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions between a lipid vesicle and substrates having complex shapes. Examples are also provided for the adhesion process in the presence of gravitational and point pulling forces. A comparison with experimental results demonstrates the effectiveness of the two phase field approach. Similarly to simulating vesiclesubstrate adhesion, we construct a multiphasefield model for simulating the adhesion between two vesicles. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accounts for the elastic bending energy of each vesicle and the contact potential energy between the two vesicles; the vesicle volume and surface area constraints are imposed using a penalty method. Numerical results are provided to verify the efficacy of our model and to provide visual illustrations of the different types of contact. The method can be adjusted to solve endocytosis problems by modifying the bending rigidity coefficients of the two elastic bending energies. The method can also be extended to simulate multicell adhesions, one example of which is erythrocyte rouleaux. A comparison with laboratory observations demonstrates the effectiveness of the multiphase field approach. Coupled with fluid, we construct a phase field model for simulating vesiclevessel adhesion in a flow. Two phase field functions are introduced to simulate the vesicle and vessel respectively. The fluid is modeled and confined inside the tube by a phase field coupled NavierStokes equation. Both vesicle and vessel are transported by fluid flow inside our computational domain. An energy model regarding the comprehensive behavior of vesiclefluid interaction, vesselfluid interaction, vesiclevessel adhesion is defined. The vesicle volume and surface area constraints are imposed using a penalty method, while the vessel elasticity is modeled under Hooke's Law. Numerical results are provided to verify the efficacy of our model and to demonstrate the effectiveness of our fluidcoupled vesicle vessel adhesion phase field approach by comparison with laboratory observations.
Show less  Date Issued
 2015
 Identifier
 FSU_2015fall_Gu_fsu_0071E_12873
 Format
 Thesis
 Title
 Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic NavierStokes Equations.
 Creator

Zhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and...
Show moreZhao, Wenju, Gunzburger, Max D., Sussman, Mark, Peterson, Janet S., Quaife, Bryan, Huang, Chen (Professor of Scientific Computing), Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Stochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and...
Show moreStochastic NavierStokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic NavierStokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic NavierStokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic NavierStokes equations with additive noises; and the stochastic NavierStokes equations constrained stochastic boundary optimal control problems. We systemically provide numerical simulation methods for the stochastic NavierStokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the NavierStokes system are presented. To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic NavierStokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified NavierStokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the pathwise modified NavierStokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of nonregularized and regularized noises for the NavierStokes system are presented to further demonstrate the efficiency of our numerical scheme. As a consequence of the above work, we consider a stochastic optimal control problem constrained by the NavierStokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H¹(∂D). To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. Oneshot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Zhao_fsu_0071E_14002
 Format
 Thesis
 Title
 Ensemble Proper Orthogonal Decomposition Algorithms for the Incompressible NavierStokes Equations.
 Creator

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

The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the timedependent NavierStokes equations for which a recently developed ensemblebased method allows for...
Show moreThe definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the timedependent NavierStokes equations for which a recently developed ensemblebased method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. In this work we incorporate a proper orthogonal decomposition (POD) reducedorder model into the ensemblebased method to further reduce the computational cost; in total, three algorithms are developed. Initially a first order accurate in time scheme for low Reynolds number flows is considered. Next a second order algorithm useful for applications that require longterm time integration, such as climate and ocean forecasting is developed. Lastly, in order to extend this approach to convection dominated flows a model incorporating a POD spatial filter is presented. For all these schemes stability and convergence results for the ensemblebased methods are extended to the ensemblePOD schemes. Numerical results are provided to illustrate the theoretical stability and convergence results which have been proven.
Show less  Date Issued
 2018
 Identifier
 2018_Su_Schneier_fsu_0071E_14687
 Format
 Thesis
 Title
 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
 Mass Conserving HamiltonianStructurePreserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme.
 Creator

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

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

Sockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of...
Show moreSockwell, K. Chadwick (Kenneth Chadwick), Gunzburger, Max D., Peterson, Janet S., Burkardt, John V., Florida State University, College of Arts and Sciences, Department of Scientific Computing
Show less  Abstract/Description

Superconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to...
Show moreSuperconductivity is a phenomenon characterized by two hallmark properties, zero electrical resistance and the Meissner effect. These properties give great promise to a new generation of resistance free electronics and powerful superconducting magnets. However this possibility is limited by the extremely low critical temperature the superconductors must operate under, typically close to 0K. The recent discovery of high temperature superconductors has brought the critical temperature closer to room temperature than ever before, making the realization of room temperature superconductivity a possibility. Simulations of superconducting technology and materials will be necessary to usher in the new wave of superconducting electronics. Unfortunately these new materials come with new properties such as effects from multiple electron bands, as is the case for magnesium diboride. Moreover, we must consider that all high temperature superconductors are of a Type II variety, which possess magnetic tubes of flux, known as vortices. These vortices interact with transport currents, creating an electrical resistance through a process known as flux flow. Thankfully this process can be prevented by placing impurities in the superconductor, pinning the vortices, making vortex pinning a necessary aspect of our model. At this time there are no other models or simulations that are aimed at modeling vortex pinning, using impurities, in twoband materials. In this work we modify an existing GinzburgLandau model for twoband superconductors and add the ability to model normal inclusions (impurities) with a new approach which is unique to the twoband model. Simulations in an attempt to model the material magnesium diboride are also presented. In particular simulations of vortex pinning and transport currents are shown using the modified model. The qualitative properties of magnesium diboride are used to validate the model and its simulations. One main goal from the computational end of the simulations is to enlarge the domain size to produce more realistic simulations that avoid boundary pinning effects. In this work we also implement the numerical software library Trilinos in order to parallelize the simulation to enlarge the domain size. Decoupling methods are also investigated with a goal of enlarging the domain size as well. The OneBand GinzburgLandau model serves as a prototypical problem in this endeavor and the methods shown that enlarge the domain size can be easily implemented in the twoband model.
Show less  Date Issued
 2016
 Identifier
 FSU_FA2016_Sockwell_fsu_0071N_13577
 Format
 Thesis
 Title
 Numerical Analysis of Nonlocal Problems.
 Creator

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

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

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

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

Sampson, James P, Osborn, Debra S, BullockYowell, Emily, Lenz, Janet G, Peterson, Gary W, Reardon, Robert C, Dozier, V Casey, Leierer, Stephen J, Hayden, Seth C W, Saunders,...
Show moreSampson, James P, Osborn, Debra S, BullockYowell, Emily, Lenz, Janet G, Peterson, Gary W, Reardon, Robert C, Dozier, V Casey, Leierer, Stephen J, Hayden, Seth C W, Saunders, Denise E
Show less  Abstract/Description

The primary purpose of this paper is to introduce essential elements of cognitive information processing (CIP) theory, research, and practice as they existed at the time of this writing. The introduction that follows describes the nature of career choices and career interventions, and the integration of theory, research, and practice. After the introduction, the paper continues with three main sections that include CIP theory related to vocational behavior, research related to vocational...
Show moreThe primary purpose of this paper is to introduce essential elements of cognitive information processing (CIP) theory, research, and practice as they existed at the time of this writing. The introduction that follows describes the nature of career choices and career interventions, and the integration of theory, research, and practice. After the introduction, the paper continues with three main sections that include CIP theory related to vocational behavior, research related to vocational behavior and career intervention, and CIP theory related to career interventions. The first main section describes CIP theory, including the evolution of CIP theory, the nature of career problems, theoretical assumptions, the pyramid of information processing domains, the CASVE Cycle, and the use of the pyramid and CASVE cycle. The second main section describes CIP theorybased research in examining vocational behavior and establishing evidencebased practice for CIP theorybased career interventions. The third main section describes CIP theory related to career intervention practice, including theoretical assumptions, readiness for career decision making, readiness for career intervention, the differentiated service delivery model, and critical ingredients of career interventions. The paper concludes with regularly updated sources of information on CIP theory.
Show less  Date Issued
 20200625
 Identifier
 FSU_libsubv1_scholarship_submission_1593091156_c171f50a
 Format
 Citation