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University of Colorado
1.
Sturdevant, Benjamin.
Fully Kinetic Ion Models for Magnetized Plasma Simulations.
Degree: PhD, Applied Mathematics, 2016, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/82 
► This thesis focuses on the development of simulation models, based on fully resolving the gyro-motion of ions with the Lorentz force equations of motion, for…
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▼ This thesis focuses on the development of simulation models, based on fully resolving the gyro-motion of ions with the Lorentz force equations of motion, for studying low-frequency phenomena in well-magnetized plasma systems. Such models, known as fully kinetic ion models, offer formal simplicity over higher order gyrokinetic ion models and may provide an important validation tool or replacement for gyrokinetic ion models in applications where the gyrokinetic ordering assumptions are in question. Methods for dealing with the added difficulty of resolving the short time scales associated with the ion gyro-motion in fully kinetic ion models are explored with the use of graphics processing units (GPUs) and advanced time integration algorithms, including sub-cycling, orbit averaging and variational integrators. Theoretical work is performed to analyze the effects of the ion Bernstein modes, which are known to cause difficulties in simulations based on fully kinetic ion models. In addition, the first simulation results for the ion temperature gradient driven instability in toroidal geometry using a fully kinetic ion model are presented. Finally, during the course of this work, a method for analyzing the effects of a finite time step size and spatial grid in the delta-f approach to the particle-in-cell method was developed for the first time. This method was applied to an implicit time integration scheme and has revealed some unusual numerical properties related to the delta-f method.
Advisors/Committee Members: Scott Parker, Thomas Manteuffel, Scott Parker, Thomas Manteuffel, Yang Chen.
Subjects/Keywords: gyro-motion; ion; gyrokinetic model; plasma; uniform magnetic field; ITG instability; ITG model; magnetized ion acoustic waves; Applied Mathematics; Plasma and Beam Physics
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APA (6th Edition):
Sturdevant, B. (2016). Fully Kinetic Ion Models for Magnetized Plasma Simulations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/82
Chicago Manual of Style (16th Edition):
Sturdevant, Benjamin. “Fully Kinetic Ion Models for Magnetized Plasma Simulations.” 2016. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/82.
MLA Handbook (7th Edition):
Sturdevant, Benjamin. “Fully Kinetic Ion Models for Magnetized Plasma Simulations.” 2016. Web. 28 Jan 2021.
Vancouver:
Sturdevant B. Fully Kinetic Ion Models for Magnetized Plasma Simulations. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/82.
Council of Science Editors:
Sturdevant B. Fully Kinetic Ion Models for Magnetized Plasma Simulations. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/82
University of Colorado
2.
Young, Patrick McKendree.
Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting.
Degree: PhD, Applied Mathematics, 2010, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/8
► Digital microfluidics is a rapidly growing field wherein droplets are manipulated for use in small-scale applications such as variable focus lenses, display technology, fiber…
(more)
▼ Digital microfluidics is a rapidly growing field wherein droplets are manipulated for use in small-scale applications such as variable focus lenses, display technology, fiber optics, and lab-on-a-chip devices. There has been considerable interest in digital microfluidics and the various methods for liquid actuation by thermal, chemical, and electrical means, where each of the actuation methods make use of the favorable scaling relationship of surface tension forces at the micro scale.
Another increasingly important field is addressing the ever growing need for improved heat transfer techniques in the next generation of electronic devices. As device size decreases and device efficiency increases, high heat flux removal capabilities (100 - 1000 W/cm2) are critical to achieve the lower device operating temperatures necessary to ensure reliably and performance.
In this thesis, we investigate the nature of the forcing that occurs in the transport of liquid drops by electrical means. The effects of system parameters on the force density and its net integral are considered in the case of dielectrophoresis (insulating fluids) and electrowetting-on-dielectric (conductive fluids). Moreover, we explore the effectiveness of a new heat transfer technique called digitized heat transfer (DHT), where droplets are utilized to enhance the removal of heat from electronic devices. Numerical computations of the Nusselt number for these types of flows provide strong evidence of the effectiveness of DHT in comparison to continuous flows.
These two physical phenomena are but two examples that illustrate the growing need for numerical techniques that simply and efficiently handle problems on irregular domains. We present two algorithms appropriate in this environment. The first extends the recently introduced Immersed Boundary Projection Method (IBPM), originally developed for the incompressible Navier-Stokes equations, to elliptic and parabolic problems on irregular domains in a second-order accurate manner. The second algorithm employs a boundary integral approach to the solution of elliptic problems in three-dimensional axisymmetric domains with non-axisymmetric boundary conditions. By using Fourier transforms to reduce the three-dimensional problem to a series of problems defined on the generating curve of the surface, a Nyström discretization employing generalized Gaussian quadratures can be applied to rapidly compute the solution with high accuracy. We demonstrate the high order nature of the discretization. An accelerated technique for computing the kernels of the reduced integral equations is developed for those kernels arising from Laplace's equation, overcoming what was previously the major obstacle in the solution to such problems. We extend this technique to a wide class of kernels, with a particular emphasis on those arising from the Helmholtz equation, and provide strong numerical evidence of the efficiency of this approach. By combining the above approach with the Fast Multipole Method, we develop an…
Advisors/Committee Members: Per-Gunnar Martinsson, Kamran Mohseni, Thomas Manteuffel.
Subjects/Keywords: Dielectrophoresis; Digital Microfluidics; Digitized Heat Transfer; Electrowetting; Immersed Boundary Methods; Integral Equations; Applied Mathematics
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APA (6th Edition):
Young, P. M. (2010). Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/8
Chicago Manual of Style (16th Edition):
Young, Patrick McKendree. “Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting.” 2010. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/8.
MLA Handbook (7th Edition):
Young, Patrick McKendree. “Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting.” 2010. Web. 28 Jan 2021.
Vancouver:
Young PM. Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting. [Internet] [Doctoral dissertation]. University of Colorado; 2010. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/8.
Council of Science Editors:
Young PM. Numerical Techniques for the Solution of Partial Differential and Integral Equations on Irregular Domains with Applications to Problems in Electrowetting. [Doctoral Dissertation]. University of Colorado; 2010. Available from: https://scholar.colorado.edu/appm_gradetds/8
University of Colorado
3.
Glugla, Andrew.
Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies.
Degree: MS, Applied Mathematics, 2011, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/12
► The application of multilevel methods to solving large algebraic systems obtained by discretization of PDEs has seen great success. However, these methods often perform…
(more)
▼ The application of multilevel methods to solving large algebraic systems obtained by discretization of PDEs has seen great success. However, these methods often perform sub-optimally when treating problems with anisotropies. For problems posed over unstructured meshes, optimal automatic multigrid coarsening is not a fully solved problem for the smoothed aggregation multigrid.
The focus of this thesis is on enhancing robustness of the coarsening in the Smoothed Aggregation (SA) multigrid. We focus on improving the standard detection of coupling, on which the coarsening decisions in SA are based. Our approach takes the form of a two-pass test which allows for a more robust local control over the coupling detection, as well as added exibility permitting utilization of new coupling detection measures in a more systematic way.
For isotropic problems, smoothed aggregation coarsening is known to offer very favorable operator complexity, but achieving similar behavior in the presence of anisotropy is more challenging. Special attention is paid to addressing the issue of controlling the complexity of the method. We discuss several existing approaches to curbing coarse-level operator fill-in, and offer generalizations and improvements.
Numerical experiments are provided to demonstrate the performance of the improved coarsening on model examples of anisotropic problems featuring both cases where anisotropies are aligned with the grid, as well as cases where they are not.
Advisors/Committee Members: Marian Brezina, Thomas Manteuffel, Stephen McCormick.
Subjects/Keywords: anisotropies; multigrid coarsening; coupling detection; Applied Mathematics
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APA (6th Edition):
Glugla, A. (2011). Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/12
Chicago Manual of Style (16th Edition):
Glugla, Andrew. “Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies.” 2011. Masters Thesis, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/12.
MLA Handbook (7th Edition):
Glugla, Andrew. “Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies.” 2011. Web. 28 Jan 2021.
Vancouver:
Glugla A. Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies. [Internet] [Masters thesis]. University of Colorado; 2011. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/12.
Council of Science Editors:
Glugla A. Improving Robustness of Smoothed Aggregation Multigrid for Problems with Anisotropies. [Masters Thesis]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/12
University of Colorado
4.
Reeger, Jonah A.
A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods.
Degree: PhD, Applied Mathematics, 2013, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/38
► This thesis explores two unrelated research topics. The first is a numerical study of the fourth Painleve equation, while the second is a characterization…
(more)
▼ This thesis explores two unrelated research topics. The first is a numerical study of the fourth Painleve equation, while the second is a characterization of the stability domains of Adams predictor-corrector methods. First, the six Painleve equations were introduced over a century ago, motivated by theoretical considerations. Over the last several decades these equations and their solutions have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast `pole field solver' (Fornberg and Weideman, J. Comp. Phys. 230 (2011), 5957-5973). This study adapts this numerical method to allow for either extended precision or faster numerical solutions to explore the solution space of the fourth Painleve (PIV) equation. This equation has two free parameters in its coefficients, as well as two free initial conditions. After summarizing key analytical results for PIV , the present study applies this new computational tool to the fundamental domain and a surrounding region of the parameter space. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature. In particular, solutions with the special characteristic of having adjacent pole-free sectors, but with no closed form, are identified. Second, the extent that the stability domain of a numerical method reaches along the imaginary axis indicates the utility of the method for approximating solutions to certain differential equations. This maximum value is called the imaginary stability boundary (ISB). It has previously been shown that exactly half of Adams-Bashforth (AB), Adams-Moulton (AM), and staggered Adams-Bashforth methods have nonzero stability ordinates. In the last chapter of this thesis, two categories of Adams predictor-corrector methods are considered, and it is shown that they have a nonzero ISB when (for a method of order p) p = 1,2, 5,6, 9,10,... for ABp-AMp and p = 3,4, 7,8, 11,12,... in the case of and AB(p-1)-AMp.
Advisors/Committee Members: Bengt Fornberg, Mark Ablowitz, Thomas Manteuffel, Barbara Prinari, Harvey Segur.
Subjects/Keywords: connection formula; Painleve equation; Painleve transcendent; pole field; Applied Mathematics
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APA (6th Edition):
Reeger, J. A. (2013). A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/38
Chicago Manual of Style (16th Edition):
Reeger, Jonah A. “A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/38.
MLA Handbook (7th Edition):
Reeger, Jonah A. “A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods.” 2013. Web. 28 Jan 2021.
Vancouver:
Reeger JA. A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/38.
Council of Science Editors:
Reeger JA. A Computational Study of the Fourth Painleve Equation and a Discussion of Adams Predictor-Corrector Methods. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/38
University of Colorado
5.
Sen, Amrik.
A Tale of Waves and Eddies in a Sea of Rotating Turbulence.
Degree: PhD, Applied Mathematics, 2014, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/46
► In this thesis, we investigate several properties of rotating turbulent flows. First, we ran several computer simulations of rotating turbulent flows and performed statistical…
(more)
▼ In this thesis, we investigate several properties of rotating turbulent flows. First, we ran several computer simulations of rotating turbulent flows and performed statistical analysis of the data produced by an established computational model using Large Eddy Simulations (LES). This enabled us to develop deeper phenomenological understanding of such flows, e.g. the effect of anisotropic injection in the power laws of energy and helicity spectral densities, development of shear in specific rotating flows and evidence of wave-vortex coupling. This served as a motivation for detailed theoretical investigations. Next, we undertook a theoretical study of nonlinear resonant wave interactions to deduce new understanding of rotating flow dynamics. The latter analysis pertained to the highly anisotropic regime of rotating flows. To the best of our knowledge, the application of wave-turbulence theory to asymptotically reduced equations in the limit of rapidly rotating hydrodynamic flows is presented here for the first time and aims to further our understanding of highly anisotropic turbulent flows. A coupled set of equations, known as the wave kinetic equations, for energy and helicity is derived using a novel symmetry argument in the canonical description of the wave field sustained by the flow. A modified wave turbulence schematic is proposed and includes scaling law solutions of the flow invariants that span a hierarchy of slow manifold regions where slow inertial waves are in geostrophic balance with non-linear advection processes. A brief summary of the key findings of this thesis is presented in Table 1.
Advisors/Committee Members: Annick Pouquet, Keith Julien, Pablo D. Mininni, Thomas Manteuffel, Bengt Fornberg.
Subjects/Keywords: Hamiltonian Dynamics; Nonlinear waves; perturbation Theory; Rotating Turbulence; Symmetries and reduction; Applied Mathematics; Geophysics and Seismology; Physics
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APA (6th Edition):
Sen, A. (2014). A Tale of Waves and Eddies in a Sea of Rotating Turbulence. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/46
Chicago Manual of Style (16th Edition):
Sen, Amrik. “A Tale of Waves and Eddies in a Sea of Rotating Turbulence.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/46.
MLA Handbook (7th Edition):
Sen, Amrik. “A Tale of Waves and Eddies in a Sea of Rotating Turbulence.” 2014. Web. 28 Jan 2021.
Vancouver:
Sen A. A Tale of Waves and Eddies in a Sea of Rotating Turbulence. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/46.
Council of Science Editors:
Sen A. A Tale of Waves and Eddies in a Sea of Rotating Turbulence. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/46
University of Colorado
6.
Allen, Jeffery M.
What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow.
Degree: PhD, 2017, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/84
► This research involves a few First-Order System Least Squares (FOSLS) formulations of a nonlinear-Stokes flow model for ice sheets. In Glen's flow law, a…
(more)
▼ This research involves a few First-Order System Least Squares (FOSLS) formulations of a nonlinear-Stokes flow model for ice sheets. In Glen's flow law, a commonly used constitutive equation for ice rheology, the viscosity becomes infinite as the velocity gradients approach zero. This typically occurs near the ice surface or where there is basal sliding. The computational difficulties associated with the infinite viscosity are often overcome by an arbitrary modification of Glen's law that bounds the maximum viscosity. The FOSLS formulations developed in this thesis are designed to overcome this difficulty.
The first FOSLS formulation is just the first-order representation of the standard nonlinear, full-Stokes and is known as the viscosity formulation and suffers from the problem above. To overcome the problem of infinite viscosity, two new formulation exploit the fact that the deviatoric stress, the product of viscosity and strain-rate, approaches zero as the viscosity goes to infinity. Using the deviatoric stress as the basis for a first-order system results in the the basic fluidity system. Augmenting the basic fluidity system with a curl-type equation results in the augmented fluidity system, which is more amenable to the iterative solver, Algebraic MultiGrid (AMG). A Nested Iteration (NI) Newton-FOSLS-AMG approach is used to solve the nonlinear-Stokes problems.
Several test problems from the ISMIP set of benchmarks is examined to test the effectiveness of the various formulations. These test show that the viscosity based method is more expensive and less accurate. The basic fluidity system shows optimal finite-element convergence. However, there is not yet an efficient iterative solver for this type of system and this is the topic of future research. Alternatively, AMG performs better on the augmented fluidity system when using specific scaling. Unfortunately, this scaling results in reduced finite-element convergence.
Advisors/Committee Members: Thomas Manteuffel, Harihar Rajaram, Robert Anderson, Stephen Becker, John Ruge.
Subjects/Keywords: fluid flow; FOSLS; glaciers; ice sheets; multigrid; nonlinear stokes equations; Fashion Design; Mathematics
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APA (6th Edition):
Allen, J. M. (2017). What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/84
Chicago Manual of Style (16th Edition):
Allen, Jeffery M. “What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/84.
MLA Handbook (7th Edition):
Allen, Jeffery M. “What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow.” 2017. Web. 28 Jan 2021.
Vancouver:
Allen JM. What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/84.
Council of Science Editors:
Allen JM. What's Cooler Than Being Cool? Ice-Sheet Models Using a Fluidity-Based FOSLS Approach to Nonlinear-Stokes Flow. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/84
University of Colorado
7.
O'Neill, Ben.
Multigrid Reduction in Time for Nonlinear Parabolic Problems.
Degree: PhD, Applied Mathematics, 2017, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/92
► The need for parallelism in the time dimension is being driven by changes in computer architectures, where recent performance increases are attributed to greater…
(more)
▼ The need for parallelism in the time dimension is being driven by changes in computer architectures, where recent performance increases are attributed to greater concurrency rather than faster clock speeds. Sequential time integration limits parallelism to the spatial domain, introducing a parallel scaling bottleneck. Multigrid Reduction in Time (MGRIT) is an iterative parallel-in-time algorithm that permits temporal concurrency by applying time-integration on a multilevel hierarchy of temporal grids. The overarching goal of this thesis is to maximize the accuracy per computational cost (APCC) of the MGRIT algorithm in the context of both linear and nonlinear parabolic partial differential equations (PDE's). The cost of the MGRIT algorithm is directly proportional to the cost of a time-integration step. For a linear problem with implicit time-stepping, each time step equates to solving one linear system. If an optimal spatial solver is used, the work required for a time-step evaluation is independent of the time-step size. For nonlinear problems, each time integration step involves an iterative nonlinear solve, the cost of which often increases with time-step size. This thesis develops a library of MGRIT optimizations, most importantly an alternate initial guess for the nonlinear time-step solver and delayed spatial coarsening, that reduce the cost of the algorithm for nonlinear parabolic problems. This allows nonlinear problems to be solved with parallel scaling behavior comparable to a corresponding linear problem. An alternative approach towards maximizing the APCC is to increase the accuracy of the method. MGRIT uses multigrid reduction techniques and a multilevel hierarchy of coarse time grids to ``parallelize'' the time dimension. Richardson extrapolation (RE) uses a similar multilevel hierarchy of time-grids to improve the accuracy of those same time-integration methods. In this thesis we develop the RE-MGRIT algorithm, a non-intrusive parallel-in-time algorithm that uses RE with MGRIT to improve the convergence order of the underlying time integration scheme, all with almost no extra cost when compared to the standard MGRIT algorithm. In addition to increasing the convergence order of the time integration scheme, RE can also be used as a means of temporal error estimation. This thesis introduces and tests a Richardson based estimate of the local truncation error. When used in conjunction with an adaptive RE-MGRIT based algorithm, these estimates, available for free at each time-point, allow for automatic error control without the requirement for an additional temporal error estimation procedure.
Advisors/Committee Members: Thomas Manteuffel, John Ruge, Jacob Schroder, Rob Falgout, Xiao-Chuan Cai.
Subjects/Keywords: high performance computing; multigrid; parallel in time; Applied Mathematics; Theory and Algorithms
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APA (6th Edition):
O'Neill, B. (2017). Multigrid Reduction in Time for Nonlinear Parabolic Problems. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/92
Chicago Manual of Style (16th Edition):
O'Neill, Ben. “Multigrid Reduction in Time for Nonlinear Parabolic Problems.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/92.
MLA Handbook (7th Edition):
O'Neill, Ben. “Multigrid Reduction in Time for Nonlinear Parabolic Problems.” 2017. Web. 28 Jan 2021.
Vancouver:
O'Neill B. Multigrid Reduction in Time for Nonlinear Parabolic Problems. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/92.
Council of Science Editors:
O'Neill B. Multigrid Reduction in Time for Nonlinear Parabolic Problems. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/92
University of Colorado
8.
Garcia, Jose Humberto.
Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation.
Degree: PhD, Applied Mathematics, 2014, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/51
► An alternative First Order Least-Squares (FOSLS) Finite Element formulation for the numerical solution of the stationary linear problem posed by the Beta-Plane approximation of…
(more)
▼ An alternative First Order Least-Squares (FOSLS) Finite Element formulation for the numerical solution of the stationary linear problem posed by the Beta-Plane approximation of wind driven ocean circulation is presented. The equations of interest are incompressible Navier-Stokes in a rotational frame of reference. The formulation is considered in all three spatial dimensions, i.e., without the hydrostatic hypothesis most often used in other numerical models based on the hydrostatic primitive equations. In order to validate the computer model, a classical experiment for Beta-Plane approximation of wind driven ocean circulation is utilized. To this extent, the sliced cylinder model introduced by Pedlosky & Greenspan (1967) defines the physical domain. The focus of the validation process is the analysis of the solution for two expected phenomena, the thin Ekman layers induced in the system as well as the east-west asymmetry of the pressure field for the interior circulation.
Advisors/Committee Members: Thomas Manteuffel, Steve McCormick, John Ruge, Frank Bryan, Marian Brezina.
Subjects/Keywords: Beta-Plane; FOSLS; Geophysics; Navier-Stokes; Applied Mathematics; Geophysics and Seismology
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APA (6th Edition):
Garcia, J. H. (2014). Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/51
Chicago Manual of Style (16th Edition):
Garcia, Jose Humberto. “Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/51.
MLA Handbook (7th Edition):
Garcia, Jose Humberto. “Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation.” 2014. Web. 28 Jan 2021.
Vancouver:
Garcia JH. Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/51.
Council of Science Editors:
Garcia JH. Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/51
University of Colorado
9.
Lipinski, Douglas Martin.
Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.
Degree: PhD, Applied Mathematics, 2012, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/56
► Lagrangian coherent structures (LCS) are recently defined structures used to analyze transport in dynamical systems with general time dependence. LCS techniques have seen increasing…
(more)
▼ Lagrangian coherent structures (LCS) are recently defined structures used to analyze transport in dynamical systems with general time dependence. LCS techniques have seen increasing use over the past decade, but several factors have limited their application to highly complex and three-dimensional flows. In this dissertation, I study the computation of LCS in the context of fluid dynamics applications. The primary examples used here are axisymmetric simulations of swimming jellyfish, a three-dimensional ocean current simulation, a three-dimensional hurricane simulation, and various test cases and analytically defined flows. All these flows involve complicated dynamics and fluid transport that can be analyzed using LCS to reveal the ow structures and underlying transport behavior.
The main contribution of this dissertation is the development and application of a class of efficient algorithms for computing LCS in a given velocity field. Large computational time has been a major hurdle to the widespread adoption of LCS techniques, especially in three dimensions. The ridge tracking algorithms presented here take advantage of the definition of LCS as codimension-one manifolds by avoiding computations in parts of the domain away from the LCS surfaces. By detecting and tracking LCS through the domain of interest, the computational order is reduced from O (1/δx
n) to O (1/δx
n-1) in n-dimensional problems. In three dimensions, this algorithm is used to compute the LCS in a warm-core ring in the Gulf of Mexico and a hurricane simulation, revealing a new type of LCS structure in the boundary layers of these geophysical vortices. The transport of these structures is analyzed and found to enhance the potential for diffusive mixing in these ow regions through the generation of small length scales.
Advisors/Committee Members: Kamran Mohseni, Keith Julien, James Meiss, Thomas Manteuffel, Sedat Biringen.
Subjects/Keywords: dynamical systems; fast algorithms; Lagrangian coherent structures; Applied Mathematics
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APA (6th Edition):
Lipinski, D. M. (2012). Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/56
Chicago Manual of Style (16th Edition):
Lipinski, Douglas Martin. “Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.” 2012. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/56.
MLA Handbook (7th Edition):
Lipinski, Douglas Martin. “Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.” 2012. Web. 28 Jan 2021.
Vancouver:
Lipinski DM. Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/56.
Council of Science Editors:
Lipinski DM. Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/56
University of Colorado
10.
Kaslovsky, Daniel N.
Geometric Sparsity in High Dimension.
Degree: PhD, Mathematics, 2012, University of Colorado
URL: https://scholar.colorado.edu/math_gradetds/15
► While typically complex and high-dimensional, modern data sets often have a concise underlying structure. This thesis explores the sparsity inherent in the geometric structure…
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▼ While typically complex and high-dimensional, modern data sets often have a concise underlying structure. This thesis explores the sparsity inherent in the geometric structure of many high-dimensional data sets.
Constructing an efficient parametrization of a large data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach, guided by geometry, consists in recovering a local parametrization (a chart) using the local tangent plane. In practice, the data are noisy and the estimation of a low-dimensional tangent plane in high dimension becomes ill posed. Principal component analysis (PCA) is often the tool of choice, as it returns an optimal basis in the case of noise-free samples from a linear subspace. To process noisy data, PCA must be applied locally, at a scale small enough such that the manifold is approximately linear, but at a scale large enough such that structure may be discerned from noise.
We present an approach that uses the geometry of the data to guide our definition of locality, discovering the optimal balance of this noise-curvature trade-off. Using eigenspace perturbation theory, we study the stability of the subspace estimated by PCA as a function of scale, and bound (with high probability) the angle it forms with the true tangent space. By adaptively selecting the scale that minimizes this bound, our analysis reveals the optimal scale for local tangent plane recovery. Additionally, we are able to accurately and efficiently estimate the curvature of the local neighborhood, and we introduce a geometric uncertainty principle quantifying the limits of noise-curvature perturbation for tangent plane recovery. An algorithm for partitioning a noisy data set is then studied, yielding an appropriate scale for practical tangent plane estimation.
Next, we study the interaction of sparsity, scale, and noise from a signal decomposition perspective. Empirical Mode Decomposition is a time-frequency analysis tool for nonstationary data that adaptively defines modes based on the intrinsic frequency scales of a signal. A novel understanding of the scales at which noise corrupts the otherwise sparse frequency decomposition is presented. The thesis concludes with a discussion of future work, including applications to image processing and the continued development of sparse representation from a geometric perspective.
Advisors/Committee Members: Francois G. Meyer, James H. Curry, Per-Gunnar Martinsson, Gregory Beylkin, Thomas Manteuffel.
Subjects/Keywords: Geometry; High-dimensional data; Noise; Sparsity; Applied Mathematics
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APA ·
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MLA ·
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APA (6th Edition):
Kaslovsky, D. N. (2012). Geometric Sparsity in High Dimension. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/15
Chicago Manual of Style (16th Edition):
Kaslovsky, Daniel N. “Geometric Sparsity in High Dimension.” 2012. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/math_gradetds/15.
MLA Handbook (7th Edition):
Kaslovsky, Daniel N. “Geometric Sparsity in High Dimension.” 2012. Web. 28 Jan 2021.
Vancouver:
Kaslovsky DN. Geometric Sparsity in High Dimension. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/math_gradetds/15.
Council of Science Editors:
Kaslovsky DN. Geometric Sparsity in High Dimension. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/math_gradetds/15
University of Colorado
11.
Monnig, Nathan D.
From Nonlinear Embedding to Graph Distances: A Spectral Perspective.
Degree: PhD, Applied Mathematics, 2015, University of Colorado
URL: https://scholar.colorado.edu/appm_gradetds/64
► In this thesis, we explore applications of spectral graph theory to the analysis of complex datasets and networks. We consider spectral embeddings of general…
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▼ In this thesis, we explore applications of spectral graph theory to the analysis of complex datasets and networks. We consider spectral embeddings of general graphs, as well as data sampled from smooth manifolds in high dimension. We specifically focus on the development of algorithms that require minimal user input. Given the inherent difficulty in parameterizing these types of complex datasets, an ideal algorithm should avoid poorly-defined user-selected parameters.
A significant limitation of nonlinear dimensionality reduction embeddings computed from datasets is the absence of a mechanism to compute the inverse map. We address the problem of computing a stable inverse using radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale.
The definition of metrics to compare networks is an open research area. In this thesis, we define the resistance perturbation distance on the space of weighted connected undirected graphs with the same vertex set. This novel distance quantifies changes in the network connectivity, as measured by the respective effective resistance matrices in each graph. We introduce a fast algorithm to approximate this novel distance in linear time. We demonstrate on simple graph models and real networks that the resistance perturbation distance is able to ignore random fluctuation of edges that are irrelevant to the network connectivity, while remaining sensitive to changes triggered by edges that play a fundamental role in the graph connectivity.
Advisors/Committee Members: Francois G. Meyer, Bengt Fornberg, Jem N. Corcoran, Thomas Manteuffel, Juan Restrepo.
Subjects/Keywords: effective resistance; graph distances; graph theory; nonlinear dimension reduction; radial basis functions; spectral algorithms; Numerical Analysis and Computation; Set Theory
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APA ·
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MLA ·
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APA (6th Edition):
Monnig, N. D. (2015). From Nonlinear Embedding to Graph Distances: A Spectral Perspective. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/64
Chicago Manual of Style (16th Edition):
Monnig, Nathan D. “From Nonlinear Embedding to Graph Distances: A Spectral Perspective.” 2015. Doctoral Dissertation, University of Colorado. Accessed January 28, 2021.
https://scholar.colorado.edu/appm_gradetds/64.
MLA Handbook (7th Edition):
Monnig, Nathan D. “From Nonlinear Embedding to Graph Distances: A Spectral Perspective.” 2015. Web. 28 Jan 2021.
Vancouver:
Monnig ND. From Nonlinear Embedding to Graph Distances: A Spectral Perspective. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Jan 28].
Available from: https://scholar.colorado.edu/appm_gradetds/64.
Council of Science Editors:
Monnig ND. From Nonlinear Embedding to Graph Distances: A Spectral Perspective. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/appm_gradetds/64
. 
Open Access Theses and Dissertations