Numerical Analysis and Computation Commons^™

Lyubeznik Numbers Of Unmixed Edge Ideals, Sara Rae Jones

Graduate Theses and Dissertations

Lyubeznik numbers, defined in terms of local cohomology, are invariants of local rings that are able to detect many algebraic and geometric properties. Notably they recognize topological behaviors of various structures associated to their rings. We will discuss computations of these numbers for unmixed edge ideals by giving a completely combinatorial construction which realizes the connectedness information captured by these numbers.

Cohomology Of The Symmetric Group With Twisted Coefficients And Quotients Of The Braid Group, Trevor Nakamura

Graduate Theses and Dissertations

In 2014 Brendle and Margalit proved the level $4$ congruence subgroup of the braid group, $B_{n}[4]$, is the subgroup of the pure braid group generated by squares of all elements, $PB_{n}^{2}$. We define the mod $4$ braid group, $\Z_{n}$, to be the quotient of the braid group by the level 4 congruence subgroup, $B_{n}/B_{n}[4]$. In this dissertation we construct a group presentation for $\Z_{n}$ and determine a normal generating set for $B_{n}[4]$ as a subgroup of the braid group. Further work by Kordek and Margalit in 2019 proved $\Z_{n}$ is an extension of the symmetric group, $S_{n}$, by $\mathbb{Z}_{2}^{\binom{n}{2}}$. A …

Grid Homology Invariants For Singular Legendrian Links, Richard Michael Shumate

Graduate Theses and Dissertations

If $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ are two oriented singular Legendrian links that are Legendrian isotopic, we first construct front diagram representations of $\Lambda_{1}^{\ast}$ and $\Lambda_{2}^{\ast}$ that have a natural allowable singular gird diagram associated to them. These allowable singular grid diagrams will always correspond to singular Legendrian links. The grid Legendrian invariants, $\lambda^{\pm}$, in the nonsingular grid homology theory have a natural extension to the singular grid theory, and are natural under the newly defined singular grid moves. This gives an invariant of singular Legendrian links, and in fact, a broader class of singular links.

Lecture 05: The Convergence Of Big Data And Extreme Computing, David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 09: Hierarchically Low Rank And Kronecker Methods, Rio Yokota

Mathematical Sciences Spring Lecture Series

Exploiting structures of matrices goes beyond identifying their non-zero patterns. In many cases, dense full-rank matrices have low-rank submatrices that can be exploited to construct fast approximate algorithms. In other cases, dense matrices can be decomposed into Kronecker factors that are much smaller than the original matrix. Sparsity is a consequence of the connectivity of the underlying geometry (mesh, graph, interaction list, etc.), whereas the rank-deficiency of submatrices is closely related to the distance within this underlying geometry. For high dimensional geometry encountered in data science applications, the curse of dimensionality poses a challenge for rank-structured approaches. On the other …

Lecture 08: Partial Eigen Decomposition Of Large Symmetric Matrices Via Thick-Restart Lanczos With Explicit External Deflation And Its Communication-Avoiding Variant, Zhaojun Bai

Mathematical Sciences Spring Lecture Series

There are continual and compelling needs for computing many eigenpairs of very large Hermitian matrix in physical simulations and data analysis. Though the Lanczos method is effective for computing a few eigenvalues, it can be expensive for computing a large number of eigenvalues. To improve the performance of the Lanczos method, in this talk, we will present a combination of explicit external deflation (EED) with an s-step variant of thick-restart Lanczos (s-step TRLan). The s-step Lanczos method can achieve an order of s reduction in data movement while the EED enables to compute eigenpairs in batches along with a number …

Lecture 04: Spatial Statistics Applications Of Hrl, Trl, And Mixed Precision, David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 14: Randomized Algorithms For Least Squares Problems, Ilse C.F. Ipsen

Mathematical Sciences Spring Lecture Series

The emergence of massive data sets, over the past twenty or so years, has lead to the development of Randomized Numerical Linear Algebra. Randomized matrix algorithms perform random sketching and sampling of rows or columns, in order to reduce the problem dimension or compute low-rank approximations. We review randomized algorithms for the solution of least squares/regression problems, based on row sketching from the left, or column sketching from the right. These algorithms tend to be efficient and accurate on matrices that have many more rows than columns. We present probabilistic bounds for the amount of sampling required to achieve a …

Lecture 13: A Low-Rank Factorization Framework For Building Scalable Algebraic Solvers And Preconditioners, X. Sherry Li

Mathematical Sciences Spring Lecture Series

Factorization based preconditioning algorithms, most notably incomplete LU (ILU) factorization, have been shown to be robust and applicable to wide ranges of problems. However, traditional ILU algorithms are not amenable to scalable implementation. In recent years, we have seen a lot of investigations using low-rank compression techniques to build approximate factorizations.
A key to achieving lower complexity is the use of hierarchical matrix algebra, stemming from the H-matrix research. In addition, the multilevel algorithm paradigm provides a good vehicle for a scalable implementation. The goal of this lecture is to give an overview of the various hierarchical matrix formats, such …

Lecture 07: Nonlinear Preconditioning Methods And Applications, Xiao-Chuan Cai

Mathematical Sciences Spring Lecture Series

We consider solving system of nonlinear algebraic equations arising from the discretization of partial differential equations. Inexact Newton is a popular technique for such problems. When the nonlinearities in the system are well-balanced, Newton's method works well, but when a small number of nonlinear functions in the system are much more nonlinear than the others, Newton may converge slowly or even stagnate. In such a situation, we introduce some nonlinear preconditioners to balance the nonlinearities in the system. The preconditioners are often constructed using a combination of some domain decomposition methods and nonlinear elimination methods. For the nonlinearly preconditioned problem, …

Lecture 03: Hierarchically Low Rank Methods And Applications, David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 02: Tile Low-Rank Methods And Applications (W/Review), David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 11: The Road To Exascale And Legacy Software For Dense Linear Algebra, Jack Dongarra

Mathematical Sciences Spring Lecture Series

In this talk, we will look at the current state of high performance computing and look at the next stage of extreme computing. With extreme computing, there will be fundamental changes in the character of floating point arithmetic and data movement. In this talk, we will look at how extreme-scale computing has caused algorithm and software developers to change their way of thinking on implementing and program-specific applications.

Lecture 00: Opening Remarks: 46th Spring Lecture Series, Tulin Kaman

Mathematical Sciences Spring Lecture Series

Opening remarks for the 46th Annual Mathematical Sciences Spring Lecture Series at the University of Arkansas, Fayetteville.

Lecture 06: The Impact Of Computer Architectures On The Design Of Algebraic Multigrid Methods, Ulrike Yang

Mathematical Sciences Spring Lecture Series

Algebraic multigrid (AMG) is a popular iterative solver and preconditioner for large sparse linear systems. When designed well, it is algorithmically scalable, enabling it to solve increasingly larger systems efficiently. While it consists of various highly parallel building blocks, the original method also consisted of various highly sequential components. A large amount of research has been performed over several decades to design new components that perform well on high performance computers. As a matter of fact, AMG has shown to scale well to more than a million processes. However, with single-core speeds plateauing, future increases in computing performance need to …

Lecture 01: Scalable Solvers: Universals And Innovations, David Keyes

Mathematical Sciences Spring Lecture Series

As simulation and analytics enter the exascale era, numerical algorithms, particularly implicit solvers that couple vast numbers of degrees of freedom, must span a widening gap between ambitious applications and austere architectures to support them. We present fifteen universals for researchers in scalable solvers: imperatives from computer architecture that scalable solvers must respect, strategies towards achieving them that are currently well established, and additional strategies currently being developed for an effective and efficient exascale software ecosystem. We consider recent generalizations of what it means to “solve” a computational problem, which suggest that we have often been “oversolving” them at the …

Lecture 10: Preconditioned Iterative Methods For Linear Systems, Edmond Chow

Mathematical Sciences Spring Lecture Series

Iterative methods for the solution of linear systems of equations – such as stationary, semi-iterative, and Krylov subspace methods – are classical methods taught in numerical analysis courses, but adapting these methods to run efficiently at large-scale on high-performance computers is challenging and a constantly evolving topic. Preconditioners – necessary to aid the convergence of iterative methods – come in many forms, from algebraic to physics-based, are regularly being developed for linear systems from different classes of problems, and similarly are evolving with high-performance computers. This lecture will cover the background and some recent developments on iterative methods and preconditioning …

Development Of An Effect Size To Classify The Magnitude Of Dif In Dichotomous And Polytomous Items, James D. Weese

Graduate Theses and Dissertations

A standardized effect size for the SIBTEST/POLYSIBTEST procedure is proposed, allowing for Differential Item Functioning (DIF) to be classified with a single set of DIF heuristics regardless of whether data are dichotomous or polytomous. This proposed standardized effect size accounts for both variability in responses and whether participants are included in the SIBTEST/POLYSIBTEST calculations. First, a new set of unstandardized effect size heuristics are established for dichotomous data that are more aligned with Educational Testing Service (ETS) standards using two and three parameter logistic (2PL and 3PL) models. Second, a standardized effect size is proposed and compared to other DIF …

Hydrodynamic Instability Simulations Using Front-Tracking With Higher-Order Splitting Methods, Dillon Trinh

Mathematical Sciences Undergraduate Honors Theses

The Rayleigh-Taylor Instability (RTI) is an instability that occurs at the interface of a lighter density fluid pushing onto a higher density fluid in constant or time-dependent accelerations. The Richtmyer-Meshkov Instability (RMI) occurs when two fluids of different densities are separated by a perturbed interface that is accelerated impulsively, usually by a shock wave. When the shock wave is applied, the less dense fluid will penetrate the denser fluid, forming a characteristic bubble feature in the displacement of the fluid. The displacement will initially obey a linear growth model, but as time progresses, a nonlinear model is required. Numerical studies …

Closed Range Composition Operators On Bmoa, Kevser Erdem

Graduate Theses and Dissertations

Let φ be an analytic self-map of the unit disk D. The composition operator with symbol φ is denoted by Cφ. Reverse Carleson type conditions, counting functions and sampling sets are important tools to give a complete characterization of closed range composition operators on BMOA and on Qp for all p ∈ (0,∞).

Let B denote the Bloch space, let H2 denote the Hardy space. We show that if Cφ is closed range on B or on H2 then it is also closed range on BMOA. Closed range composition operators Cφ : B → BMOA are also characterized. Laitila found …

Developing A Risk Analysis Model To Improve Study Abroad Awareness, Tyler Spain

Industrial Engineering Undergraduate Honors Theses

As international education opportunities increase in popularity among U.S. college students (McMurtrie, 2007), it is becoming more and more necessary for study abroad organizations to be aware of the risks students face as they travel abroad. While some international cities are riskier than others, it can be difficult to distinguish between cities which truly carry a high degree of risk for visiting students, and which cities are only perceived to be risky based on various personal misconceptions. The University of Arkansas Office of Study Abroad & International Exchange currently lacks a way to quantifiably analyze the risk of study abroad …

Cfd Model For Ventilation In Broiler Holding Sheds, Christian Heymsfield

Biological and Agricultural Engineering Undergraduate Honors Theses

Broiler production in Arkansas was valued at over $3.6 billion in 2013 (University of Arkansas Extension of Agriculture). Consequently, improvement in any phase of the production process can have significant economic impact and animal welfare implications. From the time poultry leave the farm and until they are slaughtered, they can be exposed to harsh environmental conditions, both in winter and in summer. After road transportation, birds are left to wait in holding sheds once they arrive at the processing plant, for periods of approximately 30 minutes to two hours. This project was interested in this holding shed waiting time during …

General Sampling Schemes For The Bergman Spaces, Newton Foster

Graduate Theses and Dissertations

A characterization of sampling sequences for the Bergman spaces was originally provided by Seip and later expanded upon by Schuster. We consider a generalized notion of sampling using the infimum norm of the quotient space. Adapting some old techniques, we provide a characterization of general sampling sequences in terms of the lower uniform density.

Hardy Space Properties Of The Cauchy Kernel Function For A Strictly Convex Planar Domain, Belen Espinosa Lucio

Graduate Theses and Dissertations

This work is based on a paper by Edgar Lee Stout, where it is shown that for every strictly pseudoconvex domain $D$ of class $C^2$ in $\mathbb{C}^N$, the Henkin-Ram\'irez Kernel Function belongs to the Smirnov class, $E^q(D)$, for every $q\in(0,N)$.

The main objective of this dissertation is to show an analogous result for the Cauchy Kernel Function and for any strictly convex bounded domain in the complex plane. Namely, we show that for any strictly convex bounded $D\subset\mathbb{C}$ of class $C^2$ if we fix $\zeta$ in the boundary of $D$ and consider the Cauchy Kernel Function

\mathcal{K}(\zeta,z)=\frac{1}{2\pi i}\frac{1}{\zeta-z}

as a …

Pointwise Schauder Estimates Of Parabolic Equations In Carnot Groups, Heather Arielle Griffin

Graduate Theses and Dissertations

Schauder estimates were a historical stepping stone for establishing uniqueness and smoothness of solutions for certain classes of partial differential equations. Since that time, they have remained an essential tool in the field. Roughly speaking, the estimates state that the Holder continuity of the coefficient functions and inhomogeneous term implies the Holder continuity of the solution and its derivatives. This document establishes pointwise Schauder estimates for second order parabolic equations where the traditional role of derivatives are played by vector fields generated by the first layer of the Lie algebra stratification for a Carnot group. The Schauder estimates are shown …

The Beauty Of Mathematics And The Mathematics Of Beauty: Continued Fractions And The Golden Ratio, Jessica Tush

Inquiry: The University of Arkansas Undergraduate Research Journal

This project begins with a look at the history of simple continued fractions and how we have arrived where we are today. We then move through a study of simple continued fractions, beginning first with rational numbers and moving to irrational numbers. Continuing further in the pursuit of joining mathematics and art, we define the specific continued fraction that gives rise to the Fibonacci sequence and the Golden Ratio~ (phi, pronounced 'Jai"). These two notions form a direct link to art and the properties that we hope to examine. I have taken an analytic approach to showing that the Golden …

Parallel Algorithms For Multicriteria Shortest Path Problems, David L. Sonnier

Journal of the Arkansas Academy of Science

This paper presents two strategies for solving multicriteria shortest path problems with more than two criteria. Given an undirected graph within vertices, medges, and a set of K weights associated with each edge, we define a path as a sequence of edges from vertex s to vertex t. We want to find the Pareto-optimal set of paths from s to t. The solutions proposed herein are based on cluster computing using the Message-Passing Interface (MPI) extensions to the C programming language. We solve problems with 3 and 4 criteria, using up to 8 processors in parallel and using solutions based …