Reception and poster session
Friday, February 4, 2005 - 4:15pm - 5:30pm
- Discrete network approximation for highly-packed composites with irregular geometry in three dimensions
Yuliya Gorb (The Pennsylvania State University)
In this poster, a discrete network approximation to the problem of the effective conductivity of a high contrast, densely packed composite in three dimensions is introduced. The inclusions are irregularly (randomly) distributed in a host medium. For this class of arrays of inclusions a discrete network approximation for effective conductivity is derived and a priori error estimates are obtained. A variational duality approach is used to provide a rigorous mathematical justification for the approximation and its error estimate.
- Rigorous numerical computations in complex dynamical systems
Suzanne (Hruska) Boyd (Indiana University)
We demonstrate our work in establishing rigorously, via controlled computer
arithmetic, certain phenomena of interest in discrete dynamical systems of two complex variables.
In particular, we study the family of Henon Mappings f(x,y) = (x^2+c-ay, x), first studied by the
Astronomer Henon in the late 1960s, which shares some qualitative similarities to the famed Lorenz
differential equations. This family of maps has been widely studied as a diffeomorphism of two
real variables, and has a rich variety of chaotic behavior. We extend to consider x,y complex
variables, and a,c complex parameters, with the goal of using the extra tools and structure
provided by complex analysis to gain insights about the real system contained in the complex
- Uniform convergence of a multigrid energy-based quantization scheme
Maria Emelianenko (The Pennsylvania State University)
We propose a new multigrid quantization scheme in a nonlinear
energy-based optimization setting. The problem of constructing an
optimal vector quantizer based on the Centroidal Voronoi
Tesselation is nonlinear in nature and hence cannot in general be
analyzed using standard linear multigrid approach. We try to
overcome this difficulty by essentially relying on the energy
minimization. Since the energy functional is in general
non-convex, a dynamic nonlinear preconditioner is proposed to
relate our problem to a sequence of convex optimization problems.
In the case of the one-dimensional problem, we have shown that for
a large class of density functions, the nonlinear multigrid
algorithm enjoys uniform convergence properties independent of
k, the problem size, thus a significant speedup comparing to the
traditional Lloyd-Max iteration is achieved. We show some results
of numerical experiments and discuss analytical extensions of our
theoretical framework to higher dimensions.
- Decoding Algebraic Geometric Codes over Rings
Katherine Bartley (University of Nebraska)
Many techniques of algebraic geometry have been applied to study of linear codes over finite fields, beginning with the definition of algebraic geometry codes by Goppa in 1977. In 1996 Walker defined algebraic geometric codes over rings after it had been shown that certain nonlinear binary codes are nonlinear projections of liner codes over Z/4.
Many algorithms have been developed for the efficient decoding of algebraic-geometric codes over fields. We will show that we can modify the 'Basic Algorithm' to decode algebraic geometric codes over rings with respect to the Hamming distance. We would also like to find a decoding algorithm that decodes algebraic geometric codes over rings with respect to the squared Euclidean distance.
- A story about Yangians
Natalie Rojkovskaia (University of Wisconsin, Madison)
We describe the connection between some remarkable matrices with non-commutative coefficients and the quantum groups Yangians.
- New perspective for simulating incompressible fluid flows with free boundary
Giovanna Guidoboni (University of Houston)
The investigation of a fast way of performing numerical simulation of fluid flow
with free boundary is motivated by many applications in sciences. The main difficulty lies in the
fact that the computational domain is not given a priori but it is another unknown of the problem.
Taking advantage of operator splitting techniques, we have been able to avoid the iteration
between the solution of the fluid flow and the position of the boundary at each time step and as a
consequence our solver is very simple and fast.
- Numerical analysis of the Exponential Euler method and its suitability for dynamic
Maeve McCarthy (Murray State University)
Real-time systems are frequently used as an experimental tool, whereby simulated
models interact in real-time with neurophysiological experiments. The most demanding of these
techniques is known as the dynamic clamp, where simulated ion channel conductances are
artificially injected into a neuron via intracellular electrodes for measurement and stimulation.
Methodologies for implementing the numerical integration of the gating variables in real-time
typically employ first-order numerical methods, either Euler (E) or Exponential Euler (EE). EE is
often used for rapidly integrating ion channel gating variables. We find via simulation studies
that for small time-steps, both methods are comparable, but at larger time-steps, EE performs
worse than Euler. We derive error bounds for both methods, and find that the error can be
characterized in terms of two ratios: time-step over time-constant, and voltage measurement error
over the slope-factor of the steady-state activation curve of the voltage-dependent gating
variable. These ratios reliably bound the simulation error and yield results consistent with the
simulation analysis. Our bounds quantitatively illustrate how measurement error restricts the
accuracy that can be obtained by using smaller step-sizes. Finally, we demonstrate that Euler can
be computed with identical computational efficiency as EE.
- On efficient high-order schemes for acoustic waveform simulation
Hyeona Lim (Mississippi State University)
We present new high-order implicit time-stepping schemes for the numerical solution of the acoustic wave equation, as a variant of the conventional modified equation method. For an efficient simulation, the schemes incorporate a locally one-dimensional (LOD) procedure having the fourth-order splitting error. It has been observed from various experiments for 2D problems that (a) the computational cost of the implicit LOD algorithms is only about 40% higher than that of the explicit methods, for the problems of the same size, (b) the implicit LOD methods produce less dispersive solutions in heterogeneous media, and (c) their numerical stability and accuracy match well those of the explicit methods.
- Stochastic modeling of macroevolution
Lea Popovic (University of Minnesota, Twin Cities)
The use of stochastic models of evolution has been extensively applied on each level of taxonomy (species, genera, families, etc) separately. It is however desirable to ensure hierarchical consistency between them, so that the phylogenetic tree on species is consistent with the phylogenetic tree on genera containing those species. We present the fundamental model that allows for such hierarchical structure. We start with a stochastic model for evolution of species and extend it to higher taxonomic levels allowing for several different grouping schemes.We illustrate the wide range of probabilistic calculations possible
within such model: for the shape of trees at each taxonomic level, the fluctuations of population sizes at each level, etc.
- Scalable conceptual interfaces in hypre
Allison Baker (Lawrence Livermore National Laboratory)
The hypre software library provides high performance preconditioners
and solvers for massively parallel computers. For ease of use,
hypre's conceptual interfaces allow users to describe a problem in a
natural way, such as in terms of grids and stencils. In anticipation
of machines with tens or hundreds of thousands of processors, we
recently re-examined these interfaces and made substantial design
changes to improve scalability. In this poster, we describe the
challenges we faced and present solutions.
- Graph-theoretic method for the discretization of gene expression measurements
Elena Dimitrova (Virginia Polytechnic Institute and State University)
The poster introduces a method for the discretization of experimental data into a
finite number of states. While it is of interest in various fields, this method is particularly
useful in bioinformatics for reverse engineering of gene regulatory networks built from gene
expression data. Many of these applications require discrete data, but gene expression
measurements are continuous. Statistical methods for discretization are not applicable due to the
prohibitive cost of obtaining sample sets of sufficient size. We have developed a new method of
discretizing the variables of a network into the same optimal number of states while at the same
time preserving maximum information. We employ graph-theoretic method to affect the discretization
of gene expression measurements. Our C++ program takes as an input one or more time series of gene
expression data and discretizes these values into a number of states that best fits the data. The
method is being validated on a recently published computational algebra approach to the reverse
engineering of gene regulatory networks by Laubenbacher and Stigler.
- An Optimization Algorithm for the Identification of Biochemical Network Models
Martha Vera-Licona (Virginia Polytechnic Institute and State University)
An important problem in computational biology is the modeling of several types of networks, ranging from gene regulatory networks and metabolic networks to neural response networks. In [LS], Laubenbacher and Stigler presented an algorithm that takes as input time series of system measurements, including certain perturbation time series, and provides as output a discrete dynamical system over a finite field. Since functions over finite fields can always be represented by polynomial functions, one can use tools from computational algebra for this purpose. The key step in the
algorithm is an interpolation step, which leads to a model that fits the given data set exactly. Due to the fact that biological data sets tend to contain noise, the algorithm leads to over-fitting.
Here we present a genetic algorithm that optimizes the model produced by the
Laubenbacher-Stigler algorithm between model complexity and data fit. This
algorithm too uses tools from computational algebra in order to provide a computationally simple description of the mutation rules.
We describe applications of the combined algorithm to the modeling of gene
regulatory networks, as well as a computational neuroscience project.
[LS] Laubenbacher, R. and B. Stigler, A computational algebra approach to the
reverse-engineering of gene regulatory networks, J. Theor. Biol. 229 (2004)
- A mathematical model for cell movement in tumor induced angiogenesis
Angiogenesis - proliferation of new capillaries from preexisting ones -
is a natural and complicated process. It is regulated by the interaction
between various cell types (e.g. endothelial cells (ECs), macrophages) and
factors (angiogenic promoters such as VEGF and inhibitors such as angiostatin,
extracellular matrix). It involves a series of changes in expression of genes,
enzymes, and signaling molecules in tumor cells and ECs, as well as changes
in the motility of ECs. In recent years, tumor-induced angiogenesis has
become an important field of research since it represents a crucial step
in the development of malignant tumors.
In this poster, a biologically realistic model for motile endothelial cells
is proposed. A new reaction-diffusion system is used to incorporate
the signaling mechanism in early stages of tumor angiogenesis
(signal transduction as well as cell-cell signaling). The ECs are being
modeled as deformable viscoelastic ellipsoids. We present preliminary results
that mimic the experiments done in endothelial cell cultures placed on
Matrigel film. Also, the model gives further insides into the aggregation
patterns by investigating factors that influence stream formation.
- The Numerical Solution of Linear Quadratic Optimal Control Problems by Time-Domain Decomposition
Agata Comas (Rice University)
Optimal control problems governed by time--dependent partial differential equations (PDEs) lead to large-scale optimization problems. While a single PDE can be solved marching forward in time, the optimality system for time-dependent PDE constrained optimization problems introduces a strong coupling in time of the governing PDE, the so-called adjoint PDE, which has to be solved backward in time, and the gradient equation. This coupling in time introduces huge storage requirements for solution algorithms. We study a time-domain decomposition based method that addresses the problem of storage and additionally introduces parallelism into the optimization algorithm. The method reformulates the original problem as an equivalent optimization problem using ideas from multiple shooting methods for PDEs. For convex linear--quadratic problems, the optimality conditions of the reformulated problems lead to a linear system in state and adjoint variables at time--domain interfaces and in the original control variables. This linear system is solved using a preconditioned Krylov subspace method.
- The effect of gravity modulation on the onset of filtrational convection
Natalya Popova (University of Illinois, Chicago)
The effect of vertical harmonic oscillations on the onset of convection in an
infinite horizontal layer of fluid saturating a porous medium is investigated. Constant
temperature distribution is assigned on the rigid impermeable boundaries. The mathematical model
is described by equations of filtrational convection in the Darcy-Oberbeck-Boussinesq
approximation. Linear analysis of the stability of the quasi-equilibrium state is performed by
Floquet method. Employment of the continued fractions method allows derivation of the dispersion
equation for the Floquet exponent in the explicit form. The Floquet spectrum is investigated
analytically and numerically for different values of oscillation frequency and amplitude, and the
Rayleigh number. The neutral curves of the Rayleigh number as a function of the horizontal wave
number are constructed for the synchronous and subharmonic resonant modes. The regions of
parametric instability contoured by these neutral curves are investigated under different values
of oscillation frequency and amplitude. Asymptotes for the neutral curves are constructed for the
case of high frequency using the method of averaging and, for the case of low frequency, using the
WKB method. Analytical, asymptotic and numerical investigation of the system indicates that
vertical vibration can be used to control convective instability in a layer of fluid saturating a
- Multifidelity optimization using asynchronous parallel pattern search and space mapping
Genetha Gray (Sandia National Laboratories)
We present a new method designed to improve optimization efficiency using
interactions between multifidelity models. It optimizes a high fidelity model
over a reduced design space using a direct search algorithm and a specialized
oracle. The oracle employs a space mapping technique to map the design space
of this high fidelity model to that of a computationally cheaper low fidelity
model. Then, in the low fidelity space, an optimum is obtained using gradient
based optimization and is mapped back to the high fidelity space. We will
review our algorithm, discuss the suitability of APPSPACK for multifidelity optimization, and
present some preliminary results.
- A convergence analysis of generalized iterative methods in finite-dimensional lattice-normed spaces
Elena Nagaeva (NONE)
This poster introduces a lattice-normed space approach to study convergence of iterative methods for solving systems of nonlinear operatorequations. Systems of nonlinear operator equations appear in various fields of
applied science, e.g. magnetohydrodynamics. A numerical solution of such a system is a multidimensional real vector, which is formed of several
subvectors. Each subvector corresponds to a certain physical quantity of the
problem in hand (pressure, temperature, etc.). We formulate local and semilocal convergence conditions for generalized two-step iterative methods in finite-dimensional lattice-normed spaces. Using the lattice-normed space
approach makes it possible to determine the convergence domain for each physical quantity of the problem separately.
- Evans function for periodic waves in infinite cylindrical domain
Myunghyun Oh (The Ohio State University)
n infinite dimensional Evans function theory is developed for the elliptic eigenvalue problem. We consider an elliptic equation with periodic boundary conditions and define a stability index with Evans function. The key for defining the index is exponential dichotomies for the system. This system has infinite dimensional stable and unstable spaces. We need to address the issue of how to determine Evans function if two infinite dimensional subspaces have nontrivial intersections. We use Galerkin approximation to reduce down these dimensions to finite and show persistence of dichotomies. Our work reveals a geometric criterion, the relative orientation of the linear unstable subspace, and relation to the momentum for instability of periodic waves in infinite cylindrical domain.
- A symbolic dynamical system for reconstructing repetitive DNA
Suzanne Sindi (University of Maryland)
The task of assembling a genome is a complicated lengthy process. When a genome is first published it is usually little more than a draft of the regions of the genome that can be uniquely reconstructed. The repetitive regions of the genome are much harder to assemble and are usually finished at later phases with more expensive processes. Here we describe a method for using a Symbolic Dynamical System to reconstruct sufficiently complex regions of repetitive DNA. We demonstrate the ability of our method to reconstruct repetitive DNA using only information available in the early stages of genome assembly.