SIAM Undergraduate Research Online (SIURO)
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Undergraduate Student Research
The Krein Matrix and an Interlacing Theorem
Published electronically January 16, 2014.
Authors: Shamuel Auyeung, Eric Yu, (Calvin College, Grand Rapids, MI)
Sponsor: Todd Kapitula (Calvin College, Grand Rapids, MI)
Abstract: Consider the linear general eigenvalue problem Ay = λBy, where A and B are both invertible and Hermitian N × N matrices. In this paper we construct a set of meromorphic functions, the Krein eigenvalues, whose zeros correspond to the real eigenvalues of the general eigenvalue problem. The Krein eigenvalues are generated by the Krein matrix, which is constructed through projections on the positive and negative eigenspaces of B. The number of Krein eigenvalues depends on the number of negative eigenvalues for B. These constructions not only allow for us to determine solutions to the general eigenvalue problem, but also to determine the Krein signature for each real eigenvalue. Furthermore, by applying our formulation to the simplest case of the general eigenvalue problem (where B has one negative eigenvalue), we are able to show an interlacing theorem between the eigenvalues for the general problem and the eigenvalues of A.
Sponsor: Pranay Goel, Indian Institute of Science Education and Research - Pune (IISER- Pune)
Abstract: Pedersen and Sherman have recently developed a multi-pool model of insulin vesicle secretion from pancreatic beta-cells . In the Pedersen-Sherman model, pool sizes are reported as concentrations; however, concentrations of the different pools vary by as much as seven orders of magnitude. Very low concentrations indicate there could be discrete numbers of vesicles in some of the pools, leading naturally to the questions regarding stochasticity in the signalling pathway. We therefore simulate a discrete counterpart of the deterministic model with a (hybrid) Gillespie-style stochastic simulation algorithm. The stochastic simulations are calibrated to the deterministic model in the mean. We estimate the variances in pool sizes numerically and show it closely tracks the mean, indicating a Poisson-like result. Our numerical results demonstrate that the mean behavior indicated in the Pedersen-Sherman model is evident with just one islet.
Locally Chosen Grad-Div Stabilization Parameters for Finite Element Discretizations of Incompressible Flow Problems
Published electronically March 6, 2014.
Author: Nathan D Heavner (Clemson University)
Sponsor: Leo G Rebholz (Clemson University)
Abstract: Grad-div stabilization has recently been found to be an important tool for finite element method simulations of incompressible flow problems, acting to improve mass conservation in solutions and reducing the effect of the pressure error on the velocity error. Typically, the associated stabilization parameter is chosen globally, but herein we consider local choices. We show that, both for an analytic test problem and for lift and drag calculations of flow around a cylinder, local choices of the grad-div stabilization parameter can provide more accurate solutions.
Published electronically March 18, 2014.
Author: Justin Dong (Rice University)
Sponsor: Beatrice Riviere (Rice University)
Abstract: The modeling of three-phase fluid flow has many important applications in reservoir simulation. In this paper, we introduce a high order method for sequentially solving the phase pressure-saturation formulation of three-phase flow. The sequential approach, while less stable than a fully coupled approach, is more computationally efficient. We present a discontinuous Galerkin method in space, suitable for the sequential approach due to its high-order accuracy. We consider coarser meshes with high degree polynomials to minimize the dimension of the system while maximizing accuracy. Numerical results are given for homogeneous media.
Published electronically March 25, 2014.
Authors: Eric Jones and Peter Roemer (Colorado School of Mines, Golden, CO)
Sponsors: Stephen Pankavich and Mrinal Raghupathi (Colorado School of Mines, Golden, CO)
Abstract: Mathematical modeling of biological systems is crucial to effectively and efficiently developing treatments for medical conditions that plague humanity. Often, systems of ordinary differential equations are a traditional tool used to describe the spread of disease within the body. We consider the dynamics of the Human Immunodeficiency Virus (HIV) in vivo during the initial stages of infection. In particular, we examine the well-known three-component model and prove the existence, uniqueness, and boundedness of solutions. Furthermore, we prove that solutions remain biologically meaningful, i.e., are positivity preserving, and perform a thorough, local stability analysis for the equilibrium states of the system. Finally, we incorporate random coefficients into the model, selected from uniform, triangular, and truncated normal probability distributions, and obtain numerical results to predict the probability of infection given the transmission of the virus to a new individual.
Published electronically April 2, 2014.
Authors: Louis Joslyn, Casey Croson, and Sara Reed (Simpson College)
Sponsor: Debra Czarneski (Simpson College)
Abstract: Critical locations in infrastructure are roads that, if damaged, would cause a large disruption in the ability of vehicles to navigate a city. In this paper, we consider critical locations in the road network of Indianola, Iowa. The presence of cut vertices and values of betweenness for a given road segment are used in determining the importance of that given road segment. We present a model that uses these critical factors to order the importance of separate road segments. Finally, we explore models that focus on betweenness to improve accuracy when discovering critical locations.