Abstracts

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Parallel Research Symposia 1

• 2:00 pm
  
  Systems of Conservation Laws in Continuum Physics
  
  Konstantina Trivisa
  Department of Mathematics
  University of Maryland
  College Park, MD 20742
  trivisa@math.umd.edu
  http://www.math.umd.edu/~trivisa
  
  Quasilinear systems in divergence form, commonly known as "hyperbolic conservation laws", govern a broad spectrum of physical phenomena in compressible fluid dynamics, nonlinear materials science, particle physics, semiconductors, combustion and other applied areas. Such equations admit solutions that may exhibit various kinds of shocks and other nonlinear waves, which play an significant role in multiple areas of physics. In recent years, major progress has been made in both the theoretical and the numerical aspects of this field. In this talk we present results on the qualitative behavior of solutions to hyperbolic conservation laws, in particular we explore the existence, uniqueness and asymptotic analysis of solutions.
   
• 2:30 pm
  
  Numerical Methods for Conservation Laws on Cartesian Meshes with Embedded Geometry
  
  Marsha Berger
  Courant Institute
  251 Mercer St.
  New York, NY 10012
  berger@cims.nyu.edu
  
  Cartesian mesh methods with embedded geometry can easily and robustly produce a mesh in a very complicated domain. In exchange, these methods push the difficulty of dealing with the cut cells at the boundary onto the flow solver. There is an accuracy issue since the mesh is extremely irregular at the boundary; and there is a stability issue for explicit schemes, since the cell volumes can be orders of magnitude smaller than the regular cell volumes. In this talk we discuss some of these technical issues and some proposed solutions to the problems that arise in solving time dependent systems of conservation laws.
  
  This is joint work with Michael Aftosmis and Scott Murman of NASA Ames Research Center, and Christiane Helzel from the University of Bonn.
   
• 3:00 pm
  
  A Projection-Based Approach to General Form Tikhonov Regularization
  
  Misha E. Kilmer
  113 Bromfield-Pearson Building
  Mathematics Dept.
  Tufts University
  Medford, MA 02115
  misha.kilmer@tufts.edu
  http://www.tufts.edu/~mkilme01/
  
  Linear ill-posed problems occur frequently in the physical sciences. The Fredholm equation that is typically used to mathematically describe the blurring of an image by, say, a faulty lens, is an example of a linear ill-posed problem. The term discrete ill-posed problem refers to a discretization of linear ill-posed problem. The difficulty in solving discrete ill-posed problems is that the measured data actually contains noise. Further, the matrix is typically very ill-conditioned. The ill-conditioning of the matrix, together with the noise in the data, renders the least squares solution worthless. A regularization technique must be used to lessen the effect of noise on the solution. A standard technique for approximating the true solution in the presence of noise is to solve the Tikhonov regularized problem. The quality of this solution relative to the noise-free solution depends on the value of a regularization parameter whose value is usually not known a priori. In this work we consider a projection-based variant of the well-known Tikhonov regularization method for the solution of large-scale, linear discrete ill-posed problems when the regularization operator is not the identity. Our method is appropriate when it is infeasible, from a computational point of view, to transform the regularized problem to standard form. Further, by considering the projected problem, we show how estimates of the corresponding optimal regularization parameter can be efficiently obtained. Numerical results illustrate the promise our projection-based approach.
  
  This is joint work with Per Christian Hansen and Michael Jacobsen, both of whom are affiliated with the Informatics and Mathematical Modelling Section for Numerical Analysis at the Technical University of Denmark.
   
• 4:00 pm
  
  Determining the Earth's Microstructure by Statistical Seismic Imaging
  
  Susan E. Minkoff
  Department of Mathematics and Statistics
  University of Maryland, Baltimore County
  1000 Hilltop Circle
  Baltimore, MD 21250
  sminkoff@math.umbc.edu
  http://www.math.umbc.edu/~sminkoff/
  
  The mechanical properties of the earth's interior vary over a range of spatial scales spanning many orders of magnitude. Seismic waves provide a means of probing the earth's interior, and hopefully illuminating the important statistical character of both deep crustal structures and near surface features. I will describe an extension to an operator-based upscaling technique, previously developed for elliptic flow problems, applied to finite difference solutions of the wave equation. The upscaling algorithm will be used to correlate earth model statistics with material properties.
  
  This work is joint with Oksana Korostyshevskaya and Tetyana Vdovina at UMBC as well as Bill Symes, Christian Poppeliers, and Alan Levander at Rice University.
   
• 4:30 pm
  
  Title TBA
  
  Mary Wheeler
  University of Texas at Austin
  
  Abstract TBA
   
• 5:00 pm
  
  The Identification of a Time Dependent Parameter in a Soil Column Study
  
  Maeve L. McCarthy
  Mathematics & Statistics
  Murray State University
  6C Faculty Hall
  Murray, KY 42071-3341
  maeve.mccarthy@murraystate.edu
  
  Soil column studies are used frequently in seeking to understand the behavior of a particular contaminant in a saturated homogeneous soil of a given type. The concentration of the contaminant is modelled by a parabolic partial differential equation. We seek to identify the sorption partitioning coefficient as a function of time from limited boundary data. We discuss two approaches to the problem. The first is an output least squares formulation with Tikhonov regularization. This method can be used to explicitly characterize the unknown partitioning coefficient in terms of a co-state equation. However, computational issues arise in the solution of the co-state equation and its use in the solution of the inverse problem. In the second approach, we use a mass balance law to determine the initial value of the partitioning coefficient. A particular discretization of the pde allows us to develop a time marching scheme to solve for the unknown partitioning coefficient.
  
  This is joint work with K. Renee Fister (Murray State University) and Seth F. Oppenheimer (Mississippi State University).
   

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Parallel Research Symposia 2

• 2:00 pm
  
  Global Analysis of the Forced van der Pol Equation: The Slow Flow and Canard Solutions
  
  Kathleen Hoffman
  Dept. of Math. and Stat.
  UMBC
  1000 Hilltop Circle
  Baltimore MD 21250
  khoffman@math.umbc.edu
  http://www.math.umbc.edu/~khoffman/
  
  The global bifurcation structure of multi-timescale systems is poorly understood despite the pervasiveness of these systems throughout science and engineering. The forced van der Pol oscillator serves as a classic example of a multi-timescale system as well as the basis for many models in neurobiology. I will discuss a hybrid system that approximates the forced van der Pol equation. The hybrid system consists of the trajectories that follow the ``slow flow'' and ``jumps'' to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system. In addition, I will describe a modified hybrid system that includes the canard solutions and briefly describe the additional bifurcations that occur in this extended map.
  
  This is joint work with John Guckenheimer from Cornell University, Warren Weckesser at Colgate University.
   
• 2:30 pm
  
  Integrative Models of Swimming Organisms
  
  Lisa J. Fauci
  Tulane University
  Department of Mathematics
  New Orleans, LA 70118
  fauci@tulane.edu
  http://johann.math.tulane.edu/~ljf/
  
  Problems in biological fluid dynamics typically involve the interaction of an elastic structure with the surrounding fluid. Motile spermatozoa in the reproductive tract and swimming leeches are examples of such fluid-structure interactions. We will describe a unified computational approach, based upon an immersed boundary framework, that couples internal force-generating mechanisms of organisms and cells with an external, viscous, incompressible fluid. This approach can be applied to model low, moderate, and high Reynolds number fl ow regimes. We will present recent progress on coupling internal molecular motor mechanisms of beating cilia and flagella with an external fluid, as well as coupling muscle mechanics with fluid dynamics in three dimensional undulatory swimming of nematodes and leeches.
   
• 3:00 pm
  
  Measuring Noise-Sensitivity in Dynamical Systems
  
  Rachel Kuske
  University of British Columbia
  121-1984 Mathematics Road
  Vancouver, BC V6K 1T5
  Canada
  rachel@math.ubc.ca
  http://www.math.ubc.ca/~rachel/
  
  We outline asymptotic approaches for analyzing and quantifying the effects of small noise in certain types of dynamics. To illustrate the methods, we will focus on two main areas: metastable interfaces and stochastic effects in models with delay. In these systems, small noise can result in significant qualitative variation in behavior. Although very different models, they have many similarities from the viewpoint of stochastic dynamics. Generalizations to other areas will also be discussed.
  
  This is joint work with Malgorzata Klosek at NIH and Michael Ward at UBC.
   
• 4:00 pm
  
  Modeling Immunotherapy and Chemotherapy Cancer Treatments
  
  Lisette de Pillis
  Harvey Mudd College
  Department of Mathematics
  1250 N. Dartmouth Avenue
  Claremont, CA 91711
  depillis@hmc.edu
  http://www.math.hmc.edu/~depillis/
  
  We develop a mathematical model that describes tumor-immune interactions on a cell-population level. The goal of this model is to understand the dynamics of tumor growth and rejection, as well as to describe a tumor's response to immunotherapy and chemotherapy treatments. The model focuses on the interaction of the NK and CD8⁺ T cells with various tumor cell lines using a system of ordinary differential equations. Recent experimental studies have brought to light new information about the immune response to tumor in human subjects. Our mathematical model is validated by comparing computer simulations to the data provided in these studies.
  
  This is joint work with A.E. Radunskaya of Pomona College.
   
• 4:30 pm
  
  Towards the Multiscale Numerical Simulation of Biochemical Networks
  
  Linda Petzold
  Department of Computer Science
  University of California
  Santa Barbara, CA 93106
  petzold@engineering.ucsb.edu
  
  In microscopic systems formed by living cells, small numbers of reactant molecules can result in dynamical behavior that is discrete and stochastic rather than continuous and deterministic. In simulating and analyzing such behavior it is essential to employ methods that directly take into account the underlying discrete stochastic nature of the molecular events. This leads to an accurate description of the system that in many important cases is impossible to obtain through deterministic continuous modeling (e.g. ODE's). Gillespie's Stochastic Simulation Algorithm (SSA) has been widely used to treat these problems. However as a procedure that simulates every reaction event, it is prohibitively inefficient for most realistic problems. We report on our progress in developing a multiscale computational framework for the numerical simulation of chemically reacting systems, where each reaction will be treated at the appropriate scale. The framework is based on a sequence of approximations ranging from SSA at the smallest scale, through a ``birth-death'' Markov process approximation, Gillespie's recently-developed tau-leaping approximation, a continuous stochastic differential equation (SDE) approximation, and finally to the familiar reaction rate equations (ODE's) at the coarsest scales.
   
• 5:00 pm
  
  Bosonic Fock Space on Infinite-Dimensional Lie Groups
  
  Masha Gordina
  Department of Mathematics
  University of Connecticut
  Storrs, CT 06269
  gordina@math.uconn.edu
  http://www.math.uconn.edu/~gordina/
  
  We describe an infinite-dimensional analogue of different representations of a bosonic Fock space. The bosonic Fock space can be described as the state space of a system of boson particles. The transfer of results from commutative (flat) spaces to curved infinite-dimensional spaces (such as spaces of loops) is of interest in the string theory. The main ingredient of our description is construction of the Gaussian (heat kernel) measure on infinite-dimensional Lie groups. Often physicists use the ``Lebesgue'' measure on such spaces. This measure usually does not exist as a mathematical object, therefore one needs a suitable replacement. Once the measure is defined, it is possible to prove the equivalence of two representations of the bosonic Fock space, namely, as the space of holomorphic L²-functions and a space of tensors. The infinite-dimensional groups we consider are described as invertible elements of von Neumann algebras, and the heat kernel measure is defined by using stochastic differential equations on noncommutative L²-spaces. As examples we present the construction for infinite-dimensional orthogonal and symplectic groups.
   

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Parallel Research Symposia 3

• 10:00 am
  
  Multiphase Fluid Mechanics in Self-Deforming Tissues
  
  Sharon R. Lubkin
  Dept. of Mathematics
  North Carolina State University
  Box 8205
  Raleigh, NC 27695-8205
  lubkin@eos.ncsu.edu
  http://www4.ncsu.edu/~lubkin/
  
  Tissues that are growing and changing shape, whether in development, maintenance, or disease, are deformed by forces which they generate themselves. Ongoing work in our group aims to understand how tissues deform themselves. The approach involves a continuum approximation of tissue properties, and results in multiphase fluid models of various kinds. We present two problems in tissue dynamics, one in development and one in disease.
   
• 10:30 am
  
  Pattern and Paradox: Shock Interactions in the Nonlinear Wave System
  
  Barbara Lee Keyfitz
  Department of Mathematics
  University of Houston
  651 P. G. Hoffman Hall
  Houston, TX 77204-3008
  blk@math.uh.edu
  http://www.math.uh.edu/~blk/
  
  An approach to analysing hyperbolic conservation laws in several space variables is to examine two-dimensional Riemann problems. Use of self-similar coordinates reduces the problem to a system of conservation laws in two variables; however, the system now changes type, and a complete analysis requires solving unusual boundary-value problems for degenerate elliptic and degenerate hyperbolic equations, as well as free-boundary problems for such equations. Recent work is beginning to resolve some of these difficulties.
  
  This talk combines analytical and numerical results to study a bifurcation sequence of interacting shocks in the nonlinear wave system, a simplified system related to the isentropic gas dynamics equations. As the shock angle is varied, one sees transitions beginning with regular reflection and ending with von Neumann reflection.
  
  The theory is joint work with Suncica Canic and Eun Heui Kim; numerical simulations have been performed by Alexander Kurganov, Maria Lukacova, Richard Sanders and Manuel Torrilhon.
   
• 11:00 am
  
  Pattern Prediction in Frontal Polymerization
  
  Laura K. Gross
  Dept. of Theoretical and Applied Mathematics
  The University of Akron
  Akron, OH 44303-4002
  gross@math.uakron.edu
  http://www.math.uakron.edu/~gross/
  
  In free-radical frontal polymerization, a reaction front progresses through a mixture of monomer and initiator, leaving a polymer in its wake. The behavior of the front depends upon the geometry of the system, material properties, and the reaction mechanism. In stable cases, perturbations to a traveling wave die out. In unstable regimes, pulsating, spinning, and standing waves emerge. In this talk, I will explain the essential mechanism of pattern selection based on linear and nonlinear stability analyses of a moving-boundary-value problem.
  
  This is joint work with Donna Comissiong and Vladimir Volpert from Northwestern University.
   
• 11:30 am
  
  Numerical Solution of the Shallow Water Equations in Lagrangian Coordinates
  
  Jodi L. Mead
  Department of Mathematics
  Boise State University
  1910 University Dr.
  Boise, ID 83725-1555
  mead@math.boisestate.edu
  http://www.math.boisestate.edu/~mead/
  
  The Shallow Water Equations are frequently used as a model for both atmospheric and oceanographic circulation. They are a simple form of the equations of motion that describe the evolution of an incompressible fluid in response to gravitational and rotational accelerations. In this work we write these equations in Lagrangian coordinates, and the positions of fluid particles are identified for all time. Lagrangian coordinates are useful for following contaminant transport and for assimilation of Lagrangian data. They also give a well posed statement of the full Navier-Stokes equations in a region. These coordinates do no vary with time because the independent variables are the particles' fixed initial position (compare with Eulerian coordinates where the independent variables are the fixed spatial position), thus the numerical solution can be found with traditional methods. Both a low order finite difference method, and a high order Chebyshev pseudospectral method in space with a fourth order Runge-Kutta method in time, are used to solve the Lagrangian shallow water equations. The test cases include linear solutions (center, source, spiral sink) and a nonlinear solution. It will be shown that the particles' trajectories can be accurately computed by solving the Lagrangian shallow water equations for realistic temporal and spatial scales.
   

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Parallel Research Symposia 4

• 10:00 am
  
  Massively-Parallel Integer Programming
  
  Cynthia A. Phillips
  Sandia National Laboratories
  Mail Stop 1110
  P.O. Box 5800
  Albuquerque, NM 87185-1110
  caphill@sandia.gov
  http://www.cs.sandia.gov/~caphill/
  
  Integer programming is the maximization or minimization of a linear objective function subject to linear constraints with the added restriction that variables must take integer values. Integer programs (IPs) are a general way to express a huge class of discrete optimization problems such as scheduling, vehicle routing, protein folding, and network vulnerability analysis. Integer programming is NP-complete, meaning it is formally intractible. However, it is the workhorse method for solving hard resource allocation problems in practice. Also, the linear-programming relaxation of an IP, obtained by relaxing the integrality constraints, is solvable. In this talk I'll describe the techniques for massively-parallel solution of integer programs that are implemented in the PICO (Parallel Integer and Combinatorial Optimizer) massively-parallel integer programming solver. At the highest level PICO uses a classic approach, (approximately) solving IPs using intelligent search based on branch-and-bound and branch-and-cut (constraint generation). I will describe various sources of parallelism in IP, beyond the obvious subproblem parallelism, including parallel gradient evalution, parallel bounding, parallel model improvement, and parallel cooperative search. I will also describe how to use linear-programming relaxations to efficiently compute provably-good approximate solutions to integer programs.
  
  This is joint work with Jonathan Eckstein of Rutgers University and William Hart of Sandia National Laboratories.
   
• 10:30 am
  
  Ferment in Direct Search Methods
  
  Margaret H. Wright
  Computer Science Department
  Courant Institute, New York University
  251 Mercer Street
  New York, NY 10012
  mhw@cs.nyu.edu
  
  Direct search methods, which optimize without derivatives, have come a long way from their once-scorned state of 25 years ago. Recent research has given rise to interesting and even contentious issues, ranging from the nature of convergence proofs and the associated assumptions to the proper role of these methods in modern optimization practice. This talk will examine the current status and implications of selected questions about direct search methods.
   
• 11:00 am
  
  Optimal Control Applied to Cell-Kill Strategies
  
  K. Renee Fister
  6C-7 Faculty Hall
  Department of Mathematics and Statistics
  Murray State University
  Murray, KY 42071
  renee.fister@murraystate.edu
  
  Optimal control techniques are used to develop optimal strategies for chemotherapy. In particular, we investigate the qualitative differences between three different cell-kill models: log-kill hypothesis (cell-kill is proportional to mass); Norton-Simon hypothesis (cell-kill is proportional to growth rate); and, E_max hypothesis (cell-kill is proportional to a saturable function of mass). For each hypothesis, an optimal drug strategy is characterized that minimizes the cancer mass and the cost (in terms of total amount of drug). The cost of the drug is nonlinearly defined in one objective functional and linearly defined in the other. Existence and uniqueness for the optimal control problems are analyzed. Each of the optimality systems, which consists of the state system coupled with the adjoint system, is characterized. Finally, numerical results show that there are qualitatively different treatment schemes for each model studied. In particular, the log-kill hypothesis requires less drug compared to the Norton-Simon hypothesis to reduce the cancer an equivalent amount over the treatment interval. Therefore, understanding the dynamics of cell-kill for specific treatments is of great importance when developing optimal treatment strategies.
  
  This is joint work with J. Carl Panetta from St. Jude Children's Research Hospital in Memphis, Tennessee.
   
• 11:30 am
  
  Systems Hard to Control: an Adaptive Approach
  
  Naira Hovakimyan
  Department of Aerospace and Ocean Engineering
  Virginia Polytechnic Institute and State University
  Blacksburg, VA 24061
  nhovakim@vt.edu
  
  Linear systems theory provides a firm foundation for design tools in optimal control and estimation problems, which constitute the basis for many of the synthesis approaches up today. This has to do with the fact that most of the physical systems are nearly linear over some operating range, and as a consequence the performance of linear based designs is often satisfactory, particularly when these designs are extended by gain scheduling, or by performing recursive linearizations. The problems of nonlinear estimation and control have been therefore predominantly considered in the framework of "extending" linear designs from one operational point to another. Such an approach works as long as the time variation of the state variables upon which the nonlinearities depend is slow in comparison to the evolution of regulated outputs and measurements. The presentation will give a brief overview of observer/controller design problems for nonlinear systems from the perspective of augmenting linear designs. In both problems, an adaptive element is added to the nominal design to provide a correction that accounts for modeling errors. The error signal for adaptive laws is generated through a linear observer of the nominal system's error dynamics. Ultimate boundedness of error signals is proven through Lyapunov's direct method. Several benchmark problems will illustrate the theoretical results. The future research avenues will summarize the presentation.
   

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Parallel Research Symposia 5

• 2:30 pm
  
  Issues in High-Performance Numerical Libraries
  
  Lois Curfman McInnes
  Mathematics and Computer Science Division
  Argonne National Laboratory
  9700 South Cass Avenue
  Argonne, IL 60439-4844
  mcinnes@mcs.anl.gov
  http://www.mcs.anl.gov/~mcinnes
  
  Computational scientists have benefited from the encapsulation of expertise in numerical libraries for many years. However, the complexity and scale of today's high-fidelity, multidisciplinary scientific simulations imply that development work must often be leveraged over many groups, with numerous software updates during an application's lifetime to address algorithmic advances and changing computational environments. This presentation will discuss an effective approach for addressing these challenges in parallel numerical software for nonlinear PDEs and optimization, namely the development of component-based numerical libraries. We will introduce the ideas of the Common Component Architecture (CCA) Forum (http://www.cca-forum.org) for building component interfaces and implementations specifically targeted at high-performance scientific computing. We will also illustrate this approach with numerical applications that employ the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO).
   
• 3:00 pm
  
  Blending Systems and Optimal Alphabetic Trees
  
  Maria Klawe
  Dean of Engineering and Applied Science
  Princeton University
  Princeton, NJ 08544
  klawe@princeton.edu
  
  The question of whether there is a linear time algorithm for the construction of optimal alphabetic binary trees is a long-standing open problem in theoretical computer science. The fastest known algorithm for the general case, due to Hu and Tucker, runs in O(n log n) time and is over thirty years old. This talk presents a new approach to a key sub-problem occurring in the construction of optimal trees. The new approach yields a linear time algorithm for a wider subclass of cases than previously known.
   
• 3:30 pm
  
  Load Balancing for Emerging Applications
  
  Karen Devine
  Sandia National Laboratories
  PO Box 5800
  Albuquerque, NM 87185-1111
  kddevin@sandia.gov
  http://www.cs.sandia.gov/~kddevin/
  
  With the creation of robust graph-partitioning software, many people felt load balancing was a "solved problem." Indeed, graph partitioners have excelled in enabling scalability in mesh-based PDE applications. However, in many other application areas (e.g., circuit, biology, contact, and nanotechnology simulations), straightforward graph partitioning is not the entire answer. Unlike mesh-based PDE applications, whose square, symmetric relationships are well represented by graphs, these emerging applications are characterized by non-homogeneous topologies, highly connected systems, geometric relationships, and non-square, non-symmetric data dependencies. In this talk, we look at partitioning models and techniques that can improve the performance and scalability of these new applications.
  
  This is joint work with Erik Boman, Robert Heaphy, and Bruce Hendrickson (Sandia National Laboratories) and Robert Preis (U. Paderborn).
   
• 4:30 pm
  
  Solution Methods for Large-Scale Nonlinear Diffusion Problems
  
  Carol S. Woodward
  Center for Applied Scientific Computing
  Lawrence Livermore National Laboratory
  P.O. Box 808, L-561
  Livermore, CA 94551
  cswoodward@llnl.gov
  http://www.llnl.gov/CASC/people/woodward
  
  Many mathematical models found in DOE applications result in nonlinear diffusion problems. In the Center for Applied Scientific Computing at Lawrence Livermore National Laboratory, we have been researching various methods for solving the nonlinear systems arising at each time step in an implicit integration of these problems. In this talk, I will overview our work on solution methods for these problems. In particular, I will discuss a novel but small twist on Newton-Krylov methods which may help reduce time required for each iteration, a scalability theory for Newton-multigrid methods, some developments in extensions of algebraic multigrid methods to nonlinear problems, and the application of a two-grid method which transfers the nonlinearity to a coarse grid leaving only a linear system on the original (fine) grid. Where possible, numerical results will be given showing how these methods work on nonlinear diffusion problems found in radiative transport and groundwater flow.
  
  Various parts of this work were done in collaboration with Peter Brown (LLNL), Homer Walker (WPI), Panayot Vassilevski (LLNL), Miguel Dumett (USC), and Todd Coffey (SNL).
  
  This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Release number: UCRL-200025-ABS.
   
• 5:00 pm
  
  Convergence to Equilibrium in Nonlocal Dispersal
  
  Gwendolen Hines
  Department of Mathematics
  University of Nebraska
  Lincoln, NE 68588-0323
  ghines@math.unl.edu
  
  In the modeling of dispersal of organisms and genes, it is often assumed that diffusion is the main mechanism for dispersal, and hence reaction-diffusion equations are commonly used. In recent literature however, it has been suggested that diffusion might not always be a very accurate model of dispersal, and more general models have been suggested. In this talk, we discuss a model where the dispersal is given by a nonlocal convolution term and we compare the long-term dynamics of the reaction-diffusion model with those of the nonlocal model. It is well known that in the reaction-diffusion model, the population converges to a constant distribution, but we will prove that this is not the case for the nonlocal model, provided the dispersal rate is fairly small.
  
  This is joint work with Michael Grinfeld, Vivian Hutson, Konstantin Mischaikow and Glenn Vickers.
   
• 5:30 pm
  
  Impacts of Spatial Heterogeneity on Plague
  
  Holly Gaff
  Dynamics Technology, Inc. 1555 Wilson Boulevard, Suite 700 Arlington, Virginia 22209
  hgaff@dynatec.com
  
  Plague is often thought of as an extinct disease, but plague is a naturally occurring disease today that is also one of the top biowarfare threats. Plague, caused by Yersinia pestis, is a complex vector-borne disease involving fleas, rodents and humans. We developed a spatially-explicit mathematical model by extending the standard epidemiological SEIR model to a landscape of patches. We use this model to evaluate how the disease spreads across a landscape naturally as well as from potential biological attacks. We also analyze the effectiveness of different control measures, such as quarantine and culling, in an attempt to understand how spatial patterns can be used most effectively to stop the spread of the disease. Finally, we are beginning apply this model, which was developed in a general mathematical framework, to other diseases.
   

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Parallel Research Symposia 6

• 2:30 pm
  
  Mesh Quality Improvement
  
  Lori Freitag Diachin
  Center for Advanced Scientific Computing
  Lawrence Livermore National Laboratory
  Livermore, CA 94551
  diachin1@llnl.gov
  
  In this presentation, I give an overview of the mesh quality improvement problem. Mesh quality is well known to affect both the solution time and accuracy of numerical solutions to PDE-based applications. I discuss my research in this area and focus on several aspects of the problem. First, I show the relationship between mesh quality and solution efficiency and present several examples that clearly demonstrate the need for mesh quality improvement algorithms. Second, I discuss an optimization-based approach to the mesh quality improvement problem that focuses on improving the worst quality elements. I give several examples of its use to improve and untangle simplicial meshes and discuss strategies that combine this method with related approaches designed to improve average element quality. Finally, I describe the Mesquite mesh quality improvement toolkit, a software project designed to incorporate state-of-the-art quality metrics, objective functions and algorithms into one, easily-accessible software library.
  
  Various aspects of this talk are the result of joint work with the following collaborators: Patrick Knupp from Sandia National Laboratories, Carl Ollivier-Gooch at the University of British Columbia, and Paul Plassmann at the Pennsylvania State University.
   
• 3:00 pm
  
  Creating a Medical Diagnostic Tool using Tissue Stiffness Properties
  
  Joyce R. McLaughlin
  Rensselaer Polytechnic Institute
  Math Department A. E. 415
  110 8th Street
  Troy, New York 12180
  mclauj@rpi.edu
  
  An impulse is imparted on the surface of the body initiating an elastic wave. An ultrafast ultrasound imaging system, developed by Mathias Fink, measures the space and time dependent downward component of the displacement of this wave as it propagates in the interior of the body. This talk will show how to use the propagating front of this wave to create an image of shear wavespeed variations. Shear wavespeed is chosen since it increases 2-4 times in abnormal tissue.
   
• 3:30 pm
  
  Optimal Finite Difference Grids for Targeted Approximations
  
  Shari Moskow
  358 Little Hall
  University of Flordia
  P.O. Box 118105
  Gainesville, FL 32611-8105
  moskow@math.ufl.edu
  http://www.math.ufl.edu/~moskow
  
  Most numerical methods for PDEs are intended to give equally good results at all points on the domain. However, in geophysical simulations, for example, one really only cares about the accuracy of the solution at prespecified receiver locations. Here we describe an optimal grid approach which is based on rational approximation. We show how it can be used for a variety of physical problems.
  
  This is joint work with Vladimir Druskin of Schlumberger-Doll Research.
   
• 4:30 pm
  
  An Algebraic Multigrid Solver for Meshfree Discretizations
  
  Suely Oliveira
  Department of Computer Science
  The University of Iowa
  14 MLH
  Iowa City, IA 52246
  oliveira@cs.uiowa.edu
  http://www.cs.uiowa.edu/~oliveira/
  
  Meshfree discretizations construct approximate solutions to partial differential equation based on particles, not on meshes, so that it is well suited to solve the problems on irregular domains. Since the nodal basis property is not satisfied in meshfree discretizations, it is difficult to handle essential boundary conditions. We employ the Lagrange multiplier approach to solve this problem, but this will result in an indefinite linear system. As a solver for this system, we propose a new Algebraic Multigrid (AMG) method which is a modification of aggregation which uses neighborhood and boundary information. We verify the effectiveness of our new AMG method by theoretical and numerical results.
   
• 5:00 pm
  
  Hierarchical Matrices and Iterative Solvers
  
  Sabine Le Borne
  Department of Mathematics
  Box 5054
  Tennessee Technological University
  Cookeville, TN 38505
  sleborne@tntech.edu
  http://math.tntech.edu/~sleborne/
  
  Hierarchical matrices (H-matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method (BEM) and finite element method (FEM) applications. In the latter case, it is the inverse stiffness matrix that is fully populated and approximated by an H-matrix. The approximation of a dense matrix by an H-matrix is, in short, a consequence of the exponential convergence of a separable function to certain parts G(x,y)|_{w1 x w2} of Green's function. The significance of H-matrices lies in their (nearly) optimal storage and computational complexities O(n log^α n) which are required not only for the matrix-vector multiplication but also, which distinguishes the H-matrix technique from related methods such as panel clustering or multipole methods, for matrix-matrix multiplication and (approximate) matrix inversion. Given a matrix A and a vector b, we compute an (explicit!) H-matrix approximant (in nearly linear time) to the inverse A^-1 which will be used as a preconditioner/smoother in an iteration to find the solution x = A^-1b. The importance of such fast techniques is justified with today's large scale applications - the faster the computers, the stronger becomes the need for such (nearly) optimal solution algorithms. In this talk, we will introduce the construction of H-matrices in the standard setting together with some complexity and approximation properties. We will then demonstrate the shortcomings of applying the standard H-matrix technique to the singularly perturbed convection-diffusion equation. We will develop modifications of the standard H-matrix construction that depend on the convection direction which yield significantly improved H-matrix approximants. We will use the resulting H-matrix approximants as preconditioners in iterative methods and provide numerical illustrations.
   
• 5:30 pm
  
  Representability of Rank 3 Combinatorial Geometries
  
  Sandra Kingan
  Department of Mathematical and Computer Sciences
  Penn State Harrisburg
  Middletown, PA 17057
  srkingan@psu.edu
  http://cs.hbg.psu.edu/~srk1
  
  A rank 3 combinatorial geometry (simple matroid) consists of a set of points and a family of distinguished subsets of points called lines that satisfy two properties: any two distinct points belong to exactly one line; and any line has at least two distinct points. A geometry is said to be representable over a finite field GF(q), where q is a power of a prime, if there is a rank preserving map from the elements of the geometry to a matrix with entries from GF(q). We show how to use automated matroid generation to study the representability problem for rank 3 geometries. We verify that up to 9 elements all except four rank 3 geometries are representable and the bound on the order of the field is 11.
   

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