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 Title
 Adaptive Observations in a 4DVar Framework Applied to the Nonlinear Burgers Equation Model.
 Creator

Hossen, Md. Jakir, Navon, Ionel Michael, Peterson, Janet, Erlebacher, Gordon, Department of Scientific Computing, Florida State University
 Abstract/Description

In 4DVar data assimilation for geophysical models, the goal is to reduce the lack of fit between model and observations (strong constraint approach assuming perfect model). In the last two decades four dimensional variational technique has been extensively used in the numerical weather prediction due to the fact that time distributed observations are assimilated to obtain a better initial condition thus leading to more accurate forecasts using the above 4DVar approach. The use of large...
Show moreIn 4DVar data assimilation for geophysical models, the goal is to reduce the lack of fit between model and observations (strong constraint approach assuming perfect model). In the last two decades four dimensional variational technique has been extensively used in the numerical weather prediction due to the fact that time distributed observations are assimilated to obtain a better initial condition thus leading to more accurate forecasts using the above 4DVar approach. The use of largescale unconstrained minimization routines to minimize a cost functional measuring lack of fit between observations and model forecast requires availability of the gradient of the cost functional with respect to the control variables. Nonlinear Burgers equation model is used as numerical forecast model. First order adjoint model can be used to find the gradient of the cost functional. The use of targeted observations supplementing routine observations contributes to the reduction of the forecast analysis error and can provide improved forecast of weather events of critical societal impact, for instance, hurricanes, tornadoes, sharp fronts etc. The optimal space and time locations of the adaptive observations can be determined by using a singular vector approach. In our work we use both adjoint sensitivity and sensitivity to observation approaches to identify the optimal space and time locations for targeted observations at future time aimed at providing an improved forecast. Both approaches are compared in this work and some conclusions are outlined.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd3765
 Format
 Thesis
 Title
 Analysis and Approximation of a TwoBand GinzburgLandau Model of Superconductivity.
 Creator

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

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

Saka, Yuki, Gunzburger, Max D., Wang, Xiaoming, ElAzab, 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 NavierStokes 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 NavierStokes equations. In this thesis, we examine the spectral viscosity models in which only the highfrequency spectral modes are regularized. The objective is to retain the largescale 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 nontrivial. Compared to the constant viscosity case, variable viscosity introduces a secondorder nonlinearity which makes the mathematical question of wellposedness more challenging. Here, we prove this using tools from the regularity theory of parabolic partial differential equations.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd2108
 Format
 Thesis
 Title
 Centroidal Voronoi Tessellations for Mesh Generation: from Uniform to Anisotropic Adaptive Triangulations.
 Creator

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

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

Larrea, Veronica G. Vergara, Gunzburger, Max, MeyerBaese, Anke, Peterson, Janet, Wilgenbusch, Jim, Department of Scientific Computing, Florida State University
 Abstract/Description

This thesis explores possible improvements in the construction of Delaunay Triangulations on the Sphere by designing and implementing a parallel alternative to the software package STRIPACK. First, it gives an introduction to Delaunay Triangulations on the plane and presents current methods available for their construction. Then, these concepts are mapped to the spherical case: Spherical Delaunay Triangulation (SDT). To provide a better understanding of the design choices, this document...
Show moreThis thesis explores possible improvements in the construction of Delaunay Triangulations on the Sphere by designing and implementing a parallel alternative to the software package STRIPACK. First, it gives an introduction to Delaunay Triangulations on the plane and presents current methods available for their construction. Then, these concepts are mapped to the spherical case: Spherical Delaunay Triangulation (SDT). To provide a better understanding of the design choices, this document includes a brief overview of parallel programming, that is followed by the details of the implementation of the SDT generation code. In addition, it provides examples of resulting SDTs as well as benchmarks to analyze its performance. This project was inspired by the concepts presented in Robert Renka's work and was implemented in C++ using MPI.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd4557
 Format
 Thesis
 Title
 Counterflow Heat Transfer in He II Contained in Porous Media.
 Creator

DalbanCanassy, Matthieu, Sciver, Steven W. Van, Peterson, Janet, Luongo, Cesar, Ordonez, Juan, Englander, Ongi, Department of Mechanical Engineering, Florida State University
 Abstract/Description

This dissertation presents a study of steady He II (superfluid helium) counter flow heat transfer in porous media. Porous insulation were suggested as potential alternatives to conventional fully impregnated insulations in superconducting magnet technology. Superconducting magnets are usually cooled with He II. Use of porous insulation requires thus a good knowledge of the behavior of He II within porous materials, when set in motion or exposed to a heat source. The present work was focused...
Show moreThis dissertation presents a study of steady He II (superfluid helium) counter flow heat transfer in porous media. Porous insulation were suggested as potential alternatives to conventional fully impregnated insulations in superconducting magnet technology. Superconducting magnets are usually cooled with He II. Use of porous insulation requires thus a good knowledge of the behavior of He II within porous materials, when set in motion or exposed to a heat source. The present work was focused on the design of an apparatus capable of performing both steady and transient counterflow measurements in He II saturating a porous material with a geometry similar to potential candidate porous insulations. Those will most likely be composed of tapes of preimpregnated woven ceramic fibers, forming a highly anisotropic compound, with a wide pore size distribution. The samples were provided by Composite Technology Development Inc. and are circular pellets (3.08 mm thick and 28.58 mm in diameter) of 20 compressed layers of preimpregnated woven magnet insulation. The porous material was carefully characterized prior to experimental runs in He II. The samples exhibit a porosity and a permeability of respectively 20+1% and 0.95x10^14 m^2 for water measurements. The woven fiber rovings, composing the insulation, were found to be 0.04 mm^2 of average cross sectional area with fibers of average diameter of 10.6 micron. The He II experimental apparatus is composed of a vacuum insulated open channel whose top extremity is closed to a Minco heater. The temperature differences and pressure drops across the porous plug were measured by two Lakeshore barechip Cernox 1050BC thermometers and a Validyne DP1020 differential pressure sensor. Applied heat fluxes ranged up to 0.5 kW/m^2 of sample cross section. Steady temperature differences, up to 570 mK, and pressure drops, up to 1800 Pa (limit of the sensor), measurements were performed at bath temperatures ranging from 1.6 to 2.1 K. In the low heat flux regime, the permeability data corroborate room temperature measurements. In the high heat flux regime however, we show evidence of the failure of previous models based on the inclusion of the tortuosity in the turbulent equation. We propose to include a constriction factor denoting an average maximum change in cross section in the heat path in addition to the increased path length denoted by the tortuosity. In the turbulent regime, this constriction factor is predominant as it enters in the model with a cubic power. Measurements of the critical characteristics, corresponding to the point of transition from the laminar regime, where Darcy law is applicable to the nonlinear regime, where the heat flux adopts its characteristic cubic relationship, corresponding to the appearance of turbulence within He II are also reported. We obtained critical heat fluxes ranging from 20 to 70 W/m^2, Reynolds numbers of 0.5 to 4 and normal fluid velocities from 0.5 to 2.5 mm/s, varying with bath temperature. To confirm the room temperature measurements of permeability, we also conducted a forced flow experiment. Unfortunately, the flow range covered is outside of the laminar regime and does not permit an accurate estimation of the permeability. The results are however favorably comparable to earlier data recorded in the turbulent regime in similar flow conditions but with very different materials.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd0855
 Format
 Thesis
 Title
 EdgeWeighted Centroidal Voronoi Tessellation Based Algorithms for Image Segmentation.
 Creator

Wang, Jie, Wang, Xiaoqiang, Wang, Xiaoming, Gunzburger, Max, Peterson, Janet, ElAzab, Anter, Department of Scientific Computing, Florida State University
 Abstract/Description

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations whose generators are also the centers of mass (centroids) of the Voronoi regions with respect to a given density function. CVTbased algorithms have been proved very useful in the context of image processing. However when dealing with the image segmentation problems, classic CVT algorithms are sensitive to noise. In order to overcome this limitation, we develop an edgeweighted centroidal Voronoi Tessellation (EWCVT)...
Show moreCentroidal Voronoi tessellations (CVTs) are special Voronoi tessellations whose generators are also the centers of mass (centroids) of the Voronoi regions with respect to a given density function. CVTbased algorithms have been proved very useful in the context of image processing. However when dealing with the image segmentation problems, classic CVT algorithms are sensitive to noise. In order to overcome this limitation, we develop an edgeweighted centroidal Voronoi Tessellation (EWCVT) model by introducing a new energy term related to the boundary length which is called "edge energy". The incorporation of the edge energy is equivalent to add certain form of compactness constraint in the physical space. With this compactness constraint, we can effectively control the smoothness of the clusters' boundaries. We will provide some numerical examples to demonstrate the effectiveness, efficiency, flexibility and robustness of EWCVT. Because of its simplicity and flexibility, we can easily embed other mechanisms with EWCVT to tackle more sophisticated problems. Two models based on EWCVT are developed and discussed. The first one is "local variation and edgeweighted centroidal Voronoi Tessellation" (LVEWCVT) model by encoding the information of local variation of colors. For the classic CVTs or its generalizations (like EWCVT), pixels inside a cluster share the same centroid. Therefore the set of centroids can be viewed as a piecewise constant function over the computational domain. And the resulting segmentation have to be roughly the same with respect to the corresponding centroids. Inspired by this observation, we propose to calculate the centroids for each pixel separately and locally. This scheme greatly improves the algorithms' tolerance of withincluster feature variations. By extensive numerical examples and quantitative evaluations, we demonstrate the excellent performance of LVEWCVT method compared with several stateofart algorithms. LVEWCVT model is especially suitable for detection of inhomogeneous targets with distinct color distributions and textures. Based on EWCVT, we build another model for "Superpixels" which is in fact a "regularization" of highly inhomogeneous images. We call our algorithm for superpixels as "VCells" which is the abbreviation of "Voronoi cells". For a wide range of images, VCells is capable to generate roughly uniform subregions and meanwhile nicely preserves local image boundaries. The undersegmentation error is effectively limited in a controllable manner. Moreover, VCells is very efficient. The computational cost is roughly linear in image size with small constant coefficient. For megapixel sized images, VCells is able to generate very dense superpixels in a matter of seconds. We demonstrate that VCells outperforms several stateofart algorithms through extensive qualitative and quantitative results on a wide range of complex images. Another important contribution of this work is the "DetectingSegmentBreaking" (DSB) algorithm which can be used to guarantee the spatial connectedness of resulting segments generated by CVT based algorithms. Since the metric is usually defined on the color space, the resulting segments by CVT based algorithms are not necessarily spatially connected. For some applications, this feature is useful and conceptually meaningful, e.g., the foreground objects are not spatially connected. But for some other applications, like the superpixel problem, this "good" feature becomes unacceptable. By simple "extractingconnectedcomponent" and "relabeling" schemes, DSB successfully overcomes the above difficulty. Moreover, the computational cost of DSB is roughly linear in image size with a small constant coefficient. From the theoretical perspective, the innovative idea of EWCVT greatly enriches the methodology of CVTs. (The idea of EWCVT has already been used for variational curve smoothing and reconstruction problems.) For applications, this work shows the great power of EWCVT for image segmentation related problems.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1244
 Format
 Thesis
 Title
 Effects of Vertical Mixing Closures on North Atlantic Overflow Simulations.
 Creator

Jacobsen, Douglas, Gunzburger, Max, Erlebacher, Gordon, Peterson, Janet, Department of Scientific Computing, Florida State University
 Abstract/Description

We are exploring the effect of using various vertical mixing closures on resolving the physical process known as overflow. This is when cold dense water overflows from a basin in the ocean. This process is responsible for the majority of the Ocean's dense water transport, and also creates many of the dense water currents that are part of what is known as the Ocean Conveyor Belt. One of the main places this happens is in the North Atlantic, in the Denmark strait and the Faroe Bank Sea Channel....
Show moreWe are exploring the effect of using various vertical mixing closures on resolving the physical process known as overflow. This is when cold dense water overflows from a basin in the ocean. This process is responsible for the majority of the Ocean's dense water transport, and also creates many of the dense water currents that are part of what is known as the Ocean Conveyor Belt. One of the main places this happens is in the North Atlantic, in the Denmark strait and the Faroe Bank Sea Channel. To simulate this process, two ocean models are used, the Parallel Ocean Program (POP) and the hybridcoordinate Parallel Ocean Program (HyPOP). Using these models, differences are observed in three main vertical mixing schemes Constant, Richardson Number, and KPP. Though, not included in this thesis the research also explores three different vertical griding schemes, ZGrid, Sigma Coordinate, and Isopycnal grids. The goal is to attempt to determine which combination gives the most acceptable results for resolving the overflow process. This is motivated by the large role this process plays in the ocean, as well as the difficulty in modeling this process. If an ocean model cannot accurately simulate overflow, then a large portion of the ocean model will be incorrect and one cannot hope to get reasonable results for long simulations out of it.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd3745
 Format
 Thesis
 Title
 A GISBased Model for Estimating Nitrate Fate and Transport from Septic Systems in Surficial Aquifers.
 Creator

Rios, J. Fernando, Ye, Ming, Peterson, Janet, Shanbhag, Sachin, Wilgenbusch, James, Department of Scientific Computing, Florida State University
 Abstract/Description

Estimating groundwater nitrate fate and transport is an important task in water resources and environmental management because excess nitrate loads may have negative impacts on human and environmental health. This work discusses the development of a simplified nitrate transport model and its implementation as a geographic information system (GIS)based screening tool, whose purpose is to estimate nitrate loads to surface water bodies from onsite wastewatertreatment systems (OWTS). Key...
Show moreEstimating groundwater nitrate fate and transport is an important task in water resources and environmental management because excess nitrate loads may have negative impacts on human and environmental health. This work discusses the development of a simplified nitrate transport model and its implementation as a geographic information system (GIS)based screening tool, whose purpose is to estimate nitrate loads to surface water bodies from onsite wastewatertreatment systems (OWTS). Key features of this project are the reduced data demands due to the use of a simplified model, as well as ease of use compared to traditional groundwater flow and transport models, achieved by embedding the model within a GIS. The simplified conceptual model consists of a simplified groundwater flow model in the surficial aquifer, and a simplified transport model that makes use of an analytical solution to the advectiondispersion equation, used for determining nitrate fate and transport. Denitrification is modeled using first order decay in the analytical solution with the decay constant obtained from literature and/or sitespecific data. The groundwater flow model uses readily available topographic data to approximate the hydraulic gradient, which is then used to calculate seepage velocity magnitude and direction. The flow model is evaluated by comparing the results to a previous numerical modeling study of the U.S. Naval Air Station, Jacksonville (NAS) performed by the USGS. The results show that for areas in the vicinity of the NAS, the model is capable of predicting groundwater travel times from a source to a surface water body to within ±20 years of the USGS model, 75% of the time. The transport model uses an analytical solution based on the one by Domenico and Robbins (1985), the results of which are then further processed so that they may be applied to more general, realworld scenarios. The solution, as well as the processing steps are tested using artificially constructed scenarios, each meant to evaluate a certain aspect of the solution. For comparison purposes, each scenario is solved using a well known numerical contaminant transport model. The results show that the analytical solution provides a reasonable approximation to the numerical result. However, it generally underestimates the concentration distribution to varying degrees depending on choice of parameters, especially along the plume centerline. These results are in agreement with previous studies (Srinivasan et al., 2007; West et al., 2007). The adaptation of the analytical solution to more realistic scenarios results in an adequate approximation to the numerically calculated plume, except in areas near the advection front, where the model produces a plume whose shape differs noticeably from the numerical solution. Load calculations are carried out using a mass balance approach where the system is considered to be in the steady state. The steadystate condition allows for a load estimate by subtracting the mass removal rate due to denitrification from the input mass rate. The input mass rate is calculated by taking into account advection and dispersion while the mass removal rate due to denitrification is calculated from the definition of a first order reaction. Comparison with the synthetic scenarios of the transport model shows that for the test cases, when decay rates are low, the model agrees well with the load calculation from the numerical model. As decay rates increase and the plume becomes shorter, the input load is overestimated by about 9% in the test cases and the mass removed due to denitrification is underestimated by 30% in the worst case. These results are likely due to the underestimation of concentration values by the analytical solution of the transport model.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1851
 Format
 Thesis
 Title
 Improvements in Metadynamics Simulations: The Essential Energy Space Random Walk and the WangLandau Recursion.
 Creator

Liu, Yusong, Yang, Wei, Erlebacher, Gordon, Peterson, Janet, Department of Scientific Computing, Florida State University
 Abstract/Description

Metadynamics is a popular tool to explore free energy landscapes and it has been use to elucidate various chemical or biochemical processes. The height of updating Gaussian function is very important for proper free energy convergence to the target free energy surface. Both higher and lower Gaussian heights have advantages and disadvantages, a balance is required. This thesis presents the implementation of the WangLandau recursion scheme in metadynamics simulations to adjust the height of...
Show moreMetadynamics is a popular tool to explore free energy landscapes and it has been use to elucidate various chemical or biochemical processes. The height of updating Gaussian function is very important for proper free energy convergence to the target free energy surface. Both higher and lower Gaussian heights have advantages and disadvantages, a balance is required. This thesis presents the implementation of the WangLandau recursion scheme in metadynamics simulations to adjust the height of the unit Gaussian function. Compared with classical fixed Gaussian heights, this dynamic adjustable method was demonstrated to efficiently yield better converged free energy surfaces. In addition, through combination with the realization of an energy space random walk, the WangLandau recursion scheme can be readily used to deal with the pseudoergodicity problem in molecular dynamic simulations. The use of this scheme is proven to efficiently and robustly obtain a biased free energy function within this thesis.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1161
 Format
 Thesis
 Title
 Level Set and Conservative Level Set Methods on Dynamic Quadrilateral Grids.
 Creator

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

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

Bollig, Evan F., Erlebacher, Gordon, Sussman, Mark, Flyer, Natasha, Slice, Dennis, Ye, Ming, Peterson, Janet, Department of Scientific Computing, Florida State University
 Abstract/Description

Many numerical methods based on Radial Basis Functions (RBFs) are gaining popularity in the geosciences due to their competitive accuracy, functionality on unstructured meshes, and natural extension into higher dimensions. One method in particular, the Radial Basis Functiongenerated Finite Differences (RBFFD), is drawing attention due to its comparatively low computational complexity versus other RBF methods, highorder accuracy (6th to 10th order is common), and parallel nature. Similar to...
Show moreMany numerical methods based on Radial Basis Functions (RBFs) are gaining popularity in the geosciences due to their competitive accuracy, functionality on unstructured meshes, and natural extension into higher dimensions. One method in particular, the Radial Basis Functiongenerated Finite Differences (RBFFD), is drawing attention due to its comparatively low computational complexity versus other RBF methods, highorder accuracy (6th to 10th order is common), and parallel nature. Similar to classical Finite Differences (FD), RBFFD computes weighted differences of stencil node values to approximate derivatives at stencil centers. The method differs from classical FD in that the test functions used to calculate the differentiation weights arendimensional RBFs rather than onedimensional polynomials. This allows for generalization tondimensional space on completely scattered node layouts. Although RBFFD was first proposed nearly a decade ago, it is only now gaining a critical mass to compete against well known competitors in modeling like FD, Finite Volume and Finite Element. To truly contend, RBFFD must transition from single threaded MATLAB environments to largescale parallel architectures. Many HPC systems around the world have made the transition to Graphics Processing Unit (GPU) accelerators as a solution for added parallelism and higher throughput. Some systems offer significantly more GPUs than CPUs. As the problem size,N, grows larger, it behooves us to work on parallel architectures, be it CPUs or GPUs. In addition to demonstrating the ability to scale to hundreds or thousands of compute nodes, this work introduces parallelization strategies that span RBFFD across multiGPU clusters. The stability and accuracy of the parallel implementation is verified through the explicit solution of two PDEs. Additionally, a parallel implementation for implicit solutions is introduced as part of continued research efforts. This work establishes RBFFD as a contender in the arena of distributed HPC numerical methods.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd8531
 Format
 Thesis
 Title
 Numerical Methods for Deterministic and Stochastic Nonlocal Problem in Diffusion and Mechanics.
 Creator

Chen, Xi, Gunzburger, Max, Wang, Xiaoming, Peterson, Janet, Wang, Xiaoqiang, Ye, Ming, Burkardt, John, Department of Scientific Computing, Florida State University
 Abstract/Description

In this dissertation, the recently developed peridynamic nonlocal continuum model for solid mechanics is extensively studied, specifically, the numerical methods for the deterministic and stochastic steadystate peridynamics models. In contrast to the classical partial differential equation models, peridynamic model is an integrodifferential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump...
Show moreIn this dissertation, the recently developed peridynamic nonlocal continuum model for solid mechanics is extensively studied, specifically, the numerical methods for the deterministic and stochastic steadystate peridynamics models. In contrast to the classical partial differential equation models, peridynamic model is an integrodifferential equation that does not involve spatial derivatives of the displacement field. As a result, the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to the fracture problems. This dissentation consists of three major parts. The first part focuses on the onedimensional steadystate peridynamics model. Based on a variational formulation, continuous and discontinuous Galerkin finite element methods are developed for the peridynamic model. Optimal convergence rates for different continuous and discontinuous manufactured solutions are obtained. A strategy for identifying the discontinuities of the solution is developed and implemented. The convergence of peridynamics model to classical elasticity model is studied. Some relevant nonlocal problems are also considered. In the second part, we focus on the twodimensional steadystate peridynamics model. Based on the numerical strategies and results from the onedimensional peridynamics model, we developed and implemented the corresponding approaches for the twodimensional case. Optimal convergence rates for different continuous and discontinuous manufactured solutions are obtained. In the third part, we study the stochastic peridynamics model. We focus on a version of peridynamics model whose forcing terms are described by a finitedimensional random vector, which is often called the finitedimensional noise assumption. Monte Carlo methods, stochastic collocation with full tensor product and sparse grid methods based on this stochastic peridynamics model are implemented and compared.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4753
 Format
 Thesis
 Title
 An Optimal Control Problem for a TimeDependent GinzburgLandau 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 TypeII 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 TypeII 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 GinzburgLandau 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.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1334
 Format
 Thesis
 Title
 Parallel Grid Generation and MultiResolution Methods for Climate Modeling Applications.
 Creator

Jacobsen, Douglas W. (Douglas William), Gunzburger, Max, Nof, Doron, Peterson, Janet, Erlebacher, Gordon, Navon, Michael, Burkardt, John, Ringler, Todd, Department of Scientific...
Show moreJacobsen, Douglas W. (Douglas William), Gunzburger, Max, Nof, Doron, Peterson, Janet, Erlebacher, Gordon, Navon, Michael, Burkardt, John, Ringler, Todd, Department of Scientific Computing, Florida State University
Show less  Abstract/Description

Spherical centroidal Voronoi tessellations (SCVT) are used in many applications in a variety of fields, one being climate modeling. They are a natural choice for spatial discretizations on the surface of the Earth. New modeling techniques have recently been developed that allow the simulation of ocean and atmosphere dynamics on arbitrarily unstructured meshes, including SCVTs. Creating ultrahigh resolution SCVTs can be computationally expensive. A newly developed algorithm couples current...
Show moreSpherical centroidal Voronoi tessellations (SCVT) are used in many applications in a variety of fields, one being climate modeling. They are a natural choice for spatial discretizations on the surface of the Earth. New modeling techniques have recently been developed that allow the simulation of ocean and atmosphere dynamics on arbitrarily unstructured meshes, including SCVTs. Creating ultrahigh resolution SCVTs can be computationally expensive. A newly developed algorithm couples current algorithms for the generation of SCVTs with existing computational geometry techniques to provide the parallel computation of SCVTs and spherical Delaunay triangulations. Using this new algorithm, computing spherical Delaunay triangulations shows a speed up on the order of 4000 over other well known algorithms, when using 42 processors. As mentioned previously, newly developed numerical models allow the simulation of ocean and atmosphere systems on arbitrary Voronoi meshes providing a multiresolution modeling framework. A multiresolution grid allows modelers to provide areas of interest with higher resolution with the hopes of increasing accuracy. However, one method of providing higher resolution lowers the resolution in other areas of the mesh which could potentially increase error. To determine the effect of multiresolution meshes on numerical simulations in the shallowwater context, a standard set of shallowwater test cases are explored using the Model for Prediction Across Scales (MPAS), a new modeling framework jointly developed by the Los Alamos National Laboratory and the National Center for Atmospheric Research. An alternative approach to multiresolution modeling is Adaptive Mesh Refinement (AMR). AMR typically uses information about the simulation to determine optimal locations for degrees of freedom, however standard AMR techniques are not well suited for SCVT meshes. In an effort to solve this issue, a framework is developed to allow AMR simulations on SCVT meshes within MPAS. The resulting research contained in this dissertation ties together a newly developed parallel SCVT generator with a numerical method for use on arbitrary Voronoi meshes. Simulations are performed within the shallowwater context. New algorithms and frameworks are described and benchmarked.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3743
 Format
 Thesis
 Title
 Peridynamic Multiscale Models for the Mechanics of Materials: Constitutive Relations, Upscaling from Atomistic Systems, and Interface Problems.
 Creator

Seleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific...
Show moreSeleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, ElAzab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models,...
Show moreThis dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the statebased peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models, establishing a hierarchy that reveals that some models are special cases of others. Furthermore, this allows for the modulation of the strength of non local interactions, even for a fixed radius of interactions between material points in the peridynamics model. The multiscale aspect of peridynamics is demonstrated through its connections to molecular dynamics. Using higherorder gradient models, it is shown that peridynamics can be viewed as an upscaling of molecular dynamics, preserving the relevant dynamics under appropriate choices of length scales. The statebased peridynamic model is shown to be appropriate for the description of multiscale and multimodel systems. A formulation for nonlocal interface problems involving scalar fields is presented, and derivations of non local transmission conditions are derived. Specializations that describe local, non local, and local/non local transmission conditions are considered. Moreover, the convergence of the non local transmission conditions to their classical local counterparts is shown. In all cases, results are illustrated by numerical experiments.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd0273
 Format
 Thesis
 Title
 PVInduced Forcing of Gravity Waves in a Shallow Water Model.
 Creator

Ward, Marshall, Cunningham, Philip, Peterson, Janet, Clayson, Carol Anne, Dewar, William K., Krishnamurti, Ruby, Program in Geophysical Fluid Dynamics, Florida State University
 Abstract/Description

The influence of a geostrophically balanced or potential vorticity (PV) background flow on gravity wave propagation is examined using a rotating shallow water model. The system is analyzed in the context of a perturbative expansion that focuses on the dynamics of the resonances within the nonlinear terms of the system. The nonlinearity is reconstructed as a wavewave interaction forcing on an otherwise undisturbed linear wavefield. The principal conclusion is that while the PV flow is...
Show moreThe influence of a geostrophically balanced or potential vorticity (PV) background flow on gravity wave propagation is examined using a rotating shallow water model. The system is analyzed in the context of a perturbative expansion that focuses on the dynamics of the resonances within the nonlinear terms of the system. The nonlinearity is reconstructed as a wavewave interaction forcing on an otherwise undisturbed linear wavefield. The principal conclusion is that while the PV flow is generally undisturbed by the gravity wavefield, the gravity wavefield is forced by the geostrophic flow over moderate timescales. We numerically test these results for the interaction between a single geostrophic mode and a gravity wave, followed by propagation of a single gravity mode through a turbulent PV background. We find that the gravity mode energy is scattered into other modes of similar wavelength but different directions of propagation. The rate of dispersion is in agreement with resonant triad theory, where the rate depends primarily on the initial gravity wavenumber and background PV strength. These results are expected to have relevance to the propagation of coherent internal tides in the open ocean.}
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd1243
 Format
 Thesis
 Title
 Quasirandom Optimization.
 Creator

Azoulay, Ariel, Peterson, Janet, Gunzburger, Max, Erlebacher, Gordon, Burkardt, John, Department of Scientific Computing, Florida State University
 Abstract/Description

In this work we apply quasirandom sequences to develop a derivativefree algorithm for approximating the global maximum of a given function. This work is based on previous results which used a single type of quasirandom sequence in a Brute Force approach and in an approach called Localization of Search. In this work we present several methods for computing quasirandom sequences as well as measures for determining their properties. We discuss the shortcomings of the Brute Force and...
Show moreIn this work we apply quasirandom sequences to develop a derivativefree algorithm for approximating the global maximum of a given function. This work is based on previous results which used a single type of quasirandom sequence in a Brute Force approach and in an approach called Localization of Search. In this work we present several methods for computing quasirandom sequences as well as measures for determining their properties. We discuss the shortcomings of the Brute Force and Localization of Search methods and then present modifications which address these issues which culminate in a new algorithm which we call Modified Localization of Search. Our algorithm is applied to a test suite of problems and the results are discussed. Finally we present some comments on code development for our algorithm.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd0271
 Format
 Thesis
 Title
 Realtime Computing with the Parareal Algorithm.
 Creator

Christopherr.Harden, Peterson, Janet, Gunzburger, Max, Van Engelen, Robert, Department of Scientific Computing, Florida State University
 Abstract/Description

This thesis presents and evaluates a particular algorithm used for the real time computations of time dependent ordinary and partial differential equations which employs a parallelization strategy over the temporal domain. We also discuss the coupling of this method with another popular technique used for real time computations, model reduction, which will be shown to provide more gains than either method alone. In particular, we look at reduced order modeling based on proper orthogonal...
Show moreThis thesis presents and evaluates a particular algorithm used for the real time computations of time dependent ordinary and partial differential equations which employs a parallelization strategy over the temporal domain. We also discuss the coupling of this method with another popular technique used for real time computations, model reduction, which will be shown to provide more gains than either method alone. In particular, we look at reduced order modeling based on proper orthogonal decompositions. We present some applications in terms of solving time dependent nonlinear partial diï¬erential equations and solving these equations with a coupled approach of combining model reduction and the parareal algorithm . The performance of this method, both numerically and computationally, is discussed in terms of the gains in speedup and efficiency, and in terms of the scalability of the parallelization of the temporal domain on a larger and larger set of compute nodes or processors.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd4272
 Format
 Thesis
 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
 Reduced Order Modeling Using the WaveletGalerkin Approximation of Differential Equations.
 Creator

Witman, David, Peterson, Janet, Gunzburger, Max, Ye, Ming, Department of Scientific Computing, Florida State University
 Abstract/Description

Over the past few decades an increased interest in reduced order modeling approaches has led to its application in areas such as real time simulations and parameter studies among many others. In the context of this work reduced order modeling seeks to solve differential equations using substantially fewer degrees of freedom compared to a standard approach like the finite element method. The finite element method is a Galerkin method which typically uses piecewise polynomial functions to...
Show moreOver the past few decades an increased interest in reduced order modeling approaches has led to its application in areas such as real time simulations and parameter studies among many others. In the context of this work reduced order modeling seeks to solve differential equations using substantially fewer degrees of freedom compared to a standard approach like the finite element method. The finite element method is a Galerkin method which typically uses piecewise polynomial functions to approximate the solution of a differential equation. Wavelet functions have recently become a relevant topic in the area of computational science due to their attractive properties including differentiability and multiresolution. This research seeks to combine a waveletGalerkin method with a reduced order approach to approximate the solution to a differential equation with a given set of parameters. This work will focus on showing that using a reduced order approach in a waveletGalerkin setting is a viable option in determining a reduced order solution to a differential equation.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd8663
 Format
 Thesis
 Title
 Solution of the NavierStokes Equations by the Finite Element Method Using Reduced Order Modeling.
 Creator

Forinash, Nick, Peterson, Janet, Plewa, Tomasz, Shanbhag, Sachin, Department of Scientific Computing, Florida State University
 Abstract/Description

Reduced Order Models (ROM) provide a lowdimensional alternative form of a system of differential equations. Such a form permits faster computation of solutions. In this paper, Poisson's Equation in two dimensions, the Heat Equation in one dimension, and a Nonlinear ReactionDiffusion equation in one dimension are solved using the Galerkin formulation of the Finite Element Method (FEM) in conjunction with Newton's Method. Reduced Order Modeling (ROM) by Proper Orthogonal Decomposition (POD)...
Show moreReduced Order Models (ROM) provide a lowdimensional alternative form of a system of differential equations. Such a form permits faster computation of solutions. In this paper, Poisson's Equation in two dimensions, the Heat Equation in one dimension, and a Nonlinear ReactionDiffusion equation in one dimension are solved using the Galerkin formulation of the Finite Element Method (FEM) in conjunction with Newton's Method. Reduced Order Modeling (ROM) by Proper Orthogonal Decomposition (POD) is then used to accelerate the solution of successive linear systems required by Newton's Method. This is done to show the viability of the method on a simple problem. The NavierStokes (NS) Equations are introduced and solved by FEM. A ROM using both POD and clustering by Centroidal Voronoi Tesselation (CVT) are then used to solve the NS equations, and the results are compared with the FEM solution. The specific NS problem we consider has inhomogeneous Dirichlet boundary conditions and the treatment of the boundary conditions is explained. The resulting decrease in computation time required for solving the various equations are compared with ROM methods.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5352
 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, ISSFK, for finding the best Kterm 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 Kterm wavelet (CohenDaubechiesFauraue 97) approximation. We extends ISSF to the application of Morphological Component Analysis, which leads to the second contribution, a new algorithm MCAISSF with an adaptive thresholding strategy. The adaptive MCAISSF 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 nonconvex 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, IR1ADM and IR2ADM, to solve the ellp,0.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1399
 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 ClenshawCurtis 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 ClenshawCurtis 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 preasymptotic 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 reducedorder 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 decompositionbased ROMs, we explore these counteracting effects on the accuracy of statistical information about outputs determined from ensembles of solutions.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd1223
 Format
 Thesis
 Title
 SparseGrid Methods for Several Types of Stochastic Differential Equations.
 Creator

Zhang, Guannan, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet, Wang, Xiaoqiang, Ye, Ming, Webster, Clayton, Burkardt, John, Department of Scientific Computing, Florida...
Show moreZhang, Guannan, Gunzburger, Max D., Wang, Xiaoming, Peterson, Janet, Wang, Xiaoqiang, Ye, Ming, Webster, Clayton, Burkardt, John, Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This work focuses on developing and analyzing novel, efficient sparsegrid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification. First, we consider linear parabolic partial differential equations with random diffusion coefficients, forcing term and initial condition. Error analysis for a stochastic collocation method is carried out in a wider range of situations than previous literatures,...
Show moreThis work focuses on developing and analyzing novel, efficient sparsegrid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification. First, we consider linear parabolic partial differential equations with random diffusion coefficients, forcing term and initial condition. Error analysis for a stochastic collocation method is carried out in a wider range of situations than previous literatures, including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both semidiscrete and fullydiscrete solutions. Second, we consider multidimensional backward stochastic differential equations driven by a vector of white noise. A sparsegrid scheme are proposed to discretize the target equation in the multidimensional timespace domain. In our scheme, the time discretization is conducted by the multistep scheme. In the multidimensional spatial domain, the conditional mathematical expectations derived from the original equation are approximated using sparsegrid GaussHermite quadrature rule and adaptive hierarchical sparsegrid interpolation. Error estimates are rigorously proved for the proposed fullydiscrete scheme for multidimensional BSDEs with certain types of simplified generator functions. Third, we investigate the propagation of input uncertainty through nonlocal diffusion models. Since the stochastic local diffusion equations, e.g. heat equations, have already been well studied, we are interested in extending the existing numerical methods to solve nonlocal diffusion problems. In this work, we use sparsegrid stochastic collocation method to solve nonlocal diffusion equations with colored noise and MonteCarlo method to solve the ones with white noise. Our numerical experiments show that the existing methods can achieve the desired accuracy in the nonlocal setting. Moreover, in the white noise case, the nonlocal diffusion operator can reduce the variance of the solution because the nonlocal diffusion operator has "smoothing" effect on the random field. At last, stochastic inverse problem is investigated. We propose sparsegrid Bayesian algorithm to improve the efficiency of the classic Bayesian methods. Using sparsegrid interpolation and integration, we construct a surrogate posterior probability density function and determine an appropriate alternative density which can capture the main features of the true PPDF to improve the simulation efficiency in the framework of indirect sampling. By applying this method to a groundwater flow model, we demonstrate its better accuracy when compared to bruteforce MCMC simulation results.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5298
 Format
 Thesis
 Title
 Spherical Centroidal Voronoi Tessellation Based Unstructured Meshes for Multidomain Multiphysics Applications.
 Creator

Womeldorff, Geoffrey A., Gunzburger, Max, Peterson, Janet, Gallivan, Kyle, Erlebacher, Gordon, Wang, Xiaoqiang, Ringler, Todd, Department of Scientific Computing, Florida State...
Show moreWomeldorff, Geoffrey A., Gunzburger, Max, Peterson, Janet, Gallivan, Kyle, Erlebacher, Gordon, Wang, Xiaoqiang, Ringler, Todd, Department of Scientific Computing, Florida State University
Show less  Abstract/Description

This dissertation presents and investigates ideas for improvement of the creation of quality centroidal voronoi tessellations on the sphere (SCVT) which are to be used for multiphysics, multidomain applications. As an introduction, we discuss grid generation on the sphere in a broad fashion. Next, we discuss the theory of CVTs in general, and specifically on the sphere. Subsequently we consider the iterative processes, such as Lloyd's algorithm, which are used to construct them. Following...
Show moreThis dissertation presents and investigates ideas for improvement of the creation of quality centroidal voronoi tessellations on the sphere (SCVT) which are to be used for multiphysics, multidomain applications. As an introduction, we discuss grid generation on the sphere in a broad fashion. Next, we discuss the theory of CVTs in general, and specifically on the sphere. Subsequently we consider the iterative processes, such as Lloyd's algorithm, which are used to construct them. Following this, we describe a method for density functions via images so that we can shape generator density in an intuitive, yet arbitrary, manner, and then a method by which SCVTs can be easily adapted to conform to arbitrary sets of line segments, or shorelines. Then, we discuss sample meshes, used for various physical and nonphysical applications. Penultimately, we discuss two sample applications, as a proof of concept, where we adapt the Shallow Water Model from Model for Predictions Across Scales (MPAS) to use our grids for a more accurate border, and we also discuss elliptic interface problems both with and without hanging nodes. Finally, we share a few concluding remarks.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd5250
 Format
 Thesis
 Title
 Spherical Centroidal Voronoi Tessellations: Point Generation and Density Functions via Images.
 Creator

Womeldorff, Geoffrey A., Gunzburger, Max, Peterson, Janet, Erlebacher, Gordon, Department of Scientific Computing, Florida State University
 Abstract/Description

This thesis presents and investigates ideas for improvement of the creation of quality centroidal voronoi tessellations on the sphere (SCVT). First, we discuss the theory of CVTs in general, and specifically on the sphere. Subsequently we consider the iterative processes, such as Lloyd's algorithm, which are used to construct them. Following this, we examine and introduce different schemes for creating their input values, known as generators, and compare the effects of these different initial...
Show moreThis thesis presents and investigates ideas for improvement of the creation of quality centroidal voronoi tessellations on the sphere (SCVT). First, we discuss the theory of CVTs in general, and specifically on the sphere. Subsequently we consider the iterative processes, such as Lloyd's algorithm, which are used to construct them. Following this, we examine and introduce different schemes for creating their input values, known as generators, and compare the effects of these different initial points with respect to their ability to converge and the amount of work required to meet a given tolerance goal. In addition, we describe a method for density functions via images so that we can shape generator density in an intuitive manner and then implement this method with examples to demonstrate it's efficacy.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0843
 Format
 Thesis