# Reception and poster session

Friday, February 4, 2005 - 4:15pm - 5:30pm

Lind 400

**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

system.**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**

clamp experiments

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)

523-537.**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

using the

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

porous medium.**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.