+publisher:"University of Colorado" +contributor:("John Ruge")

University of Colorado

1. Mitchell, Wayne Bradford. Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations.

Degree: PhD, 2017, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/126

► When solving elliptic partial differential equations (PDE's) multigrid algorithms often provide optimal solvers and preconditioners capable of providing solutions with O(N) computational cost, where… (more)

▼ When solving elliptic partial differential equations (PDE's) multigrid algorithms often provide optimal solvers and preconditioners capable of providing solutions with O(N) computational cost, where N is the number of unknowns. As parallelism of modern super computers continues to grow towards exascale, however, the cost of communication has overshadowed the cost of computation as the next major bottleneck for multigrid algorithms. Typically, multigrid algorithms require O((log P)²) communication steps in order to solve a PDE problem to the level of discretization accuracy, where P is the number of processors. This has inspired the development of new algorithms that employ novel paradigms for parallelizing PDE problems, and this thesis studies and further develops two such algorithms. The nested iteration with range decomposition (NIRD) algorithm is known to achieve accuracy similar to that of traditional methods in only a single iteration with log P communication steps for simple elliptic problems. This thesis makes several improvements to the NIRD algorithm and extends its application to a much wider variety of problems, while also examining and updating previously proposed convergence theory and performance models. The second method studied is the algebraic multigrid with domain decomposition (AMG-DD) algorithm. Though previous work showed only marginal benefits when comparing convergence factors for AMG-DD against standard AMG V-cycles, this thesis studies the potential of AMG-DD as a discretization-accuracy solver. In addition to detailing the first parallel implementation of this algorithm, this thesis also shows new results that study the effect of different AMG coarsening and interpolation strategies on AMG-DD convergence and show some evidence to suggest AMG-DD may achieve discretization accuracy in a fixed number of cycles with O(log P) communication cost even as problem size increases. Advisors/Committee Members: Thomas A. Manteuffel, Stephen F. McCormick, John Ruge.

Subjects/Keywords: adaptive mesh refinement; algebraic multigrid; domain decomposition; first-order system least-squares; nested iteration; range decomposition; Applied Mathematics; Partial Differential Equations

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APA (6^th Edition):

Mitchell, W. B. (2017). Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/126

Chicago Manual of Style (16^th Edition):

Mitchell, Wayne Bradford. “Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/126.

MLA Handbook (7^th Edition):

Mitchell, Wayne Bradford. “Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations.” 2017. Web. 24 Jan 2021.

Vancouver:

Mitchell WB. Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/126.

Council of Science Editors:

Mitchell WB. Low-Communication, Parallel Multigrid Algorithms for Elliptic Partial Differential Equations. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/126

University of Colorado

2. 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 (6^th 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 (16^th 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 24, 2021. https://scholar.colorado.edu/appm_gradetds/84.

MLA Handbook (7^th 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. 24 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 24]. 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

3. 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 (6^th 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 (16^th Edition):

O'Neill, Ben. “Multigrid Reduction in Time for Nonlinear Parabolic Problems.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/92.

MLA Handbook (7^th Edition):

O'Neill, Ben. “Multigrid Reduction in Time for Nonlinear Parabolic Problems.” 2017. Web. 24 Jan 2021.

Vancouver:

O'Neill B. Multigrid Reduction in Time for Nonlinear Parabolic Problems. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 24]. 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

4. Fox, Alyson Lindsey. Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs.

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

URL: https://scholar.colorado.edu/appm_gradetds/96

► Relational datasets are often modeled as an unsigned, undirected graph due the nice properties of the resulting graph Laplacian, but information is lost if… (more)

▼ Relational datasets are often modeled as an unsigned, undirected graph due the nice properties of the resulting graph Laplacian, but information is lost if certain attributes of the graph are not represented. This thesis presents two generalizations of Algebraic Multigrid (AMG) solvers with graph Laplacian systems for different graph types: applying Gremban’s expansion to extend unsigned graph Laplacian solvers to signed graph Laplacian systems and generalizing techniques in Lean Algebraic Multigrid (LAMG) to a new multigrid solver for unsigned, directed graph Laplacian systems. Signed graphs extend the traditional notion of connections and disconnections to in- clude both favorable and adverse relationships, such as friend-enemy social networks or social networks with “likes” and “dislikes.” Gremban’s expansion is used to transform the signed graph Laplacian into an unsigned graph Laplacian with twice the number of unknowns. By using Gremban’s expansion, we extend current unsigned graph Laplacian solvers’ to signed graph Laplacians. This thesis analyzes the numerical stability and applicability of Grem- ban’s expansion and proves that the error of the solution of the original linear system can be tightly bounded by the error of the expanded system. In directed graphs, some subset of relationships are not reciprocal, such as hyperlink graphs, biological neural networks, and electrical power grids. A new algebraic multigrid algorithm, Nonsymmetric Lean Algebraic Multigrid (NS-LAMG), is proposed, which uses ideas from Lean Algebraic Multigrid, nonsymmetric Smoothed Aggregation, and multigrid solvers for Markov chain stationary distribution systems. Low-degree elimination, intro- duced in Lean Algebraic Multigrid for undirected graphs, is redefined for directed graphs. A semi-adaptive multigrid solver, inspired by low-degree elimination, is instrumented in the setup phase, which can be adapted for Markov chain stationary distributions systems. Nu- merical results shows that NS-LAMG out performs GMRES(k) for real-world, directed graph Laplacian linear systems. Both generalizations enable more choices in modeling decisions for graph Laplacian systems. Due the successfulness of NS-LAMG and other various nonsymmetric AMG (NS-AMG) solvers, a further study of theoretical convergence properties are discussed in this thesis. In particular, a necessary condition known as “weak approximation property”, and a sufficient one, referred to as “strong approximation property” as well as the “super strong approx- imation property” are generalized to nonsymmetric matrices and the various relationships between the approximation properties are proved for the nonsymmetric case. In NS-AMG, if P ̸= R the two-grid error propagation operator for the coarse-grid correction is an oblique projection with respect to any reasonable norm, which can cause the error to increase. A main focal point of this paper is a discussion on the conditions in which the error propagation operator is bounded, as the stability of the error… Advisors/Committee Members: Tom Manteuffel, Geoff Sanders, John Ruge, Christian Ketelsen, Francois Meyer.

Subjects/Keywords: Algebraic Multigrid; Directed graphs; Graph Laplacians; Gremban's expansion; Signed graphs; Applied Mechanics

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APA (6^th Edition):

Fox, A. L. (2017). Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/96

Chicago Manual of Style (16^th Edition):

Fox, Alyson Lindsey. “Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/96.

MLA Handbook (7^th Edition):

Fox, Alyson Lindsey. “Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs.” 2017. Web. 24 Jan 2021.

Vancouver:

Fox AL. Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/96.

Council of Science Editors:

Fox AL. Algebraic Multigrid(amg) for Graph Laplacian Linear Systems: Extensions of Amg for Signed, Undirected and Unsigned, Directed Graphs. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/96

University of Colorado

5. Southworth, Benjamin Scott. Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid.

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

URL: https://scholar.colorado.edu/appm_gradetds/99

► PART I: One of the most fascinating questions to humans has long been whether life exists outside of our planet. To our knowledge, water… (more)

▼ PART I: One of the most fascinating questions to humans has long been whether life exists outside of our planet. To our knowledge, water is a fundamental building block of life, which makes liquid water on other bodies in the universe a topic of great interest. In fact, there are large bodies of water right here in our solar system, underneath the icy crust of moons around Saturn and Jupiter. The NASA-ESA Cassini Mission spent two decades studying the Saturnian system. One of the many exciting discoveries was a “plume” on the south pole of Enceladus, emitting hundreds of kg/s of water vapor and frozen water-ice particles from Enceladus’ subsurface ocean. It has since been determined that Enceladus likely has a global liquid water ocean separating its rocky core from icy surface, with conditions that are relatively favorable to support life. The plume is of particular interest because it gives direct access to ocean particles from space, by flying through the plume. Recently, evidence has been found for similar geological activity occurring on Jupiter’s moon Europa, long considered one of the most likely candidate bodies to support life in our solar system. Here, a model for plume-particle dynamics is developed based on studies of the Enceladus plume and data from the Cassini Cosmic Dust Analyzer. A C++, OpenMP/MPI parallel software package is then built to run large scale simulations of dust plumes on planetary satellites. In the case of Enceladus, data from simulations and the Cassini mission provide insight into the structure of emissions on the surface, the total mass production of the plume, and the distribution of particles being emitted. Each of these are fundamental to understanding the plume and, for Europa and Enceladus, simulation data provide important results for the planning of future missions to these icy moons. In particular, this work has contributed to the Europa Clipper mission and proposed Enceladus Life Finder. PART II: Solving large, sparse linear systems arises often in the modeling of biological and physical phenomenon, data analysis through graphs and networks, and other scientific applications. This work focuses primarily on linear systems resulting from the discretization of partial differential equations (PDEs). Because solving linear systems is the bottleneck of many large simulation codes, there is a rich field of research in developing “fast” solvers, with the ultimate goal being a method that solves an n × n linear system in O(n) operations. One of the most effective classes of solvers is algebraic multigrid (AMG), which is a multilevel iterative method based on projecting the problem into progressively smaller spaces, and scales like O(n) or O(nlogn) for certain classes of problems. The field of AMG is well-developed for symmetric positive definite matrices, and is typically most effective on linear systems resulting from the discretization of scalar elliptic PDEs, such as the heat equation. Systems of PDEs can add… Advisors/Committee Members: Thomas A. Manteuffel, Sascha Kempf, Steve McCormick, John Ruge, Jacob Schroder.

Subjects/Keywords: algebraic multigrid; dust plume; Enceladus; linear system; nonsymmetric; transport; Applied Mathematics; Physics

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APA (6^th Edition):

Southworth, B. S. (2017). Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/99

Chicago Manual of Style (16^th Edition):

Southworth, Benjamin Scott. “Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid.” 2017. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/99.

MLA Handbook (7^th Edition):

Southworth, Benjamin Scott. “Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid.” 2017. Web. 24 Jan 2021.

Vancouver:

Southworth BS. Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/99.

Council of Science Editors:

Southworth BS. Seeking Space Aliens and the Strong Approximation Property: a (Disjoint) Study in Dust Plumes on Planetary Satellites and Nonsymmetric Algebraic Multigrid. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/99

University of Colorado

6. 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)

Subjects/Keywords: Beta-Plane; FOSLS; Geophysics; Navier-Stokes; Applied Mathematics; Geophysics and Seismology

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APA (6^th 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 (16^th 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 24, 2021. https://scholar.colorado.edu/appm_gradetds/51.

MLA Handbook (7^th Edition):

Garcia, Jose Humberto. “Beta-Plane Approximation of Wind Driven Ocean Circulation using a First Order System Least-Squares Formulation.” 2014. Web. 24 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 24]. 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

7. Brutz, Michael Joseph. Mathematical Modelling and Analysis of Several Diffusive Processes.

Degree: PhD, Applied Mathematics, 2014, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/60

► The underlying theme of this research is using numerical methods to develop computationally efficient algorithms for three separate problems driven by diffusive processes. The… (more)

▼ The underlying theme of this research is using numerical methods to develop computationally efficient algorithms for three separate problems driven by diffusive processes. The problems under consideration are: contaminant dispersal through fracture networks, modelling the flow of glacial ice, and community detection on networks. A common feature of containment facilities for nuclear waste is to use expansive geological formations as an added barrier to contaminant dispersal in the event of a leak. Although these formations are generally comprised of dense rock that is difficult to penetrate, fractures within them provide a potential means for contaminants to rapidly transport across the barrier. The typical width of such fractures is only on the order of millimeters whereas the typical scale of interest for contaminant transport is on the order of kilometers. When particle tracking methods are used to simulate the contaminant dispersal in fracture networks, this disparity of scales severely restricts maximum time step sizes because features at the millimeter scale need to be resolved. Our contribution to this problem is developing a coarse scale particle tracking method that allows for substantially larger time steps when particles are navigating straight fractures. With global warming comes concerns as to how the changing temperature will impact glacial systems and their contribution to sea level rise. On glacial scales, ice behaves as a very slowly moving non-Newtonian fluid, and the primary problem for numerically simulating the evolution of ice masses comes with Glen's flow law for the effective viscosity. The flow law is empirically based, and its simple form has proven useful for analytical calculations. However, its simple form also allows for the effective viscosity to become unbounded in regions of low strain rate, and has proven to be very problematic for numerical simulations. Our contribution to this problem is re-examining the datasets the flow law was originally based on to develop an alternative model that fits the data with comparable accuracy, but without the problematic singularity. When working with networks that represent real world systems, a common feature of interest is to find collections of vertices that form communities. Because the word "community" is an ambiguous term, our interpretation is that it is necessary to quantify what it means to be a community at a minimum of three scales for any given problem. These scales are at the level of: individual nodes, individual communities, and the network as a whole. Although our work focuses on detecting overlapping communities in the context of social networks, our primary contribution is developing a methodology that is highly modular and can easily be adapted to target other problem-specific notions of community. Advisors/Committee Members: Francois Meyer, Tom Manteuffel, Harihar Rajaram, John Ruge, Juan Restrepo.

Subjects/Keywords: community; glacier; multiscale; network; flow; particle tracking; Applied Mathematics

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APA (6^th Edition):

Brutz, M. J. (2014). Mathematical Modelling and Analysis of Several Diffusive Processes. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/60

Chicago Manual of Style (16^th Edition):

Brutz, Michael Joseph. “Mathematical Modelling and Analysis of Several Diffusive Processes.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/60.

MLA Handbook (7^th Edition):

Brutz, Michael Joseph. “Mathematical Modelling and Analysis of Several Diffusive Processes.” 2014. Web. 24 Jan 2021.

Vancouver:

Brutz MJ. Mathematical Modelling and Analysis of Several Diffusive Processes. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/60.

Council of Science Editors:

Brutz MJ. Mathematical Modelling and Analysis of Several Diffusive Processes. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/60

University of Colorado

8. Appelhans, David John. Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement.

Degree: PhD, Applied Mathematics, 2014, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/63

► In this thesis we propose a new algorithm for solving PDEs on massively parallel computers. The Nested Iteration Adaptive Mesh Refinement Range Decomposition (NI-AMRRD)… (more)

Subjects/Keywords: Adaptive Refinement; FOSLS; High Performance Computing; Nested Iteration; Parallel Algorithms; Range Decomposition; Applied Mathematics; Computer Sciences; Theory and Algorithms

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APA (6^th Edition):

Appelhans, D. J. (2014). Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/63

Chicago Manual of Style (16^th Edition):

Appelhans, David John. “Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/63.

MLA Handbook (7^th Edition):

Appelhans, David John. “Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement.” 2014. Web. 24 Jan 2021.

Vancouver:

Appelhans DJ. Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/63.

Council of Science Editors:

Appelhans DJ. Trading Computation for Communication: A Low Communication Algorithm for the Parallel Solution of PDEs Using Range Decomposition, Nested Iteration, and Adaptive Mesh Refinement. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/63

University of Colorado

9. Liu, Kuo. Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations.

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

URL: https://scholar.colorado.edu/appm_gradetds/30

► This thesis combines the FOSLS method with the FOSLL* method to create a Hybrid method. The FOSLS approach minimizes the error, e^h =… (more)

Subjects/Keywords: Finite Element; Least-squares; Navier Stokes; Stokes; Applied Mathematics

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APA (6^th Edition):

Liu, K. (2012). Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/30

Chicago Manual of Style (16^th Edition):

Liu, Kuo. “Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations.” 2012. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/30.

MLA Handbook (7^th Edition):

Liu, Kuo. “Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations.” 2012. Web. 24 Jan 2021.

Vancouver:

Liu K. Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/30.

Council of Science Editors:

Liu K. Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/30

University of Colorado

10. Jones, Tobias M. Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs.

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

URL: https://scholar.colorado.edu/appm_gradetds/43

► In modern large-scale supercomputing applications, Algebraic Multigrid (AMG) is a leading choice for solving linear systems. However, on the newest architectures, the relatively high… (more)

Subjects/Keywords: AMG; Graphs; Parallel; Systems; Applied Mathematics

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APA (6^th Edition):

Jones, T. M. (2013). Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/43

Chicago Manual of Style (16^th Edition):

Jones, Tobias M. “Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 24, 2021. https://scholar.colorado.edu/appm_gradetds/43.

MLA Handbook (7^th Edition):

Jones, Tobias M. “Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs.” 2013. Web. 24 Jan 2021.

Vancouver:

Jones TM. Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 24]. Available from: https://scholar.colorado.edu/appm_gradetds/43.

Council of Science Editors:

Jones TM. Algebraic Multigrid Methods for Parallel Computing, Systems, and Graphs. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/43

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