Spring 2006 GSS Schedule

Organizers: Shannon Purvis and Alex Engau

Date Speaker Topic Abstract
January 23 Sundeep Samson Performance Based Decisions Under Uncertainty For Complex Systems Decision problems under conditions of uncertainty and risk are challenging. To begin with uncertainty and risk representations in the literature are typically problem specific and the methodologies that can handle these formulations are computationally demanding.

In the methodology presented here, an attempt as been made to expand the representation and modeling of uncertainty and risk to be applicable to a wide class of decision problems. Separable stochastic approximations of random fields and functional representations of risk are the major tools of this methodology.

This methodology has the potential of resolving complex problems such as multicriteria portfolio selection, engineering design problems, etc, beyond the scope of conventional mathematical programming and data analysis.
January 30 Alan Thomas Refractive Index Based Tomography with an Application to Biomedical Diagnostics Positron emission tomography (PET scan) and computed axial tomography (CAT scan) are well known methods of medical imaging. Optical tomography is an emerging area of medical imaging that utilizes visible or near-infrared light to probe tissue. In this talk we begin by introducing the inverse problem arising in optical tomography. We then introduce a new PDE model for radiative transport and examine the inverse problem associated with this PDE. Finally, using a nonlinear optimization algorithm we will find numerical solutions to several inverse problems.
February 6 Dr. Jamison
February 13 Santosh Tiwari Genetic Algorithms: A Brief Overview Evolutionary Algorithms (EAs) are nature inspired adaptive search procedures based on the Darwin’s hypothesis of the survival of the fittest. The adaptive nature of EAs can be exploited to design optimization algorithms by choosing an appropriate fitness function. Genetic Algorithm (GA) is one of the evolutionary techniques that has been successfully used as an optimization tool. Genetic Algorithms (GAs) mimic the principles of natural genetics and natural selection to constitute search and optimization procedures. Typically a GA works with a population (a set of solutions) instead of a single solution. This property of a GA makes it an ideal candidate for solving multi-objective optimization problems where the outcome (in most cases) is a set of solutions. Genetic algorithms can easily handle non-convex, discontinuous, multi-modal functions. GA can also be easily applied to problems having multiple global optima, discrete variables, constrained optimization problems etc. In this talk, genetic algorithm as an optimization tool will be presented. The talk will focus on basic genetic make-up of an evolutionary optimization procedure and its underlying operators. Current state-of-the-art in evolutionary algorithms will be presented and few sample simulations will be showcased to demonstrate the power and efficiency of the genetic algorithms.
February 20 Cancelled
February 27 Robert Beeler Fully Automorphic Decompositions of Graphs Given a graphs $H$ and $G$, we say that $H$ has a $G$-Decomposition if we are to partition the edge set of $H$ so that for each part $\mathscr{D}$ of the partition, the subgraph of $H$ induced by $\mathscr{D}$ is isomorphic to $G$. We then define a new graph, $I(\mathcal{D})$ as follows: the vertices of $I(\mathcal{D})$ represent the parts of the partition, and two vertices in $I(\mathcal{D})$ share an edge if and only if there corresponding blocks share a common node in $H$. We say that $\mathcal{D}$ is an \emph{automorphic decomposition} of $H$ with respect to $G$ if $I(\mathcal{D}) \cong H$. In a previous talk, we gave several necessary conditions for the existence of such a decomposition, in particular we required that the chromatic number of $H$, $\chi(H)$, be greater than or equal to the number of verices in $G$, $n(G)$. We say that $\mathscr{D}$ is a \emph{fully automorphic decomposition} of $H$ with respect to $G$ if $n(G)=\chi(H)$ and $\mathscr{D}$ is an automorphic decomposition of $H$ with respect to $G$. In this talk, we will state several necessary conditions for the existence of a fully automorphic decomposition as well as give some of their implications.
March 6 Christine Kraft An Adaptive Model for Predicting Course Enrollment Predicting university course enrollment is an inherently difficult problem because of uncertainty in student retention, course pass rates, admission policies, and curriculum requirements. This talk will present a procedure for creating adaptive course prediction models. We use student characteristics to identify groups of undergraduates whose course enrollment rates are significantly different than the rest of the university population and use historical enrollment rates and current information to predict enrollment for the coming semester. We then use the prediction model to aid in a system for releasing seats to new students during summer orientation sessions. The seat release system addresses how to estimate seat need each session, how to allocate seats for multiple course sections, and how to predict seat shortage and surpluses. We illustrate the course prediction modeling procedure and the seat release system using data from Clemson University.
March 13 Megan Koehler Does the number of oocytes retrieved significantly impact the clinical pregnancy rates realized in Assisted Reproductive Technology (ART) procedures? Patients seeking a child through ART typically undergo a hormone regimen that stimulates a patient’s ovaries to produce numerous follicles. Oocytes harvested from these follicles undergo sperm insemination procedures to obtain the maximum number of viable fertilized oocytes. While this approach has produced clinical pregnancy rates between 35% and 45%, a question of interest is whether or not the number of oocytes retrieved during ART procedures affects the likelihood of realizing a clinical pregnancy. To answer this question, data collected from the ART laboratory at the Greenville Hospital System University Medical Center between 2000 and 2004 were examined. These data included 333 ART cycles from patients participating in their first ART cycle, using their own oocytes, and having their embryos transferred to themselves three days after oocyte harvest. The association of clinical pregnancy with 19 possible predictors was investigated, with special interest in the relationship of clinical pregnancy to the number of oocytes retrieved. A preliminary univariate analysis was performed to investigate the effect of each of these possible predictors on clinical pregnancy rates. Using the Kruskal-Wallis Rank test for continuous variables and Chi-square contingency tables for categorical variables, the only variables that were found to have a significant effect on clinical pregnancy were age (P = .1), the number of oocytes fertilized (P = .03), the number of embryos transferred (P = .003), and the quality of the embryos transferred (P = .01). The number of oocytes retrieved was found not to significantly affect clinical pregnancy rates (P = .4) and a comparison of the mean and range of the number of oocytes retrieved from women who became pregnant to women who did not become pregnant revealed that the two statistics were almost identical for each group (mean and range for pregnant group = 15.2; [1-40] and mean and range for non pregnant group = 14.2; [3-40]). However, a logistic regression controlling for age and body mass index indicates that the quality of the embryos transferred (P = .04) and the ratio of the number of oocytes retrieved that had a chance of fertilizing to the overall number of oocytes retrieved (P = .0007) were significant in predicting pregnancy while the number of oocytes retrieved was not predictive. Thus, the number of oocytes retrieved does not significantly impact pregnancy rates realized from undergoing ART procedures, but the percentage of those oocytes that have the ability to be fertilized does have an affect.
March 13 Nilanjana Rahman Simulation model: Can we predict patient success in an assisted reproductive technology (ART) cycles? We present a simulation model to assist couples considering the use of ART procedures to achieve a pregnancy. The model is tailored to handle the individual patient characteristics and financial resources. Patient characteristics and patient decisions are input into the model, and the model displays possible outcomes as well as how those outcomes change with different decisions. The couple’s relevant physiological details like the mother’s body mass, age, the couple’s hormone levels, etc. are some of the patient characteristics included in the model. We examine the critical points of the ART process and hypothesize the distribution of the outcomes at the various stages based on numerous sources, such as previous studies, and data modeling and analysis. Key stages, including harvesting the oocytes, fertilizing the oocytes, and pregnancy are modeled with random variables. The central output of the model is the number of successful pregnancies that result given each couple’s history, decisions, and resources. A distribution of the number of successful pregnancies is ultimately obtained from the simulation model. The purpose of a model such as this is to help patients make decisions and to inform them about the likely outcomes of their scenario. It also helps them to visualize the process, and the workings of each of their decisions. The model helps compile the results from several other studies and relates them to actual outcomes. These results can lead to new insights and to further studies. This model can help experts communicate with each other, and as well as help caregivers communicate with patients. In particular, the model provides patients with a way to understand the wide array of possible outcomes and to examine the sensitivity of those outcomes to their medical history and decision making.
March 20 Spring Break
March 27 Jason Howell A two-grid method for nonlinear reaction-diffusion equations Two-grid methods have been developed by Xu (SIAM J. Num. Anal. 1996) for application to linear and nonlinear PDEs. Of particular interest are methods that can be applied to large nonlinear problems that arise in the simulation of physical processes, such as a method due to Dawson, Wheeler, and Woodward (SIAM J. Num. Anal. 1998). This scheme solves the original nonlinear problem on a mesh coarser than originally specified to capture the nonlinear behavior of the solution, then utilizes a linearized version of the problem to correct the coarse approximation on the original problem mesh. The two-grid method potentially reduces the overall computational cost by requiring the solution of a smaller nonlinear system and a large linear system in place of the original large nonlinear problem. In this talk we investigate the application of this method to nonlinear reaction-diffusion equations. In addition, we discuss some issues that arise in the implementation of the algorithm, and perform numerical experiments on problems designed to gauge the performance of the two-grid method relative to a standard Newton iterative nonlinear solver.
April 3 Jeremy Lyle Fall Coloring of the Cartesian Products of Graphs A \textit{proper coloring} of a graph $G=(V,E)$ is a partition $\Pi = \{V_1, V_2,\dots V_k\}$ of the vertices of $G$ into independent sets $V_i$, called \textit{color classes}. A $k$-\textit{fall coloring} is a proper coloring with $k$ color classes, that have the added condition that each color class is a dominating set in $V(G)$. Deciding if a graph has a fall coloring is an NP-complete problem, even for specific values of $k$.

The cartesian product of graphs $G$ and $H$ is denoted by $G\Box H$, where $V(G\Box H) = \{(u,v):u \in G, v\in H\}$ and $E(G\Box H) = \{((u,v),(w,x)): u=w , vx \in E(H) \mbox{ or } v=x, uw \in E(G)\}$. The $n$-cubes, denoted $Q_n$, are the graphs whose vertices represent all $n$ length binary strings, where two strings are connected if they differ in one coordinate. These can be recursively generated by the cartesian product, $Q_n = Q_{n-1} \Box K_2$, and have application as Cayley graphs and as network architectures in parallel computing.

In this talk, I will consider the fall coloring of cartesian product of bipartite graphs, and in particular, the $n$-cubes, in an attempt to answer an open question about what values of $k$-fall colorings occur in the $n$-cubes.

Keywords: fall coloring, idomatic partition, domatic partition, $n$-cubes, cartesian product
March 10 Bonnie McAdoo Varieties of Centers in Trees The \emph{center} of a graph $G$ is the subgraph induced by the vertices of minimum eccentricity. Many variations on the center of a tree have been developed, often with applications in the theory of facility location. For a connected graph $G$ the \emph{harmonic weight} of a vertex $v$ is given by $h(v) = \sum_{x \in V(G)-\left\{v\right\}}\frac{1}{d(v,x)}.$ The \emph{harmonic center} $H(G)$ is the subgraph induced by the set of vertices of maximum harmonic weight.

In this talk, we will discuss proprties several classical and recently developed central sets in trees, as well as their applications. We will also present observations regarding the harmonic center of a connected graph, including a proof that for any positive integer $p$, there exists a tree whose harmonic center consists of at least $p$ vertices.
March 17 John Chrispell A Fractional Step $\theta$ -Method for Convection Diffusion Problems The accurate numerical solutions of viscoelastic fluid flow problems poses two difficulties: the large number of unknowns in the approximating algebraic system (corresponding to velocity, pressure, and stress), and the different mathematical types of the modeling equations. Specifically for viscoelastic fluid flow the conservation of momentum equation is parabolic, and the constitutive equation for the fluid is hyperbolic. An appealing approximation approach is to use an operator splitting method which decouples the conservation of momentum equations from the constitutive equation. This split thereby reduces the size of the linear systems that need to be solved, and separates the parabolic and hyperbolic equations into different substeps. Motivated by the viscoelastic fluid flow problem, we consider an operator splitting fractional step $\theta$ - scheme for the numerical approximation of time dependent convection-diffusion problems. Here the fractional step $\theta$ -method allows for the convection and diffusion operators to be considered in separate solution substeps. In this presentation we describe the approximation scheme, present numerical simulations, and give some preliminary theoretical results.
April 24 Jobby Jacob Dominator Partitions of Graphs A vertex $v\in V$ in a graph $G=(V,E)$ \emph{dominates} a set $S\subseteq V$ if it is adjacent to every vertex $w\in S$, in which case we say that $v$ is a \emph{dominator} of $S$. A partition $\Pi=\{V_1,V_2,\ldots,V_k\}$ of $V(G)$ is called a \emph{dominator partition} if every vertex $v\in V$ is a dominator of at least one class $V_j$ of $\Pi$. A dominator partiton $\Pi=\{V_1,V_2,\ldots,V_k\}$ is \emph{minimal} if any partiton $\Pi'$ obtained from $\Pi$ by forming the union of any two classes $V_i\cup V_j,~i\neq j$, as one class in $\Pi'$ is no longer a dominator partion. The \emph{dominator partition number} of a graph $G$, denoted $\pi_d(G)$, is the minimum order of a dominator partiton of $G$. The \emph{upper dominator partiton number} of a graph $G$, denoted $\Pi_d(G)$, is the maximum order of a minimal dominator partition of $G$.

In this talk, I will try to discuss most of the results regarding the domination number and the upper domination number. If time permits, I will try to convince you that finding the upper domination partition number of even some trivial graphs is not trivial, by giving an outline of a proof to find the upper domination partition number of certain trivial graphs.