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Title: Adaptive Observations in a 4D-Var 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 4D-Var 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 4D-Var approach. The use of large...
Show moreIn 4D-Var 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 4D-Var approach. The use of large-scale 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.
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Date Issued: 2008
Identifier: FSU_migr_etd-3765
Format: Thesis

Title: Construction of Delaunay Triangulations on the Sphere: A Parallel Approach.
Creator: Larrea, Veronica G. Vergara, Gunzburger, Max, Meyer-Baese, 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.
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Date Issued: 2011
Identifier: FSU_migr_etd-4557
Format: Thesis

Title: Counterflow Heat Transfer in He II Contained in Porous Media.
Creator: Dalban-Canassy, 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 pre-impregnated 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 pre-impregnated 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 DP10-20 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 non-linear 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.
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Date Issued: 2010
Identifier: FSU_migr_etd-0855
Format: Thesis

Title: Edge-Weighted Centroidal Voronoi Tessellation Based Algorithms for Image Segmentation.
Creator: Wang, Jie, Wang, Xiaoqiang, Wang, Xiaoming, Gunzburger, Max, Peterson, Janet, El-Azab, 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. CVT-based 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 edge-weighted 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. CVT-based 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 edge-weighted 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 edge-weighted 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 within-cluster feature variations. By extensive numerical examples and quantitative evaluations, we demonstrate the excellent performance of LVEWCVT method compared with several state-of-art 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 "Super-pixels" which is in fact a "regularization" of highly inhomogeneous images. We call our algorithm for super-pixels as "VCells" which is the abbreviation of "Voronoi cells". For a wide range of images, VCells is capable to generate roughly uniform sub-regions and meanwhile nicely preserves local image boundaries. The under-segmentation 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 state-of-art algorithms through extensive qualitative and quantitative results on a wide range of complex images. Another important contribution of this work is the "Detecting-Segment-Breaking" (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 "extracting-connected-component" 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.
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Date Issued: 2011
Identifier: FSU_migr_etd-1244
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 hybrid-coordinate 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, Z-Grid, 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.
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Date Issued: 2009
Identifier: FSU_migr_etd-3745
Format: Thesis

Title: A GIS-Based 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 wastewater-treatment 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 wastewater-treatment 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 advection-dispersion 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 site-specific 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, real-world 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 steady-state 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.
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Date Issued: 2010
Identifier: FSU_migr_etd-1851
Format: Thesis

Title: Improvements in Metadynamics Simulations: The Essential Energy Space Random Walk and the Wang-Landau 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 Wang-Landau 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 Wang-Landau 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 Wang-Landau 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.
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Date Issued: 2008
Identifier: FSU_migr_etd-1161
Format: Thesis

Title: Multi-GPU Solutions of Geophysical PDEs with Radial Basis Function-Generated 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 Function-generated Finite Differences (RBF-FD), is drawing attention due to its comparatively low computational complexity versus other RBF methods, high-order 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 Function-generated Finite Differences (RBF-FD), is drawing attention due to its comparatively low computational complexity versus other RBF methods, high-order accuracy (6th to 10th order is common), and parallel nature. Similar to classical Finite Differences (FD), RBF-FD 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 aren-dimensional RBFs rather than one-dimensional polynomials. This allows for generalization ton-dimensional space on completely scattered node layouts. Although RBF-FD 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, RBF-FD must transition from single threaded MATLAB environments to large-scale 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 RBF-FD across multi-GPU 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 RBF-FD as a contender in the arena of distributed HPC numerical methods.
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Date Issued: 2013
Identifier: FSU_migr_etd-8531
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 steady-state peridynamics models. In contrast to the classical partial differential equation models, peridynamic model is an integro-differential 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 steady-state peridynamics models. In contrast to the classical partial differential equation models, peridynamic model is an integro-differential 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 one-dimensional steady-state 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 two-dimensional steady-state peridynamics model. Based on the numerical strategies and results from the one-dimensional peridynamics model, we developed and implemented the corresponding approaches for the two-dimensional 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 finite-dimensional random vector, which is often called the finite-dimensional 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.
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Date Issued: 2012
Identifier: FSU_migr_etd-4753
Format: Thesis

Title: Parallel Grid Generation and Multi-Resolution 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
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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 ultra-high 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 ultra-high 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 multi-resolution modeling framework. A multi-resolution 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 multi-resolution meshes on numerical simulations in the shallow-water context, a standard set of shallow-water 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 multi-resolution 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 shallow-water context. New algorithms and frameworks are described and bench-marked.
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Date Issued: 2011
Identifier: FSU_migr_etd-3743
Format: Thesis

Title: PV-Induced 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 wave-wave 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 wave-wave 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.}
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Date Issued: 2008
Identifier: FSU_migr_etd-1243
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 derivative-free 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 derivative-free 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.
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Date Issued: 2011
Identifier: FSU_migr_etd-0271
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.
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Date Issued: 2008
Identifier: FSU_migr_etd-4272
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 time-dependent 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 high-dimensional (finite element) model. Our results indicate that one may substitute a reduced model in place of a high-dimensional model to solve the forward problem in parameter estimation with many fewer degrees of freedom.
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Date Issued: 2011
Identifier: FSU_migr_etd-5030
Format: Thesis

Title: Reduced Order Modeling Using the Wavelet-Galerkin 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 multi-resolution. This research seeks to combine a wavelet-Galerkin 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 wavelet-Galerkin setting is a viable option in determining a reduced order solution to a differential equation.
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Date Issued: 2013
Identifier: FSU_migr_etd-8663
Format: Thesis

Title: Solution of the Navier-Stokes 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 low-dimensional 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 Reaction-Diffusion 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 low-dimensional 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 Reaction-Diffusion 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 Navier-Stokes (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.
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Date Issued: 2012
Identifier: FSU_migr_etd-5352
Format: Thesis

Title: Sparse-Grid 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
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Abstract/Description: This work focuses on developing and analyzing novel, efficient sparse-grid 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 sparse-grid 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 semi-discrete and fully-discrete solutions. Second, we consider multi-dimensional backward stochastic differential equations driven by a vector of white noise. A sparse-grid scheme are proposed to discretize the target equation in the multi-dimensional time-space domain. In our scheme, the time discretization is conducted by the multi-step scheme. In the multi-dimensional spatial domain, the conditional mathematical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and adaptive hierarchical sparse-grid interpolation. Error estimates are rigorously proved for the proposed fully-discrete scheme for multi-dimensional 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 sparse-grid stochastic collocation method to solve nonlocal diffusion equations with colored noise and Monte-Carlo 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 sparse-grid Bayesian algorithm to improve the efficiency of the classic Bayesian methods. Using sparse-grid 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 brute-force MCMC simulation results.
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Date Issued: 2012
Identifier: FSU_migr_etd-5298
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
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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.
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Date Issued: 2011
Identifier: FSU_migr_etd-5250
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.
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Date Issued: 2008
Identifier: FSU_migr_etd-0843
Format: Thesis

Numerical analysis (15)	+ -
Computer science (2)	+ -
Geophysics (1)	+ -
Philosophy (1)	+ -
Seismology (1)	+ -

Florida State University (19)	+ -
Department of Scientific Computing (17)	+ -
Gunzburger, Max (10)	+ -
Erlebacher, Gordon (8)	+ -
Ye, Ming (6)	+ -

Burkardt, John (4)	+ -
Wang, Xiaoqiang (4)	+ -
Shanbhag, Sachin (3)	+ -
Wang, Xiaoming (3)	+ -
Ringler, Todd (2)	+ -
Womeldorff, Geoffrey A. (2)	+ -
Azoulay, Ariel (1)	+ -
Bollig, Evan F. (1)	+ -
Chen, Xi (1)	+ -
Christopherr.Harden (1)	+ -
Clayson, Carol Anne (1)	+ -
Cunningham, Philip (1)	+ -
Dalban-Canassy, Matthieu (1)	+ -
Department of Mechanical Engineering (1)	+ -
Dewar, William K. (1)	+ -
El-Azab, Anter (1)	+ -
Englander, Ongi (1)	+ -
Flyer, Natasha (1)	+ -
Forinash, Nick (1)	+ -
Gallivan, Kyle (1)	+ -
Gunzburger, Max D. (1)	+ -
Hossen, Md. Jakir (1)	+ -
Jacobsen, Douglas (1)	+ -
Jacobsen, Douglas W. (Douglas William) (1)	+ -

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