CRSC Participation in 2005 SIAM Southeast Atlantic Section Meeting

In March, 2005, during NCSU's spring recess, more than 20 CRSC graduate students, post-docs and professors attended the SIAM Southeast Atlantic Section Meeting in Charleston, South Carolina. Details on the CRSC contributed talks can be found below. Two of our graduate students were honored with awards: Matthew Lassater for his talk "Simulating Resonant Tunneling Diodes with the Wigner-Poisson Equations" describing his work with Tim Kelley, and Laura Ellwein for "Modeling Autonomic and Autoregulation of Blood Flow during Postural Change from Sitting to Standing", which was advised by Mette Olufsen. Professor Tim Kelley was also an invited speaker talking on "Optimal Design of Groundwater Remediation Systems with Sampling Methods". Interim Mathematics Department Head Jean-Pierre Fouque presented a short course entitled "Introduction to Financial Mathematics and Volatility Modeling". Lastly, three of the nine minisymposia (Special Sessions) were organized by CRSC members: "Information Retrieval" organized by Amy Langville and Carl Meyer, "Thin Liquid Films" organized by Michael Shearer and Rachel Levy, and "Inverse Problems in Electromagnetics and Biology" organized by H.T. Banks and Nathan Louis Gibson.

In addition to enjoying nearly two full days of mathematical exposition, we were also treated to a buffet dinner featuring local cuisine at the Citadel Beach House. However, several of us left the meal early to catch what turned out to be an unfortunate loss for NC State (and Duke) in the Sweet Sixteen.

Click the title to skip to the abstract:

Naomi Caldwell and Robert E. Funderlic--The k-Means Clustering Method and Concept Vectors Based on the Centroid Method

The k-means partitional clustering method is put into a general setting. As is known, a singular vector associated with the largest singular value has been used in place of the mean of a k-means cluster. In generalizations of k-means, sometimes called "k-means like methods", we call all concept vectors simply center objects of clusters and call all these methods k-center methods. We explore a fast discrete estimate for a center vector and present related linear algebra. These discrete center vectors are based on what statisticians call the centroid method for which the vectors are restricted to have components with absolute value one.

Prakash Chanchana--Updating the stationary distribution vector for an irreducible Markov Chain

Updating the stationary distribution vector of an irreducible Markov chain is the ongoing research problem. Using idea of aggregation/disaggregation to build the Iterative algorithm is one way to update the stationary distribution vector, especially, for the well known Google's PageRank application. The algorithm works for both states updating and elements updating problem.

John David--Optimal Design of Traveling Wave Tubes

The traveling wave tube amplifier (TWT) is a vacuum device invented in the early 1940s for amplification of radio frequency (RF) power. This device is critical for communications and electronic warfare missions in the military, as well as in commercial applications. In this talk, we will discuss the optimal design of these devices to improve essential performance criteria including efficiency, linear power range and phase distortion. The simulation code, CHRISTINE, was used to evaluate the performance of TWTs given a set of design parameters. Current capability of CHRISTINE allows only for a limited number of basic TWT designs including a limited number of design goal functions, and employs a modified steepest descent method to carry out the optimization process. However, in this type of application the landscapes of the goal functions are, in general, very noisy which can defeat most gradient based methods. The objectives of our work are two-fold: (i) to improve the design capabilities for TWT (including basic geometry and new goal functions) and (ii) to investigate optimization techniques that are better suited for problems with complex and noisy landscapes (for example, simplex type methods such as Nelder-Mead and DIRECT).

Jimena Davis--Comparison of Two Approximation Methods in the Estimation of Growth Rate Distributions in Size-Structured Mosquitofish Populations

In an effort to protect the environment, biologists are using mosquitofish in the place of chemicals to control mosquito populations in rice fields. While they have used mosquitofish in the place of pesticides for some time, they have not completely understood the control of the growth of mosquitofish populations. Biologists would like to be able to predict the growth and decline of the mosquitofish populations in order to determine the optimal amount of mosquitofish to use for control purposes. We will present the Sinko-Streifer population model modified as in the Growth Rate Distribution model of Banks-Botsford-Kappel-Wang. We will also present and compare the results of two approximation methods used in the inverse problem for estimation of distributions of growth rates in size-structured mosquitofish populations.

Laura Ellwein--Modeling Autonomic and Autoregulation of Blood Flow during Postural Change from Sitting to Standing

The "light-headedness" a person feels with a sudden posture change is caused by a lack of blood flow to the brain. In normal individuals it is overcome naturally via autonomic reflexes (ARF), which impact the entire body, and cerebral autoregulation (CA), a local control mechanism. The interaction between the two mechanisms is not well understood. To study these mechanisms we have developed a compartment model coupled with physiological models describing the ARF and CA during postural change from sitting to standing. To validate our mathematical model we use finger blood pressure and cerebral blood flow velocity data obtained non-invasively from a young subject. This model validation is carried out using nonlinear optimization methods.

Sarah Grove--HIV Modeling and Inference

During this talk we will briefly explain Monte Carlo Markov Chains (MCMC). Also we will consider Gibb's method which uses Metropolis-Hastings to create these chains. We will examine how these techniques are used to better understand the distribution of parameters across our population. We will be using the Bayesian approach to update our model on both the individual as well as the population level. Specifically the HIV model will use viral load, uninfected CD4 counts, and immune indicator data to characterize an individuals dynamics.

Matthew Scott Lasater--Simulating Resonant Tunneling Diodes with the Wigner-Poisson Equations

Resonant tunneling diodes (RTDs) are quantum size semiconductor devices, which both theory and numerical simulation predict can sustain terahertz frequency current oscillations. The electron transport in these devices are modeled by the Wigner-Poisson equations: a nonlinear PDE which describes the time-evolution of the electrons coupled with Poisson's equation to incorporate the potential effects of the electrons. Ongoing research with an RTD involves removing it from a circuit and searching for a voltage drop across the RTD that creates these high frequency current oscillations within the device. To accomplish this, we connected our simulator to LOCA (Library of Continuation Algorithms), a software library developed at Sandia National Laboratories. These algorithms enable us to trace-out the steady-state solutions to the PDE as the voltage drop across the device is varied. An eigenvalue analysis performed by LOCA allows us to predict the development of current oscillations from just steady-state calculations. Numerical results will be presented.

Nicholas Luke--Noise Reduction in a Cylindrical Duct

In this talk, we propose a method of noise reduction in a cylindrical geometry (such as a duct) which uses waveguides to cause destructive interference, resulting in the desired noise reduction. We present an outline of the proposed method, along with experimental data pertaining to the fundamental frequencies of different waveguides, representing a variety of geometries. Using the commercially available finite element software, ANSYS, we attempt to create a computational model that reproduces the experimental data and may be used to predict more complicated geometries.

Rebecca Wills and Ilse Ipsen--Different Approaches to Computing PageRank

Computing PageRank scores is a major component of the Web search ranking software used by Google. The traditional algorithm determines PageRank by computing the dominant eigenvector of a Markov matrix via the Power Method. An alternate method formulates the problem of finding the PageRank vector as a solution to a linear system. To ascertain the possible advantages of viewing the computation of PageRank as a linear system problem, iterative stationary methods, such as Jacobi, Gauss-Seidel and SOR, and pre-conditioned Krylov subspace methods are applied to the linear system. Presented is a comparison of these methods.

Karen Yokley--Physiologically Based Model Development and Parameter

Nerves in the nasal cavity of rodents can be stimulated by the presence of inhaled irritants. In order to better understand how the nervous system responds to such chemicals, we have created a model to describe how the presence of irritants affects respiration in the rat. By combining and adapting two previous models, one that evaluates the relationship between inhaled acrylic acid vapor concentration and the tissue concentration in various regions of the nasal cavity and another which describes the baroreflex-feedback mechanism regulating human blood pressure, we created a system of equations that models the sensory irritant response. The adapted model focuses on the dosimetry of these reactive gases in the respiratory tract, with particular focus on the physiology of the upper respiratory tract, and on the neurological control of respiration rate due to signaling from the irritant-responsive nerves in the nasal cavity. Further, the model is evaluated and improved through optimization of particular parameters and through sensitivity analysis.

Daniela Valdez-Jasso--Developing a viscoelastic model for arteries

When deriving one-dimensional fluid dynamic models of blood flow in arteries it is necessary to include a constitutive equation. Typically such models relate cross-sectional area of the artery to blood pressure. Most previous constitutive equations are based on elastic or empirical models. However, it has long been known that arteries display viscoelatic properties. In this work we present a viscoelastic model relating blood pressure and cross-sectional area and validate this model against data obtained from pigs. Using measured pressure data as an input, we used nonlinear optimization to compute model parameters that minimized the difference between computed and measured values of the cross-sectional area. With this optimization we were able to obtain high coherence between our model and data and, furthermore, we showed that the viscoelastic model was able to predict the data significantly better than a traditional elastic model.

Vrushali A. Bokil--Parameter Identification for Dispersive Dielectrics using Acoustooptic Material Interrogation

We consider an electromagnetic interrogation technique in two dimensions for identifying the dielectric parameters of a Debye medium. In this technique a traveling acoustic pressure wave in the Debye medium is used as a virtual reflector for an interrogating microwave electromagnetic pulse that is generated in free space. The reflections of the microwave pulse from the air-Debye interface and from the acoustic pressure wave are recorded at a remote antenna. The data is used in an inverse problem to estimate the locally pressure dependent dielectric parameters of the Debye medium. We present a time domain formulation that is solved using finite differences (FDTD) in time and in space. Perfectly matched layer (PML) absorbing boundary conditions are used to absorb outgoing waves at the finite boundaries of the computational domain, preventing spurious reflections from reentering the domain. Using the method of least squares for the parameter identification problem, we compare two different algorithms (the gradient based Levenberg-Marquardt method, and the gradient free, simplex based Nelder-Mead method) in solving an inverse problem to calculate estimates for two or more dielectric parameters. Finally we use statistical error analysis to construct confidence intervals for all the presented estimates, thereby providing a probabilistic statement about the computational procedure with uncertainty aspects of estimates.

Nathan Louis Gibson--Gap Detection with Electromagnetic Terahertz Signals

We apply an inverse problem formulation to determine characteristics of a defect from a perturbed electromagnetic interrogating signal. A defect (gap) inside of a dielectric material causes a disruption, via reflections and refractions at the material interfaces, in the windowed interrogating signal. We model the electromagnetic waves inside the material with Maxwell's equations. Using simulations as forward solves, our Newton-based, iterative optimization scheme resolves the dimensions and location of the defect. Numerical results, including standard errors, will be presented, and computational issues will be addressed. Our research is supported by NASA with the ultimate goal of designing devices for damage detection for use in applications such as preventing delamination of foam on the space shuttle fuel tanks.

Chris Kees--Infinite Speed of Propagation for Some Models of Two-Phase Flow in Porous Media

In continuum models for flow of two immiscible, incompressible fluids in a porous medium, the wetting phase saturation is usually described by a doubly degenerate nonlinear advection-diffusion equation. The degeneracies in the diffusion coefficient typically give rise to finite speed of propagation: Perturbations in saturation propagate with finite speed through regions that are either fully saturated or fully unsaturated. This qualitative property of solutions is considered physically realistic since the fluids flow at a finite speed. Under certain parameter choices for capillary pressure curves of the type described by M. Th. van Genuchten, combined with the permeability model of Y. Mualem, the finite speed of propagation property is lost, despite the fact that the equation has degenerate diffusion. We present analytical and numerical results demonstrating the loss of finite speed of propagation.

Amy N. Langville and Carl Meyer--Text Mining using the Nonnegative Matrix Factorization

Text mining usually begins by using a low rank factorization of the familiar term-by-document matrix to reduce linguistic noise and reveal hidden patterns. One of the most common low rank approximations is the singular value decomposition (SVD), which factors the nonnegative term-by-document matrix in factors that are mixed in sign. A newer technique, the nonnegative matrix factorization (NMF), instead factors the term-by-document matrix into nonnegative factors, which provide several advantages over the SVD factors. We discuss popular nonlinear optimization algorithms for computing the NMF and use several examples to demonstrate the advantages of this new factorization.

Hoan K. Nguyen--Sensitivity with respect to Probability Densities in Inverse Problems

We consider general nonlinear dynamical systems in a Banach space with dependence on parameters in a second Banach space. An abstract theoretical framework for sensitivity equations is developed. An application to measure dependent delay differential systems arising in a class of HIV models is presented.

Shuhua Hu--Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations

A least-squares technique is developed for identifying unknown parameters in a coupled system of nonlinear size-structured populations. Convergence results for the parameter estimation technique are established. Ample numerical simulations and statistical evidence are provided to demonstrate the feasibility of this approach.

Alina Chertock--Hybrid Finite-Volume Particle Method for Stiff Detonation Waves

We propose a hybrid finite-volume-particle method for hyperbolic conservation laws with stiff source terms. Such problems arise, among many other applications, in modeling chemical reactive flows, in which the reaction is fast and the time scale associated with the stiff reaction term is much smaller than that associated with the fluid advection. The main difficulty in numerically solving such equations is obtaining the correct propagation speed of detonation waves in the case of underresolved computations. Since the numerical dissipation in the computation of the mass fraction of the burnt/unburnt species may lead to an incorrect speed, a natural idea is to evolve the mass fraction field with a nonviscous numerical method. Our approach consists of two steps: solving the gas dynamics part of the system by a finite-volume method and applying a particle method to the mass fraction transport equation with the stiff reaction term. By doing so, we take an advantage of low dissipation and mesh-free feature of particle methods and demonstrate that applying particle methods to such models helps to accurately capture detonation waves even when a problem with a complicated nonlinear wave interactions is being considered.

Jean-Pierre Fouque--Introduction to Financial Mathematics and Volatility Modeling

Basic notions of risky asset, option, replicating portfolio, no-arbitrage, and risk-neutral pricing will be introduced on a very simple one-period example. We will then briefly look at multi-period tree models before introducing the continuous time geometric Brownian motion model and the associated stochastic calculus. The famous Black-Scholes PDE will be derived by a no-arbitrage self-financing replicating portfolio argument. The Black-Scholes formula will also be explained in terms of risk-neutral expectations. Still in the context of constant volatility more complicated derivative contracts will be presented. In the second part of the mini-course we will explain why varying volatilities are considered in order to match return distributions and observed option prices. Local volatility and stochastic volatility models with their associated mathematical challenges will be discussed. Finally in the last part, models for other markets, fixed income and credit in particular, will be presented.

Robert E. Funderlic--The Mathematics of k-Modes, a Qualitative k-Means Like Clustering Method

Spherical k-means takes the normalized mean of a normalized cluster of vectors as its representative vector, whereas k-modes takes its mode vector. Spherical k-means yields a rich matrix algebra context as well as its set context. There are several analogies for non-numerical categorical vectors to normalized numerical vectors. The comparison of spherical k-means and k-modes is made in the environment of a word-document matrix and a matrix that comes from requests for the manufacturing of cabinets of electronic boards.

Tim Kelley--Optimal Design of Groundwater Remediation Systems with Sampling Methods

In this talk we discuss a class of problems in optimal design of subsurface remediation and flow control systems. The objective functions and constraints are constructed from PDE solvers for the flow and species transport equations. We show how to realize the problems with well-known production codes, how these codes and the underlying physics present obstacles to the optimization algorithms, and how sampling methods, in particular implicit filtering, can overcome these problems. The objective functions and constraints in these problems are typically non-smooth in all the design parameters and discontinuous in some of them. We will discuss the physical reasons for these properties and how optimization methods can deal with them.

Michael Shearer and Rachel Levy--Waves in Thin Liquid Films

The lubrication approximation for the slow flow of a thin liquid film leads to a single partial differential equation (known as the thin film equation) for the height of the free surface. However, when the motion is driven by a nonuniform distribution of surfactant on the free surface, an equation for the surfactant concentration is coupled to the thin film equation. The coupled system has several important parameters, and we shall show how a collection of self similar solutions depends on some of these. Of primary interest is the limiting system, in which surface tension is neglected. The resulting equations sustain discontinuities in the surface height, and corresponding discontinuities in surfactant concentration. In this preliminary report, we show some numerical simulations and discuss their connection with a family of discontinuous solutions and a family of smooth traveling waves.