+publisher:"University of Colorado" +contributor:("Gregory Beylkin")

University of Colorado

1. Yang, Xinshuo. Reduction of Multivariate Mixtures and Its Applications.

Degree: PhD, Applied Mathematics, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/139

► We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation… (more)

▼ We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation with fewer terms, up to user-selected accuracy. Our algorithm employs the well-known pivoted Cholesky decomposition of the Gram matrix constructed using terms of the mixture. Importantly, the multivariate mixtures do not have to be a separated representation of a function and complexity of the algorithm is independent of the number of variables (dimensions). The algorithm requires 𝓞 ≤ ft(r²N)) operations, where N is the initial number of terms in a multivariate mixture and r is the number of selected terms. Due to the condition number of the Gram matrix, the resulting accuracy is limited to about 1/2 digits of the used floating point arithmetic. We also consider two additional reduction algorithms for the same purpose. The first algorithm is based on orthogonalization of the multivariate mixture and have a similar performance as the approach based on Cholesky factorization. The second algorithm yields a better accuracy, but currently in high dimensions is only applicable to multivariate mixtures in a separated representation. We use the reduction algorithm to develop a new adaptive numerical method for solving differential and integral equations in quantum chemistry. We demonstrate the performance of this approach by solving the Hartree-Fock equations in two cases of small molecules. We also describe a number of initial applications of the reduction algorithm to solve partial differential and integral equations and to address several problems in data sciences. For data science applications in high dimensions we consider kernel density estimation (KDE) approach for constructing a probability density function (PDF) of a cloud of points, a far-field kernel summation method and the construction of equivalent sources for non-oscillatory kernels (used in both, computational physics and data science) and, finally, show how to use the reduction algorithm to produce seeds for subdividing a cloud of points into groups. Advisors/Committee Members: Gregory Beylkin.

Subjects/Keywords: multivariate mixtures; reduction algorithms; Hartree-Fock equations; integral equations; far-field summation in high dimensions; kernel density estimation; Numerical Analysis and Computation; Partial Differential Equations

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APA (6^th Edition):

Yang, X. (2018). Reduction of Multivariate Mixtures and Its Applications. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/139

Chicago Manual of Style (16^th Edition):

Yang, Xinshuo. “Reduction of Multivariate Mixtures and Its Applications.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/139.

MLA Handbook (7^th Edition):

Yang, Xinshuo. “Reduction of Multivariate Mixtures and Its Applications.” 2018. Web. 28 Feb 2021.

Vancouver:

Yang X. Reduction of Multivariate Mixtures and Its Applications. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/139.

Council of Science Editors:

Yang X. Reduction of Multivariate Mixtures and Its Applications. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/139

University of Colorado

2. Quirin, Sean Albert. Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms.

Degree: PhD, Electrical, Computer & Energy Engineering, 2012, University of Colorado

URL: https://scholar.colorado.edu/ecen_gradetds/35

► The joint application of tailored optical Point Spread Functions (PSF) and estimation methods is an important tool for designing quantitative imaging and sensing solutions.… (more)

▼ The joint application of tailored optical Point Spread Functions (PSF) and estimation methods is an important tool for designing quantitative imaging and sensing solutions. By enhancing the information transfer encoded by the optical waves into an image, matched post-processing algorithms are able to complete tasks with improved performance relative to conventional designs. In this thesis, new engineered PSF solutions with image processing algorithms are introduced and demonstrated for quantitative imaging using information-efficient signal processing tools and/or optical-efficient experimental implementations. The use of a 3D engineered PSF, the Double-Helix (DH-PSF), is applied as one solution for three-dimensional, super-resolution fluorescence microscopy. The DH-PSF is a tailored PSF which was engineered to have enhanced information transfer for the task of localizing point sources in three dimensions. Both an information- and optical-efficient implementation of the DH-PSF microscope are demonstrated here for the first time. This microscope is applied to image single-molecules and micro-tubules located within a biological sample. A joint imaging/axial-ranging modality is demonstrated for application to quantifying sources of extended transverse and axial extent. The proposed implementation has improved optical-efficiency relative to prior designs due to the use of serialized cycling through select engineered PSFs. This system is demonstrated for passive-ranging, extended Depth-of-Field imaging and digital refocusing of random objects under broadband illumination. Although the serialized engineered PSF solution is an improvement over prior designs for the joint imaging/passive-ranging modality, it requires the use of multiple PSFs - a potentially significant constraint. Therefore an alternative design is proposed, the Single-Helix PSF, where only one engineered PSF is necessary and the chromatic behavior of objects under broadband illumination provides the necessary information transfer. The matched estimation algorithms are introduced along with an optically-efficient experimental system to image and passively estimate the distance to a test object. An engineered PSF solution is proposed for improving the sensitivity of optical wave-front sensing using a Shack-Hartmann Wave-front Sensor (SHWFS). The performance limits of the classical SHWFS design are evaluated and the engineered PSF system design is demonstrated to enhance performance. This system is fabricated and the mechanism for additional information transfer is identified. Advisors/Committee Members: Rafael Piestun, Kelvin Wagner, Gregory Beylkin.

Subjects/Keywords: Image Processing; Nanoscopy; Passive Ranging; Point Spread Function Engineering; Wavefront Sensing; Optics; Remote Sensing

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APA (6^th Edition):

Quirin, S. A. (2012). Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/ecen_gradetds/35

Chicago Manual of Style (16^th Edition):

Quirin, Sean Albert. “Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/ecen_gradetds/35.

MLA Handbook (7^th Edition):

Quirin, Sean Albert. “Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms.” 2012. Web. 28 Feb 2021.

Vancouver:

Quirin SA. Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/ecen_gradetds/35.

Council of Science Editors:

Quirin SA. Quantitative Optical Imaging and Sensing by Joint Design of Point Spread Functions and Estimation Algorithms. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/ecen_gradetds/35

University of Colorado

3. Balducci, Marc. Orbit Uncertainty Propagation with Separated Representations.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/asen_gradetds/236

► In light of recent collisions and an increasing population of objects in Earth orbit, the space situational awareness community has significant motivation to develop novel… (more)

▼ In light of recent collisions and an increasing population of objects in Earth orbit, the space situational awareness community has significant motivation to develop novel and effective methods of predicting the behavior of object states under the presence of uncertainty. Unfortunately, approaches to uncertainty quantification often make simplifying assumptions in order to reduce computation cost. This thesis proposes the method of separated representations (SR) as an efficient and accurate approach to uncertainty quantification. The properties of an orthogonal polynomial basis and a uni-directional least squares regression approach allow for the theoretical computation cost of SR to remain low when compared to Monte Carlo or other surrogate methods. Specifically, SR does not suffer from the curse of dimensionality, where computation cost increases exponentially with respect to input dimension. Benefits of this low computation cost are shown in a series of low Earth orbit test cases, where SR is used to accurately approximate non-Gaussian posterior distribution functions. Here, the dimension of the problem is increased from 6 to 20 without incurring significantly more computation time. Taking advantage of a large input dimension, this research presents a global sensitivity analysis computed via SR, which affords a more nuanced analysis of a previously examined case in the literature. By considering design variables, SR is formulated to perform optimization under uncertainty. A novel method that utilizes a Brent optimizer to create training data at unique times of closest approach is devised and implemented in order to detect low probability collision events. This methodology is leveraged to design an optimal avoidance maneuver, which would be intractable when using traditional Monte Carlo. Lastly, a multi-element algorithm is formulated and presented to estimate solutions that are challenging for unmodified SR. This multi-element SR leads to orders of magnitude in accuracy improvement when considering the ability of unmodified SR to approximate discontinuous, multimodal, or diffuse solutions. Advisors/Committee Members: Brandon A. Jones, Alireza Doostan, Gregory Beylkin, Daniel Scheeres.

Subjects/Keywords: separated representations; uncertainty quantification; earth orbit; avoidance maneuver; multi-element algorithm; Aerospace Engineering; Mathematics

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APA (6^th Edition):

Balducci, M. (2018). Orbit Uncertainty Propagation with Separated Representations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/asen_gradetds/236

Chicago Manual of Style (16^th Edition):

Balducci, Marc. “Orbit Uncertainty Propagation with Separated Representations.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/asen_gradetds/236.

MLA Handbook (7^th Edition):

Balducci, Marc. “Orbit Uncertainty Propagation with Separated Representations.” 2018. Web. 28 Feb 2021.

Vancouver:

Balducci M. Orbit Uncertainty Propagation with Separated Representations. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/asen_gradetds/236.

Council of Science Editors:

Balducci M. Orbit Uncertainty Propagation with Separated Representations. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/asen_gradetds/236

University of Colorado

4. Jones, Brandon Allan. Efficient Models for the Evaluation and Estimation of the Gravity Field.

Degree: PhD, Aerospace Engineering Sciences, 2010, University of Colorado

URL: https://scholar.colorado.edu/asen_gradetds/11

► Current astrodynamics applications require a rapid evaluation of the gravity field and an efficient approach to gravity estimation. The commonly used spherical harmonic model… (more)

▼ Current astrodynamics applications require a rapid evaluation of the gravity field and an efficient approach to gravity estimation. The commonly used spherical harmonic model does not meet either of these needs. To address these issues, this research considers two new gravity representations: the cubed-sphere and the MRQSphere models. Offering a means for rapid evaluation, the cubed-sphere model yields an effectively constant computation time for any degree of the modeled gravity field. Analyzing the model's performance in a series of Monte-Carlo-like tests characterizes its effects on both orbit propagation and determination. When compared to the spherical harmonic gravity model, the cubed-sphere model improves computational efficiency without causing any significant deviation in resulting trajectories. Using this new model in sequential orbit determination improves the computational efficiency of the time update. As a result, the measurement update now dominates the filter execution time for near real-time applications. Since cubed-sphere models of higher degree require only a slight change in computation time, orbit propagation and determination systems may now use this model to improve fidelity without any significant change in cost. To address the gravity estimation problem, combining a new multiresolution technique with nearly optimal quadratures (for the sphere) invariant under the icosahedral group defines the MRQSphere model. This new multiresolution representation allows for gravity estimation via a naturally staged approach to a celestial body with an unknown gravity field, which aids in the design of missions to small bodies. To test the new model's capabilities, this research simulates a mission to an asteroid. Tests include the characterization of a MRQSphere model derived from the asteroid's spherical harmonic model, and the estimation of a model via observations of the gravity potential. For such a simplified scenario, the results indicate that the MRQSphere model meets the estimation accuracy requirements; future work is recommended to fully explore its capabilities. Advisors/Committee Members: George H. Born, Gregory Beylkin, Daniel J. Schneeres.

Subjects/Keywords: asteroid; gravity estimation; gravity modeling; orbit determination; orbit propagation; spherical harmonics; Aerospace Engineering; Astrodynamics

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APA (6^th Edition):

Jones, B. A. (2010). Efficient Models for the Evaluation and Estimation of the Gravity Field. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/asen_gradetds/11

Chicago Manual of Style (16^th Edition):

Jones, Brandon Allan. “Efficient Models for the Evaluation and Estimation of the Gravity Field.” 2010. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/asen_gradetds/11.

MLA Handbook (7^th Edition):

Jones, Brandon Allan. “Efficient Models for the Evaluation and Estimation of the Gravity Field.” 2010. Web. 28 Feb 2021.

Vancouver:

Jones BA. Efficient Models for the Evaluation and Estimation of the Gravity Field. [Internet] [Doctoral dissertation]. University of Colorado; 2010. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/asen_gradetds/11.

Council of Science Editors:

Jones BA. Efficient Models for the Evaluation and Estimation of the Gravity Field. [Doctoral Dissertation]. University of Colorado; 2010. Available from: https://scholar.colorado.edu/asen_gradetds/11

University of Colorado

5. Damle, Anil. Near Optimal Rational Approximations of Large Data Sets.

Degree: MS, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/16

► We introduce a new computationally efficient algorithm for constructing near optimal rational approximations of large data sets. In contrast to wavelet-type approximations often used… (more)

Subjects/Keywords: Approximation by rational functions; Applied Mathematics

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APA (6^th Edition):

Damle, A. (2011). Near Optimal Rational Approximations of Large Data Sets. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/16

Chicago Manual of Style (16^th Edition):

Damle, Anil. “Near Optimal Rational Approximations of Large Data Sets.” 2011. Masters Thesis, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/16.

MLA Handbook (7^th Edition):

Damle, Anil. “Near Optimal Rational Approximations of Large Data Sets.” 2011. Web. 28 Feb 2021.

Vancouver:

Damle A. Near Optimal Rational Approximations of Large Data Sets. [Internet] [Masters thesis]. University of Colorado; 2011. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/16.

Council of Science Editors:

Damle A. Near Optimal Rational Approximations of Large Data Sets. [Masters Thesis]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/16

University of Colorado

6. Nixon, Sean David. Development and Applications of Soliton Perturbation Theory.

Degree: PhD, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/18

► This thesis examines the effects of small perturbation to soliton solutions of the nonlinear Schrödinger (NLS) equation on two fronts: the development of a… (more)

▼ This thesis examines the effects of small perturbation to soliton solutions of the nonlinear Schrödinger (NLS) equation on two fronts: the development of a direct perturbation method for dark solitons, and the application of perturbation theory to the study of nonlinear optical systems including the dynamics of ultra-short pulses in mode-locked lasers. For dark soliton solutions of the NLS equation a direct perturbation method for approximating the influence of perturbations is presented. The problem is broken into an inner region, where core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton which propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated including linear and nonlinear dissipative perturbations. In the study of mode-locking lasers the power-energy saturation (PES) equation is a variant of the nonlinear NLS equation, which incorporates gain and filtering saturated with energy, and loss saturated with power (intensity). Solutions of the PES equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by soliton solutions of the unperturbed NLS equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulses’ amplitude, velocity, position, and phase using integral relations derived from the PES equation. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters. The results from the integral relations are shown to agree with the secularity conditions found in multi-scale perturbation theory. In the normal regime both bright and dark pulses are found. Here the NLS equation does not have bright soliton solutions, and the mode-locked pulse are wide and strongly chirped. For dark pulses there are two interpretations of the PES equation. The existence and stability of mode-locked dark pulses are studied for both cases. Soliton strings are found in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution. Advisors/Committee Members: Mark J. Ablowitz, Keith Julien, Gregory Beylkin.

Subjects/Keywords: Mode-locked lasers; Nonlinear Schrodinger equation; Nonlinear waves; Perturbation theory; Solitons; Applied Mathematics; Optics

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APA (6^th Edition):

Nixon, S. D. (2011). Development and Applications of Soliton Perturbation Theory. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/18

Chicago Manual of Style (16^th Edition):

Nixon, Sean David. “Development and Applications of Soliton Perturbation Theory.” 2011. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/18.

MLA Handbook (7^th Edition):

Nixon, Sean David. “Development and Applications of Soliton Perturbation Theory.” 2011. Web. 28 Feb 2021.

Vancouver:

Nixon SD. Development and Applications of Soliton Perturbation Theory. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/18.

Council of Science Editors:

Nixon SD. Development and Applications of Soliton Perturbation Theory. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/18

University of Colorado

7. Gillman, Adrianna. Fast Direct Solvers for Elliptic Partial Differential Equations.

Degree: PhD, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/20

► The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace's and Helmholtz'… (more)

▼ The dissertation describes fast, robust, and highly accurate numerical methods for solving boundary value problems associated with elliptic PDEs such as Laplace's and Helmholtz' equations, the equations of elasticity, and time-harmonic Maxwell's equation. In many areas of science and engineering, the cost of solving such problems determines what can and cannot be modeled computationally. Elliptic boundary value problems may be solved either via discretization of the PDE (e.g., finite element methods) or by first reformulating the equation as an integral equation, and then discretizing the integral equation. In either case, one is left with the task of solving a system of linear algebraic equations that could be very large. There exist a broad range of schemes with linear complexity for solving these equations (multigrid, preconditioned Krylov methods, etc). Most of these schemes are based on ``iterative'' techniques that build a sequence of approximate solutions that converges to the exact solution. In contrast, the methods described here are ``direct'' in the sense that they construct an approximation to the inverse (or LU/Cholesky factorization) of the coefficient matrix. Such direct solvers tend to be more robust, versatile, and stable than iterative methods, but have until recently been considered prohibitively expensive for large scale problems. The objective of the dissertation is to demonstrate that in important environments it is possible to construct an approximate inverse with linear computational cost. The methods are for a single solve competitive with the best iterative methods, and can be far faster than any previously available methods in situations where the same coefficient matrix is used in a sequence of problems. In addition, a new discretization technique for elliptic boundary value problems is proposed. The idea is to first compute the solution operator of a large collection of small domains. The small domains are chosen such that the operator is easily computed to high accuracy. A global equilibrium equation is then built by equating the fluxes through all internal domain boundaries. The resulting linear system is well-suited to the newly developed fast direct solvers. Advisors/Committee Members: Per-Gunnar Martinsson, Gregory Beylkin, Bradley Alpert.

Subjects/Keywords: Fast methods; Linear algebra; Numerical Analysis; Partial Differential Equations; Applied Mathematics

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APA (6^th Edition):

Gillman, A. (2011). Fast Direct Solvers for Elliptic Partial Differential Equations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/20

Chicago Manual of Style (16^th Edition):

Gillman, Adrianna. “Fast Direct Solvers for Elliptic Partial Differential Equations.” 2011. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/20.

MLA Handbook (7^th Edition):

Gillman, Adrianna. “Fast Direct Solvers for Elliptic Partial Differential Equations.” 2011. Web. 28 Feb 2021.

Vancouver:

Gillman A. Fast Direct Solvers for Elliptic Partial Differential Equations. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/20.

Council of Science Editors:

Gillman A. Fast Direct Solvers for Elliptic Partial Differential Equations. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/20

University of Colorado

8. Mendoza, Cristian Rafael. Rays, Waves, and Separatrices.

Degree: MS, Applied Mathematics, 2015, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/77

► A qualitative study on the ray and wave dynamics of light in optical waveguides with separatrix geometry is presented herein. The thesis attempts to answer… (more)

Subjects/Keywords: Electrical Engineering; Periodically Segmented Waveguides; Ray Analysis; Separatrix; Wave Analysis; Applied Mathematics; Optics; Physics

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APA (6^th Edition):

Mendoza, C. R. (2015). Rays, Waves, and Separatrices. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/77

Chicago Manual of Style (16^th Edition):

Mendoza, Cristian Rafael. “Rays, Waves, and Separatrices.” 2015. Masters Thesis, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/77.

MLA Handbook (7^th Edition):

Mendoza, Cristian Rafael. “Rays, Waves, and Separatrices.” 2015. Web. 28 Feb 2021.

Vancouver:

Mendoza CR. Rays, Waves, and Separatrices. [Internet] [Masters thesis]. University of Colorado; 2015. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/77.

Council of Science Editors:

Mendoza CR. Rays, Waves, and Separatrices. [Masters Thesis]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/appm_gradetds/77

University of Colorado

9. Benzaken, Joseph David. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/108

► In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised… (more)

▼ In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised of two major components: (i) novel discretizations and associated solution strategies for rapid numerical solution over geometric parametrizations of the linear and nonlinear thin-shell equations, and (ii) efficient surrogate modeling techniques and algorithms towards the control of geometric variation. While the methodologies presented are in the setting of structural mechanics, particularly Nitsche's method in the context of linearized membranes, Kirchhoff-Love plates, and Kirchhoff-Love shells, they are applicable to any system of parametric partial differential equations. We present a design space exploration framework that elucidates design parameter sensitivities used to inform initial and early-stage design and a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance. Both of these methodologies rely on surrogate modeling techniques where various designs throughout the design space considered are sampled and used in the construction of approximations to the system response. The design space exploration paradigm enables the visualization of a full system response through the surrogate model approximation. The tolerance allocation algorithm poses a set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. design. The loci of these tolerance hyperrectangles with maximizers attaining the performance constraint represents the boundary to the feasible region of allocatable tolerances. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to a user-specified measure. Examples of these methodologies for problems of various complexities are presented. Advisors/Committee Members: John A. Evans, Stephen Becker, Bengt Fornberg, Gregory Beylkin, Alireza Doostan.

Subjects/Keywords: design space exploration; manifold optimization; parametric partial differential equations; thin shell structures; tolerance allocation protocols; uncertainty quantification; Aerospace Engineering; Applied Mathematics

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APA (6^th Edition):

Benzaken, J. D. (2018). Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/108

Chicago Manual of Style (16^th Edition):

Benzaken, Joseph David. “Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/108.

MLA Handbook (7^th Edition):

Benzaken, Joseph David. “Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.” 2018. Web. 28 Feb 2021.

Vancouver:

Benzaken JD. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/108.

Council of Science Editors:

Benzaken JD. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/108

University of Colorado

10. Fairbanks, Hillary Ruth. Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/114

► Characterizing and incorporating uncertainties when simulating physical phenomena is essential for improving model-based predictions. These uncertainties may stem from a lack of knowledge regarding… (more)

▼ Characterizing and incorporating uncertainties when simulating physical phenomena is essential for improving model-based predictions. These uncertainties may stem from a lack of knowledge regarding the underlying physical processes or from imprecise measurements of quantities that describe properties of the physical system. Uncertainty quantification (UQ) is a tool that seeks to characterize the impact of these uncertainties on solutions of computational models, resulting in improved predictive models. In practice, these uncertainties are either treated as random parameters to inform the statistics of the solution of interest (forward UQ), or their statistics are inferred from noisy observations of the solutions (inverse UQ). For systems exhibiting high-dimensional uncertainty, performing either forward or inverse UQ presents a significant computational challenge, as these methods require a large number forward solves of the high-fidelity model, that is, the model that accurately captures the physics of the problem. For large-scale problems, this may result in the need for a possibly infeasible number of simulations. Prominent methods have been developed to reduce the burdens related to these challenges, including multilevel Monte Carlo (for forward UQ) and low-rank approximations to the posterior covariance (for inverse UQ). However, these methods may still require many forward solves of the high-fidelity model. To reduce the cost of performing UQ on high-dimensional systems, we apply multi-fidelity strategies to both the forward problem, in order to estimate moments of the quantity of interest, and inverse problem, to approximate the posterior covariance. In particular, we formulate multi-fidelity methods that exploit the low-rank structure of the solution of interest and utilize models of lower fidelity (which are computationally cheaper to simulate) than the intended high-fidelity model, in a nonintrusive manner. Doing so results in surrogate models that may have accuracies closer to that of the high-fidelity model, yet have computational costs comparable to that of the low-fidelity models. Theoretical error analysis, cost comparisons, and numerical examples are provided to to show the promise of these novel methods. Advisors/Committee Members: Alireza Doostan, Gregory Beylkin, Stephen Becker, Jem Corcoran, Chris Ketelsen.

Subjects/Keywords: bi-fidelity approximations; low-rank approximations; multi-fidelity approximations; parametric model reduction; uncertainty quantification; Applied Mathematics; Models and Methods

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APA (6^th Edition):

Fairbanks, H. R. (2018). Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/114

Chicago Manual of Style (16^th Edition):

Fairbanks, Hillary Ruth. “Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/114.

MLA Handbook (7^th Edition):

Fairbanks, Hillary Ruth. “Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems.” 2018. Web. 28 Feb 2021.

Vancouver:

Fairbanks HR. Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/114.

Council of Science Editors:

Fairbanks HR. Low-Rank, Multi-Fidelity Methods for Uncertainty Quantification of High-Dimensional Systems. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/114

University of Colorado

11. Benzaken, Joseph D. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.

Degree: PhD, Applied Mathematics, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/137

► In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is… (more)

▼ In this dissertation, we present a methodology for understanding the propagation and control of geometric variation in engineering design and analysis. This work is comprised of two major components: (i) novel discretizations and associated solution strategies for rapid numerical solution over geometric parametrizations of the linear and nonlinear thin-shell equations, and (ii) efficient surrogate modeling techniques and algorithms towards the control of geometric variation. While the methodologies presented are in the setting of structural mechanics, particularly Nitsche's method in the context of linearized membranes, Kirchhoff-Love plates, and Kirchhoff-Love shells, they are applicable to any system of parametric partial differential equations. We present a design space exploration framework that elucidates design parameter sensitivities used to inform initial and early-stage design and a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance. Both of these methodologies rely on surrogate modeling techniques where various designs throughout the design space considered are sampled and used in the construction of approximations to the system response. The design space exploration paradigm enables the visualization of a full system response through the surrogate model approximation. The tolerance allocation algorithm poses a set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. design. The loci of these tolerance hyperrectangles with maximizers attaining the performance constraint represents the boundary to the feasible region of allocatable tolerances. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to a user-specified measure. Examples of these methodologies for problems of various complexities are presented. Advisors/Committee Members: John A. Evans, Stephen Becker, Bengt Fornberg, Alireza Doostan, Gregory Beylkin.

Subjects/Keywords: Design Space Exploration; Manifold Optimization; Parametric Partial Differential Equations; Thin Shell Structures; Tolerance Allocation Protocols; Uncertainty Quantification; Numerical Analysis and Computation; Partial Differential Equations; Structures and Materials

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APA (6^th Edition):

Benzaken, J. D. (2018). Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/137

Chicago Manual of Style (16^th Edition):

Benzaken, Joseph D. “Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/137.

MLA Handbook (7^th Edition):

Benzaken, Joseph D. “Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis.” 2018. Web. 28 Feb 2021.

Vancouver:

Benzaken JD. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/137.

Council of Science Editors:

Benzaken JD. Propagation and Control of Geometric Variation in Engineering Structural Design and Analysis. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/137

University of Colorado

12. Babb, Tracy. Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems.

Degree: PhD, 2019, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/156

► The dissertation concerns numerical methods for approximately solving certain linear partial differential equations. The foundation is a solution methodology for linear elliptic boundary value… (more)

▼ The dissertation concerns numerical methods for approximately solving certain linear partial differential equations. The foundation is a solution methodology for linear elliptic boundary value problems that we call the ``Hierarchical Poincare-Steklov (HPS)'' method. This method is based on a high-order multidomain spectral discretization that is designed to work particularly well in conjunction with nested-dissection type direct solvers. The methods presented apply in any dimension, but their efficiency deteriorates as the dimension increases, and dimensions higher than three are generally not considered. A key competitive advantage of the HPS method is that the linear system that results from discretizing an elliptic PDE is solved using a direct rather than an iterative solver. This solver is closely related to existing nested dissection and multifrontal solvers, and has a similar computational profile that involves a ``build stage'' that is reasonably efficient, and then a ``solve stage'' that is very fast. This makes the method particularly powerful for use in situations where a sequence of linear problems involving the same operator needs to be solved, as happens for instance when solving certain parabolic and hyperbolic PDEs. The use of a direct solver also enables the method to solve many problems that are intractable to iterative solvers, such as Helmholtz problems at intermediate and high frequencies. The HPS methodology was originally published as a solution method for homogeneous elliptic problems, and the core contributions of the dissertation involve the extension of the methodology to more general environments. Specifically, there are four key contributions: 1. An extension of the method to handle non homogeneous elliptic equations that involve forcing terms in the volume of the domain. 2. A generalization of the method to allow the use of refined meshes in order to resolve local singularities. 3. An efficient solver for hyperbolic equations that works by applying the HPS methodology to explicitly build highly accurate approximations to the time evolution operator. This enables the use of very long time steps, and parallel in time implementations. 4. An efficient solver for parabolic problems, where the main idea is to accelerate implicit time-stepping schemes by using the HPS methodology to pre-compute the solution operator involved in the elliptic solve. This work also includes an extension to certain non-linear problems. All techniques presented are analyzed in terms of their complexity. Accuracy and stability are demonstrated via extensive numerical examples. Advisors/Committee Members: Per-Gunnar Martinsson, Daniel Appelo, Adrianna Gillman, Bengt Fornberg, Gregory Beylkin.

Subjects/Keywords: Poincare-steklov; linear partial differential equations; multidomain spectral discretization; Applied Mathematics

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APA (6^th Edition):

Babb, T. (2019). Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/156

Chicago Manual of Style (16^th Edition):

Babb, Tracy. “Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems.” 2019. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/156.

MLA Handbook (7^th Edition):

Babb, Tracy. “Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems.” 2019. Web. 28 Feb 2021.

Vancouver:

Babb T. Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/156.

Council of Science Editors:

Babb T. Accelerated Time-Stepping of Parabolic and Hyperbolic Pdes Via Fast Direct Solvers for Elliptic Problems. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/156

University of Colorado

13. Biagioni, David Joseph. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/29

► This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Green's function for elliptic problems with variable coefficients… (more)

▼ This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Green's function for elliptic problems with variable coefficients in arbitrary dimension. The basis for our approach is the separated representation, which appears as a way of approximating functions of many variables by sums of products of univariate functions. While the differential operator we wish to invert is typically ill-conditioned, its conditioning may be improved by first applying the Green's function for the constant coefficient problem. This function may be computed either numerically or, in some case, analytically in a separated format. The variable coefficient Green's function is then computed using a quadratically convergent iteration on the preconditioned operator, with sparsity maintained via representation in a wavelet basis. Of particular interest is that the method scales linearly in the number of dimensions, a feature that very desirable in high dimensional problems in which the curse of dimensionality must be reckoned with. As a corollary to this work, we described a randomized algorithm for maintaining low separation rank of the functions used in the construction of the Green's function. For certain functions of practical interest, one can avoid the cost of using standard methods such as alternating least squares (ALS) to reduce the separation rank. Instead, terms from the separated representation may be selected using a randomized approach based on matrix skeletonization and the interpolative decomposition. The use of random projections can greatly reduce the cost of rank reduction, as well as calculation of the Frobenius norm and term-wise Gram matrices. In Part II of the thesis, we highlight three practical applications of sparse and separable approximations to the analysis of renewable energy data. In the first application, error estimates gleaned from repeated measurements are incorporated into sparse regression algorithms (LASSO and the Dantzig selector) to minimize the statistical uncertainty of the resulting model. Applied to real biomass data, this approach leads to sparser regression coefficients corresponding to improved accuracy as measured by k-fold cross validation error. In the second application, a regression model based on separated representations is fit to reliability data for cadmium telluride (CdTe) thin-film solar cells. The data is inherently multi-way, and our approach avoids artificial matricization that would typically be performed for use with standard regression algorithms. Two distinct modes of degradation, corresponding to short- and long-term decrease in cell efficiency, are identified. In the third application, some theoretical properties of a popular chemometrics algorithm called orthogonal projections to latent structures (O-PLS) are derived. Advisors/Committee Members: Gregory Beylkin, Alireza Doostan, Peter Graf, Gunnar Martinsson, Keith Julien.

Subjects/Keywords: Curse of dimensionality; Direct Poisson solver; High dimensional partial differential equations; Numerical analysis; Randomized canonical tensor decomposition; Separated representations; Applied Mathematics

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APA (6^th Edition):

Biagioni, D. J. (2012). Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/29

Chicago Manual of Style (16^th Edition):

Biagioni, David Joseph. “Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/29.

MLA Handbook (7^th Edition):

Biagioni, David Joseph. “Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.” 2012. Web. 28 Feb 2021.

Vancouver:

Biagioni DJ. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/29.

Council of Science Editors:

Biagioni DJ. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/29

University of Colorado

14. Reynolds, Matthew Jason. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/37

► This thesis consists of contributions to three topics: algorithms for computing generalized Gaussian quadratures, tomographic imaging algorithms, and reduction algorithms. Our approach is based… (more)

▼ This thesis consists of contributions to three topics: algorithms for computing generalized Gaussian quadratures, tomographic imaging algorithms, and reduction algorithms. Our approach is based on using non-linear approximations of functions. We develop a new algorithm for constructing generalized Gaussian quadratures for exponentials inte- grated against a non-sign-definite weight function. These quadratures integrate band-limited exponentials to a user-defined accuracy. We also introduce a method of computing quadrature weights via l∞ minimization. Second, we develop a new imaging algorithm for X-ray tomography. This algorithm, Polar Quadrature Inversion, uses rational approximations to approximate tomographic projections with a near optimal number of terms for a given accuracy. This rational signal model allows us to augment the measured data by extending the tomographic projection's domain in Fourier space. As the extended data from all the projections fill a disk in the Fourier domain, we use polar quadratures for band-limited exponentials and the Unequally Spaced Fast Fourier Transform to obtain our image. We demonstrate that the resulting images have significantly improved resolution without additional artifacts near sharp transitions. Finally, we develop an extension of existing reduction algorithms for functions of one variable to functions of many variables. By reduction, we understand an approximation (to a user-supplied accuracy) of a linear combination of decaying exponentials by a representation of the same form but with a minimal number of terms. While for functions of one variable there is an underlying theory based on the analysis of functions of one complex variable, no such theory is available for the multivariate case. Our approach is a first step in the development of such theory. We demonstrate our algorithm on two examples of multivariate functions, a suboptimal linear combination of real-valued, decaying exponentials, and that of complex-valued, decaying exponentials. Advisors/Committee Members: Gregory Beylkin, Gunnar Martinsson, Keith Julien, Francois Meyer, Rafael Peistun.

Subjects/Keywords: Applied Mathematics

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APA (6^th Edition):

Reynolds, M. J. (2012). Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/37

Chicago Manual of Style (16^th Edition):

Reynolds, Matthew Jason. “Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/37.

MLA Handbook (7^th Edition):

Reynolds, Matthew Jason. “Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.” 2012. Web. 28 Feb 2021.

Vancouver:

Reynolds MJ. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/37.

Council of Science Editors:

Reynolds MJ. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/37

University of Colorado

15. Lewis, Ryan D. Nonlinear Approximations in Filter Design and Wave Propagation.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/47

► This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become… (more)

▼ This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible. The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods. The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain. Advisors/Committee Members: Gregory Beylkin, Bradley Alpert, Mark Ablowitz, Per-Gunnar Martinsson, Rafael Piestun.

Subjects/Keywords: approximation by Gaussians; digital filter design; optimal rational approximation; Rayleigh-Sommerfeld integral; Applied Mathematics

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APA (6^th Edition):

Lewis, R. D. (2013). Nonlinear Approximations in Filter Design and Wave Propagation. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/47

Chicago Manual of Style (16^th Edition):

Lewis, Ryan D. “Nonlinear Approximations in Filter Design and Wave Propagation.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/47.

MLA Handbook (7^th Edition):

Lewis, Ryan D. “Nonlinear Approximations in Filter Design and Wave Propagation.” 2013. Web. 28 Feb 2021.

Vancouver:

Lewis RD. Nonlinear Approximations in Filter Design and Wave Propagation. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/47.

Council of Science Editors:

Lewis RD. Nonlinear Approximations in Filter Design and Wave Propagation. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/47

University of Colorado

16. Feldhacker, Juliana D. Incorporating Uncertainty into Spacecraft Mission and Trajectory Design.

Degree: PhD, Aerospace Engineering Sciences, 2016, University of Colorado

URL: https://scholar.colorado.edu/asen_gradetds/135

► The complex nature of many astrodynamic systems often leads to high computational costs or degraded accuracy in the analysis and design of spacecraft missions,… (more)

▼ The complex nature of many astrodynamic systems often leads to high computational costs or degraded accuracy in the analysis and design of spacecraft missions, and the incorporation of uncertainty into the trajectory optimization process often becomes intractable. This research applies mathematical modeling techniques to reduce computational cost and improve tractability for design, optimization, uncertainty quantification (UQ) and sensitivity analysis (SA) in astrodynamic systems and develops a method for trajectory optimization under uncertainty (OUU). This thesis demonstrates the use of surrogate regression models and polynomial chaos expansions for the purpose of design and UQ in the complex three-body system. Results are presented for the application of the models to the design of mid-field rendezvous maneuvers for spacecraft in three-body orbits. The models are shown to provide high accuracy with no a priori knowledge on the sample size required for convergence. Additionally, a method is developed for the direct incorporation of system uncertainties into the design process for the purpose of OUU and robust design; these methods are also applied to the rendezvous problem. It is shown that the models can be used for constrained optimization with orders of magnitude fewer samples than is required for a Monte Carlo approach to the same problem. Finally, this research considers an application for which regression models are not well-suited, namely UQ for the kinetic deflection of potentially hazardous asteroids under the assumptions of real asteroid shape models and uncertainties in the impact trajectory and the surface material properties of the asteroid, which produce a non-smooth system response. An alternate set of models is presented that enables analytic computation of the uncertainties in the imparted momentum from impact. Use of these models for a survey of asteroids allows conclusions to be drawn on the effects of an asteroid's shape on the ability to successfully divert the asteroid via kinetic impactor. Advisors/Committee Members: Brandon A. Jones, Alireza Doostan, Daniel Scheeres, Jeffrey Parker, Gregory Beylkin.

Subjects/Keywords: mission design; optimization under uncertainty; robust design; trajectory optimization; uncertainty quantification; Aerospace Engineering

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APA (6^th Edition):

Feldhacker, J. D. (2016). Incorporating Uncertainty into Spacecraft Mission and Trajectory Design. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/asen_gradetds/135

Chicago Manual of Style (16^th Edition):

Feldhacker, Juliana D. “Incorporating Uncertainty into Spacecraft Mission and Trajectory Design.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/asen_gradetds/135.

MLA Handbook (7^th Edition):

Feldhacker, Juliana D. “Incorporating Uncertainty into Spacecraft Mission and Trajectory Design.” 2016. Web. 28 Feb 2021.

Vancouver:

Feldhacker JD. Incorporating Uncertainty into Spacecraft Mission and Trajectory Design. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/asen_gradetds/135.

Council of Science Editors:

Feldhacker JD. Incorporating Uncertainty into Spacecraft Mission and Trajectory Design. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/asen_gradetds/135

University of Colorado

17. Gehly, Steven. Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods.

Degree: PhD, Aerospace Engineering Sciences, 2016, University of Colorado

URL: https://scholar.colorado.edu/asen_gradetds/148

► The use of near Earth space has increased dramatically in the past few decades, and operational satellites are an integral part of modern society. The… (more)

▼ The use of near Earth space has increased dramatically in the past few decades, and operational satellites are an integral part of modern society. The increased presence in space has led to an increase in the amount of orbital debris, which poses a growing threat to current and future space missions. Characterization of the debris environment is crucial to our continued use of high value orbit regimes such as the geosynchronous (GEO) belt. Objects in GEO pose unique challenges, by virtue of being densely spaced and tracked by a limited number of sensors in short observation windows. This research examines the use of a new class of multitarget filters to approach the problem of orbit determination for the large number of objects present. The filters make use of a recently developed mathematical toolbox derived from point process theory known as Finite Set Statistics (FISST). Details of implementing FISST-derived filters are discussed, and a qualitative and quantitative comparison between FISST and traditional multitarget estimators demonstrates the suitability of the new methods for space object estimation. Specific challenges in the areas of sensor allocation and initial orbit determination are addressed in the framework. The sensor allocation scheme makes use of information gain functionals as formulated for FISST to efficiently collect measurements on the full multitarget system. Results from a simulated network of three ground stations tracking a large catalog of geosynchronous objects demonstrate improved performance as compared to simpler, non-information theoretic tasking schemes. Further studies incorporate an initial orbit determination technique to initiate new tracks in the multitarget filter. Together with a sensor allocation scheme designed to search for new targets and maintain knowledge of the existing catalog, the method comprises a solution to the search-detect-track problem. Simulation results for a single sensor case show that the problem can be solved for multiple objects with no a priori information, even in the presence of missed detections and false measurements. Collectively, this research seeks to advance the capabilities of FISST-derived filters for use in the estimation of geosynchronous space objects; additional directions for future research are presented in the conclusion. Advisors/Committee Members: Penina Axelrad, Brandon Jones, Jay McMahon, Nisar Ahmed, Gregory Beylkin.

Subjects/Keywords: Geosynchronous Orbit; Information Gain; Initial Orbit Determination; Multitarget Filtering; Random Finite Sets; Sensor Allocation; Aerospace Engineering

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APA (6^th Edition):

Gehly, S. (2016). Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/asen_gradetds/148

Chicago Manual of Style (16^th Edition):

Gehly, Steven. “Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/asen_gradetds/148.

MLA Handbook (7^th Edition):

Gehly, Steven. “Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods.” 2016. Web. 28 Feb 2021.

Vancouver:

Gehly S. Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/asen_gradetds/148.

Council of Science Editors:

Gehly S. Estimation of Geosynchronous Space Objects Using Finite Set Statistics Filtering Methods. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/asen_gradetds/148

University of Colorado

18. Satkauskas, Ignas V. Numerical Calculus of Probability Density Functions.

Degree: PhD, Applied Mathematics, 2017, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/93

► In this thesis we construct novel functional representations for the Probability Density Functions (PDFs) of random variables and develop efficient and accurate algorithms for… (more)

▼ In this thesis we construct novel functional representations for the Probability Density Functions (PDFs) of random variables and develop efficient and accurate algorithms for computing the PDFs of their sums, products and quotients, again in the same representation. We consider two important cases of random variables: non-negative random variables and random variables taking both positive and negative values. For the first case, we use approximations by decaying exponentials with complex exponents, while for the second case we develop a Gaussian-based multiresolution analysis (GMRA). The need to represent distributions of products and quotients of random variables appear in many areas of theoretical and applied sciences. However, there are currently only limited number of numerical techniques for computing such products and quotients and this thesis presents new numerical methods for this purpose. Current methods for computing the product and quotients typically rely on a Monte Carlo type approach, where the PDFs of the product or quotient are sampled individually and the histogram of the resulting PDF is obtained from computed products or quotients of the individual samples. Although Monte Carlo methods are easy to implement, they suffer from slow convergence and therefore are not well suited for achieving high accuracy. Another method for computing the PDFs of the products and ratios of positive independent random variables relies on the Mellin transform and we describe such methods in greater detail in the thesis. Although mathematically appealing, techniques based on the Mellin transform lack in robust and stable numerical algorithms for computation of the inverse Mellin transform, hence making them not universally applicable. Our novel representations and associated numerical algorithms produce a general framework for computing of PDFs of random variables which we call numerical calculus of PDFs in functional form. The new fast algorithms of this thesis allow user to select computational accuracy; the speed of algorithms only weakly depends on such selection. We demonstrate the performance of new algorithms on multiple examples using accuracies that are well beyond the reach of Monte Carlo based methods. Advisors/Committee Members: Gregory Beylkin, Alireza Doostan, Michael Sprague, Bengt Fornberg, Keith Julien.

Subjects/Keywords: multiresolution analysis; probability density function; product of random variables; Applied Mathematics; Probability

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APA (6^th Edition):

Satkauskas, I. V. (2017). Numerical Calculus of Probability Density Functions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/93

Chicago Manual of Style (16^th Edition):

Satkauskas, Ignas V. “Numerical Calculus of Probability Density Functions.” 2017. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/93.

MLA Handbook (7^th Edition):

Satkauskas, Ignas V. “Numerical Calculus of Probability Density Functions.” 2017. Web. 28 Feb 2021.

Vancouver:

Satkauskas IV. Numerical Calculus of Probability Density Functions. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/93.

Council of Science Editors:

Satkauskas IV. Numerical Calculus of Probability Density Functions. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/93

University of Colorado

19. Yang, Xinshuo. Reduction of Multivariate Mixtures and Its Applications.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/106

► We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation… (more)

▼ We consider a fast deterministic algorithm to identify the "best" linearly independent terms in multivariate mixtures and use them to compute an equivalent representation with fewer terms, up to user-selected accuracy. Our algorithm employs the well-known pivoted Cholesky decomposition of the Gram matrix constructed using terms of the mixture. Importantly, the multivariate mixtures do not have to be a separated representation of a function and complexity of the algorithm is independent of the number of variables (dimensions). The algorithm requires 𝒪(r²N) operations, where N is the initial number of terms in a multivariate mixture and r is the number of selected terms. Due to the condition number of the Gram matrix, the resulting accuracy is limited to about 1/2 digits of the used floating point arithmetic. We also consider two additional reduction algorithms for the same purpose. The first algorithm is based on orthogonalization of the multivariate mixture and have a similar performance as the approach based on Cholesky factorization. The second algorithm yields a better accuracy, but currently in high dimensions is only applicable to multivariate mixtures in a separated representation. We use the reduction algorithm to develop a new adaptive numerical method for solving differential and integral equations in quantum chemistry. We demonstrate the performance of this approach by solving the Hartree-Fock equations in two cases of small molecules. We also describe a number of initial applications of the reduction algorithm to solve partial differential and integral equations and to address several problems in data sciences. For data science applications in high dimensions we consider kernel density estimation (KDE) approach for constructing a probability density function (PDF) of a cloud of points, a far-field kernel summation method and the construction of equivalent sources for non-oscillatory kernels (used in both, computational physics and data science) and, finally, show how to use the reduction algorithm to produce seeds for subdividing a cloud of points into groups. Advisors/Committee Members: Gregory Beylkin, Bengt Fornberg, Zydrunas Gimbutas, Ian Grooms, Per-Gunnar Martinsson.

Subjects/Keywords: far-field summation in high dimensions; hartree-fock equations; integral equations; kernel density estimation; multivariate mixtures; reduction algorithms; Applied Mathematics; Theory and Algorithms

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yang, X. (2018). Reduction of Multivariate Mixtures and Its Applications. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/106

Chicago Manual of Style (16^th Edition):

Yang, Xinshuo. “Reduction of Multivariate Mixtures and Its Applications.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/106.

MLA Handbook (7^th Edition):

Yang, Xinshuo. “Reduction of Multivariate Mixtures and Its Applications.” 2018. Web. 28 Feb 2021.

Vancouver:

Yang X. Reduction of Multivariate Mixtures and Its Applications. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/106.

Council of Science Editors:

Yang X. Reduction of Multivariate Mixtures and Its Applications. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/106

University of Colorado

20. Heavner, Nathan. Building Rank-Revealing Factorizations with Randomization.

Degree: PhD, 2019, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/155

► This thesis describes a set of randomized algorithms for computing rank revealing factorizations of matrices. These algorithms are designed specifically to minimize the amount… (more)

▼ This thesis describes a set of randomized algorithms for computing rank revealing factorizations of matrices. These algorithms are designed specifically to minimize the amount of data movement required, which is essential to high practical performance on modern computing hardware. The work presented builds on existing randomized algorithms for computing low-rank approximations to matrices, but essentially ex- tends the range of applicability of these methods by allowing for the efficient decomposition of matrices of any numerical rank, including full rank matrices. In contrast, existing methods worked well only when the numerical rank was substantially smaller than the dimensions of the matrix. The thesis describes algorithms for computing two of the most popular rank-revealing matrix decom- positions: the column pivoted QR (CPQR) decomposition, and the so called UTV decomposition that factors a given matrix A as A = UTV∗, where U and V have orthonormal columns and T is triangular. For each algorithm, the thesis presents algorithms that are tailored for different computing environments, including multicore shared memory processors, GPUs, distributed memory machines, and matrices that are stored on hard drives (“out of core”). The first chapter of the thesis consists of an introduction that provides context, reviews previous work in the field, and summarizes the key contributions. Beside the introduction, the thesis contains six additional chapters: Chapter 2 introduces a fully blocked algorithm HQRRP for computing a QR factorization with col- umn pivoting. The key to the full blocking of the algorithm lies in using randomized projections to create a low dimensional sketch of the data, where multiple good pivot columns may be cheaply computed. Nu- merical experiments show that HQRRP is several times faster than the classical algorithm for computing a column pivoted QR on a multicore machine, and the acceleration factor increases with the number of cores. Chapter 3 introduces randUTV, a randomized algorithm for computing a rank-revealing factorization of the form A = UTV∗, where U and V are orthogonal and T is upper triangular. RandUTV uses random- ized methods to efficiently build U and V as approximations of the column and row spaces of A. The result is an algorithm that reveals rank nearly as well as the SVD and costs at most as much as a column pivoted QR. Chapter 4 provides optimized implementations for shared and distributed memory architectures. For shared memory, we show that formulating randUTV as an algorithm-by-blocks increases its efficiency in parallel. The fifth chapter implements randUTV on the GPU and augments the algorithm with an over- sampling technique to further increase the low rank approximation properties of the resulting factorization. Chapter 6 implements both randUTV and HQRRP for use with matrices stored out of core. It is shown that reorganizing HQRRP as a left-looking algorithm to reduce the number of writes to the drive is in the tested cases… Advisors/Committee Members: Per-Gunnar Martinsson, Stephen Becker, Gregory Beylkin, Gregorio Quintana-Ortí, Christian Ketelsen.

Subjects/Keywords: linear algebra; matrix factorizations; randomization; rank-revealing factorizations; Applied Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Heavner, N. (2019). Building Rank-Revealing Factorizations with Randomization. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/155

Chicago Manual of Style (16^th Edition):

Heavner, Nathan. “Building Rank-Revealing Factorizations with Randomization.” 2019. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/appm_gradetds/155.

MLA Handbook (7^th Edition):

Heavner, Nathan. “Building Rank-Revealing Factorizations with Randomization.” 2019. Web. 28 Feb 2021.

Vancouver:

Heavner N. Building Rank-Revealing Factorizations with Randomization. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/appm_gradetds/155.

Council of Science Editors:

Heavner N. Building Rank-Revealing Factorizations with Randomization. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/155

University of Colorado

21. 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… (more)

▼ 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 · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th 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 (16^th Edition):

Kaslovsky, Daniel N. “Geometric Sparsity in High Dimension.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/math_gradetds/15.

MLA Handbook (7^th Edition):

Kaslovsky, Daniel N. “Geometric Sparsity in High Dimension.” 2012. Web. 28 Feb 2021.

Vancouver:

Kaslovsky DN. Geometric Sparsity in High Dimension. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 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

22. Nejadmalayeri, Alireza. Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding.

Degree: PhD, Mechanical Engineering, 2012, University of Colorado

URL: https://scholar.colorado.edu/mcen_gradetds/42

► The current work develops a wavelet-based adaptive variable fidelity approach that integrates Wavelet-based Direct Numerical Simulation (WDNS), Coherent Vortex Simulations (CVS), and Stochastic Coherent… (more)

▼ The current work develops a wavelet-based adaptive variable fidelity approach that integrates Wavelet-based Direct Numerical Simulation (WDNS), Coherent Vortex Simulations (CVS), and Stochastic Coherent Adaptive Large Eddy Simulations (SCALES). The proposed methodology employs the notion of spatially and temporarily varying wavelet thresholding combined with hierarchical wavelet-based turbulence modeling. The transition between WDNS, CVS, and SCALES regimes is achieved through two-way physics-based feedback between the modeled SGS dissipation (or other dynamically important physical quantity) and the spatial resolution. The feedback is based on spatio-temporal variation of the wavelet threshold, where the thresholding level is adjusted on the fly depending on the deviation of local significant SGS dissipation from the user prescribed level. This strategy overcomes a major limitation for all previously existing wavelet-based multi-resolution schemes: the global thresholding criterion, which does not fully utilize the spatial/temporal intermittency of the turbulent flow. Hence, the aforementioned concept of physics-based spatially variable thresholding in the context of wavelet-based numerical techniques for solving PDEs is established. The procedure consists of tracking the wavelet thresholding-factor within a Lagrangian frame by exploiting a Lagrangian Path-Line Diffusive Averaging approach based on either linear averaging along characteristics or direct solution of the evolution equation. This innovative technique represents a framework of continuously variable fidelity wavelet-based space/time/model-form adaptive multiscale methodology. This methodology has been tested and has provided very promising results on a benchmark with time-varying user prescribed level of SGS dissipation. In addition, a longtime effort to develop a novel parallel adaptive wavelet collocation method for numerical solution of PDEs has been completed during the course of the current work. The scalability and speedup studies of this powerful parallel PDE solver are performed on various architectures. Furthermore, Reynolds scaling of active spatial modes of both CVS and SCALES of linearly forced homogeneous turbulence at high Reynolds numbers is investigated for the first time. This computational complexity study, by demonstrating very promising slope for Reynolds scaling of SCALES even at constant level of fidelity for SGS dissipation, proves the argument that SCALES as a dynamically adaptive turbulence modeling technique, can offer a plethora of flexibilities in hierarchical multiscale space/time adaptive variable fidelity simulations of high Reynolds number turbulent flows. Advisors/Committee Members: Oleg V. Vasilyev, Gregory Beylkin, Henry Tufo, Kurt Karl Maute, Alireza Doostan.

Subjects/Keywords: 3D homogeneous turbulence; hierarchical multiscale space/time adaptive variable fidelity; intermittency and fractal-dimension in turbulence; Lagrangian spatially variable thresholding; Reynolds scaling and computational complexity; wavelets; Aerospace Engineering; Mechanical Engineering

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Nejadmalayeri, A. (2012). Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/mcen_gradetds/42

Chicago Manual of Style (16^th Edition):

Nejadmalayeri, Alireza. “Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 28, 2021. https://scholar.colorado.edu/mcen_gradetds/42.

MLA Handbook (7^th Edition):

Nejadmalayeri, Alireza. “Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding.” 2012. Web. 28 Feb 2021.

Vancouver:

Nejadmalayeri A. Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 28]. Available from: https://scholar.colorado.edu/mcen_gradetds/42.

Council of Science Editors:

Nejadmalayeri A. Hierarchical Multiscale Adaptive Variable Fidelity Wavelet-based Turbulence Modeling with Lagrangian Spatially Variable Thresholding. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/mcen_gradetds/42

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