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Title: Analysis and Approximation of a Two-Band Ginzburg-Landau Model of Superconductivity.
Creator: Chan, Wan-Kan, 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 Ginzburg-Landau model, namely the two-band time-dependent Ginzburg-Landau (2B-TDGL) equations, to model the phenomena of two-band superconductivity. In this work, various mathematical and numerical aspects of the two-dimensional, isothermal, isotropic 2B-TDGL equations in the presence of a time-dependent 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 Ginzburg-Landau model, namely the two-band time-dependent Ginzburg-Landau (2B-TDGL) equations, to model the phenomena of two-band superconductivity. In this work, various mathematical and numerical aspects of the two-dimensional, isothermal, isotropic 2B-TDGL equations in the presence of a time-dependent applied magnetic field and a time-dependent applied current are investigated. A new gauge is proposed to facilitate the inclusion of a time-dependent current into the model. There are three parts in this work. First, the 2B-TDGL model which includes a time-dependent 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 2B-TDGL 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 two-band superconductivity. Some novel behaviors are also identified.
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Date Issued: 2007
Identifier: FSU_migr_etd-3923
Format: Thesis

Title: Analysis of Two Partial Differential Equation Models in Fluid Mechanics: Nonlinear Spectral Eddy-Viscosity Model of Turbulence and Infinite-Prandtl-Number Model of Mantle Convection.
Creator: Saka, Yuki, Gunzburger, Max D., Wang, Xiaoming, El-Azab, Anter, Peterson, Janet, Wang, Xiaoqiang, Department of Mathematics, Florida State University
Abstract/Description: This thesis presents two problems in the mathematical and numerical analysis of partial differential equations modeling fluids. The first is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the flow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the Navier-Stokes equations. In this thesis, we examine the spectral viscosity models...
Show moreThis thesis presents two problems in the mathematical and numerical analysis of partial differential equations modeling fluids. The first is related to modeling of turbulence phenomena. One of the objectives in simulating turbulence is to capture the large scale structures in the flow without explicitly resolving the small scales numerically. This is generally accomplished by adding regularization terms to the Navier-Stokes equations. In this thesis, we examine the spectral viscosity models in which only the high-frequency spectral modes are regularized. The objective is to retain the large-scale dynamics while modeling the turbulent fluctuations accurately. The spectral regularization introduces a host of parameters to the model. In this thesis, we rigorously justify effective choices of parameters. The other problem is related to modeling of the mantle flow in the Earth's interior. We study a model equation derived from the Boussinesq equation where the Prandtl number is taken to infinity. This essentially models the flow under the assumption of a large viscosity limit. The novelty in our problem formulation is that the viscosity depends on the temperature field, which makes the mathematical analysis non-trivial. Compared to the constant viscosity case, variable viscosity introduces a second-order nonlinearity which makes the mathematical question of well-posedness more challenging. Here, we prove this using tools from the regularity theory of parabolic partial differential equations.
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Date Issued: 2007
Identifier: FSU_migr_etd-2108
Format: Thesis

Title: Anova for Parameter Dependent Nonlinear PDEs and Numerical Methods for the Stochastic Stokes Equations.
Creator: Chen, Zheng, Gunzburger, Max, Huﬀer, Fred, Peterson, Janet, Wang, Xiaoqiang, Department of Mathematics, Florida State University
Abstract/Description: This dissertation includes the application of analysis-of-variance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects...
Show moreThis dissertation includes the application of analysis-of-variance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to PDE constrained optimization is considered. The effects of the smoothness of the functional and the nonlinearity in the PDE on the decay of the higher-order ANOVA terms are studied. The concept of effective dimensions is used to determine the accuracy of the ANOVA expansions. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. The effects of several parameter sampling strategies on the accuracy of the surrogate optimization method are also considered; it is found that for this sparse sampling application, the Latin hypercube sampling method has advantages over other well-known sampling methods. Although most of the results are presented and discussed in the context of surrogate optimization problems, they also apply to other settings such as stochastic ensemble methods and reduced-order modeling for nonlinear PDEs. In the second part of the dissertation, we study the numerical analysis of the Stokes equations driven by a stochastic process. The random processes we use are white noise, colored noise and the homogeneous Gaussian process. When the process is white noise, we deal with the singularity of matrix Green's functions in the form of mild solutions with the aid of the theory of distributions. We develop finite element methods to solve the stochastic Stokes equations. In the 2D and 3D cases, we derive error estimates for the approximate solutions. The results of numerical experiments are provided in the 2D case that demonstrate the algorithm and convergence rates. On the other hand, the singularity of the matrix Green's functions necessitates the use of the homogeneous Gaussian process. In the framework of theory of abstract Wiener spaces, the stochastic integrals with respect to the homogeneous Gaussian process can be defined on a larger space than L2 . With some conditions on the density function in the definition of the homogeneous Gaussian process, the matrix Green's functions have well defined integrals. We have studied the probability properties of this kind of integral and simulated discretized colored noise.
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Date Issued: 2007
Identifier: FSU_migr_etd-3851
Format: Thesis

Title: Centroidal Voronoi Tessellations for Mesh Generation: from Uniform to Anisotropic Adaptive Triangulations.
Creator: Nguyen, Hoa V., Gunzburger, Max D., El-Azab, 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 non-uniform) 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 non-uniform) 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, non-uniform meshes result in more efficient calculations. If the solution changes more rapidly in one direction than in others, non-uniform 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 non-uniform 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 CVT-based grid generation. We also compare several methods, including CVT-based 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 high-quality uniform isotropic grids and that the CVT-based grids are at least as good as any of the others. For more general grid generation settings, e.g., non-uniform and/or anistropic grids, such quantitative comparisons are much more difficult, if not impossible, to either make or interpret. This motivates us to develop CVT-based adaptive non-uniform anisotropic mesh refinement in the context of solving the convection-diffusion equation with emphasis on convection-dominated 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 2-dimensional examples. It is robust in detecting layers and efficient in resolving non-physical oscillations in the numerical approximation.
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Date Issued: 2008
Identifier: FSU_migr_etd-2616
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 break-up of a liquid jet in a gas cross-flow, then a body-fitted 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 break-up of a liquid jet in a gas cross-flow, then a body-fitted 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 quasi-cubic interpolation is presented. Quasi-cubic interpolation has been used for local polynomial interpolation on an orthogonal mesh before, but never on a general, non-orthogonal curvilinear mesh. The new (tunnel quasi-cubic) algorithm enables one to find a global piece-wise polynomial interpolation of degree three on an orthogonal mesh, and to find a local polynomial interpolation of degree three on a curvilinear mesh.
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Date Issued: 2010
Identifier: FSU_migr_etd-1724
Format: Thesis

Title: An Optimal Control Problem for a Time-Dependent Ginzburg-Landau Model of Superconductivity.
Creator: Lin, Haomin, Peterson, Janet, Gunzburger, Max, Schwartz, Justin, Wang, Xiaoming, Horne, Rudy, Trenchea, Catalin, Department of Mathematics, Florida State University
Abstract/Description: The motion of vortices in a Type II superconductor destroys the material's superconductivity because it dissipates energy and causes resistance. When a transport current is applied to a clean Type-II superconductor in the mixed state, the vortices will go into motion due to the induced Lorentz force and thus the superconductivity of the material is lost. However, various pinning mechanisms, such as normal inclusions, can inhibit vortex motion and pin the vortices to specific sites. We...
Show moreThe motion of vortices in a Type II superconductor destroys the material's superconductivity because it dissipates energy and causes resistance. When a transport current is applied to a clean Type-II superconductor in the mixed state, the vortices will go into motion due to the induced Lorentz force and thus the superconductivity of the material is lost. However, various pinning mechanisms, such as normal inclusions, can inhibit vortex motion and pin the vortices to specific sites. We demonstrate that the placement of the normal inclusion sites has an important effect on the largest electrical current that can be applied to the superconducting material while all vortices remain stationary. Here, an optimal control problem using a time dependent Ginzburg-Landau model is proposed to seek numerically the optimal locations of the normal inclusion sites. An analysis of this optimal control problem is performed, the existence of an optimal control solution is proved and a sensitivity system is given. We then derive a gradient method to solve this optimal control problem. Numerical simulations are performed and the results are presented and discussed.
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Date Issued: 2008
Identifier: FSU_migr_etd-1334
Format: Thesis

Title: Sparse Approximation and Its Applications.
Creator: Li, Qin, Erlebacher, Gordon, Wang, Xiaoming, Hart, Robert, Peterson, Janet, Sussman, Mark, Gallivan, Kyle A., Department of Mathematics, Florida State University
Abstract/Description: In this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of...
Show moreIn this thesis, we tackle the fundamental problem of how to effectively and reliably calculate sparse solutions to underdetermined systems of equations. This class of problems is found in applied mathematics, electrical engineering, statistics, geophysics, just to name a few. This dissertation concentrates on developing efficient and robust solution algorithms, and applies them in several applications in the field of signal/image processing. The first contribution concerns the development of a new Iterative Shrinkage algorithm based on Surrogate Function, ISSF-K, for finding the best K-term approximation to an image. In this problem, we seek to represent an image with K elements from an overcomplete dictionary. We present a proof that this algorithm converges to a local minimum of the NP hard sparsity constrained optimization problem. In addition, we choose curvelets as the dictionary. The approximation obtained by our approach achieves higher PSNR than that of the best K-term wavelet (Cohen-Daubechies-Fauraue 9-7) approximation. We extends ISSF to the application of Morphological Component Analysis, which leads to the second contribution, a new algorithm MCA-ISSF with an adaptive thresholding strategy. The adaptive MCA-ISSF algorithm approximates the problem from the synthesis approach, and it is the only algorithm that incorporate an adaptive strategy to update its algorithmic parameter. Compared to the existent MCA algorithms, our method is more efficient and is parameter free in the thresdholding update. The third contribution concerns the non-convex optimization problems in Compressive Sensing (CS), which is an important extension of sparse approximation. We propose two new iterative reweighted algorithms based on Alternating Direction Method of Multiplier, IR1-ADM and IR2-ADM, to solve the ell-p,0.
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Date Issued: 2011
Identifier: FSU_migr_etd-1399
Format: Thesis

Title: Sparse Grid Stochastic Collocation Techniques for the Numerical Solution of Partial Differential Equations with Random Input Data.
Creator: Webster, Clayton G. (Clayton Garrett), Gunzburger, Max D., Gallivan, Kyle, Peterson, Janet, Tempone, Raul, Department of Mathematics, Florida State University
Abstract/Description: The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian abscissas. Even in the presence of...
Show moreThe objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms (input data of the model). These techniques consist of a Galerkin approximation in the physical domain and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian abscissas. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. The full tensor product spaces suffer from the curse of dimensionality since the dimension of the approximating space grows exponentially in the number of random variables. When this number is moderately large, we combine the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh equally in the solution; the latter approach is ideal when solving highly anisotropic problems depending on a relatively small number of random variables. We also include a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each problem. These procedures are very effective for the problems under study. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence in the asymptotic regime and algebraic convergence in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples illustrate the theoretical results and compare this approach with several others, including the standard Monte Carlo. For moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy is very efficient and superior to all examined methods. Due to the high cost of effecting each realization of the PDE this work also proposes the use of reduced-order models (ROMs) that assist in minimizing the cost of determining accurate statistical information about outputs from ensembles of realizations. We explore the use of ROMs, that greatly reduce the cost of determining approximate solutions, for determining outputs that depend on solutions of stochastic PDEs. One is then able to cheaply determine much larger ensembles, but this increase in sample size is countered by the lower fidelity of the ROM used to approximate the state. In the contexts of proper orthogonal decomposition-based ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
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Date Issued: 2007
Identifier: FSU_migr_etd-1223
Format: Thesis

Wang, Xiaoming (5)	+ -
Gunzburger, Max (3)	+ -
Gunzburger, Max D. (3)	+ -
Wang, Xiaoqiang (3)	+ -
El-Azab, Anter (2)	+ -

Sussman, Mark (2)	+ -
Chan, Wan-Kan (1)	+ -
Chen, Zheng (1)	+ -
Erlebacher, Gordon (1)	+ -
Ewald, Brian (1)	+ -
Gallivan, Kyle (1)	+ -
Gallivan, Kyle A. (1)	+ -
Hart, Robert (1)	+ -
Horne, Rudy (1)	+ -
Huﬀer, Fred (1)	+ -
Kopriva, David (1)	+ -
Li, Qin (1)	+ -
Lin, Haomin (1)	+ -
Manousakis, Efstratios (1)	+ -
Nguyen, Hoa V. (1)	+ -
Roper, Michael (1)	+ -
Saka, Yuki (1)	+ -
Schwartz, Justin (1)	+ -
Simakhina, Svetlana (1)	+ -
Tempone, Raul (1)	+ -
Trenchea, Catalin (1)	+ -
Webster, Clayton G. (Clayton Garrett) (1)	+ -

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