Natalia M. Alexandrov (Systems Analysis and Concepts Directorate , NASA Langley Research Center) http://mdob.larc.nasa.gov/staff/natalia/Personal_info/personal.html
NASA is an unending source of spectacularly interesting problems for an applied mathematician. Although it has traditionally been an "engineering shop," in recent years the growing complexity of goals and the ever increasing computational power clearly necessitate the development of sophisticated computational models and rigorous numerical procedures, thus providing an opportunity for a closer collaboration between NASA engineers, scientists and applied mathematicians. I will give an overview of some interesting problems in modeling and design, as well as some ideas of working for and with NASA.
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.
Decoding algebraic geometric codes over rings
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.
Margaret Cheney (Department of Mathematical Sciences, Rensselaer Polytechnic Institute) http://www.rpi.edu/~cheney/
This talk will survey some of the mathematical ideas behind the formation of high-resolution images from radar data, and will outline some of the open problems in the field.
The numerical solution of linear quadratic optimal control problems by time-domain decomposition
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.
We study two preconditioners. The first is a block Gauss-Seidel preconditioner for a suitable permutation of the optimality system. Unfortunately, the Gauss-Seidel preconditioners that work well in terms of reduction in the number of iterations do not parallelize. This has motivated our second preconditioner, which is based on an approximate factorization of the optimality system and has been used by Biros, et. al.(1999) in another context. It requires approximate state and adjoint solves as well as a preconditioner for the so-called reduced Hessian. We approximate state and adjoint solves using the parareal algorithm of Maday, et. al.(2001) and present new results on the spectrum of the reduced Hessian. We illustrate the performance of our preconditioners on some model problems.
Brenda L. Dietrich (Department Manager, Mathematical Sciences, IBM Thomas J. Watson Research Center) http://www.research.ibm.com/people/d/dietric
Math inside IBM
In this talk I will discuss several IBM Research projects in which advanced mathematics is used to dramatically improve IBM products and processes. Examples include product design, manufacturing process design, and supply chain operations. I will also discuss ways in which our ability to deploy mathematics, by embedding the math in automated processes or tools, has dramatically improved in the past 20 years.
Graph-theoretic method for the discretization of gene
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.
Uniform convergence of a multigrid energy-based quantization
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.
Complex fluid systems in nanotechnology, biology, and life
Complex fluids are ubiquitous in nanoscale materials, at interfaces, and in biology. They are typically modeled with either molecular simulation or molecular theory approaches. Our research has emphasized implementation of large scale algorithms for density functional theory based approaches to these problems. In density functional theories a free energy functional is minimized to determine an optimal solution. It turns out that many women also find their lives to be complex fluid systems that require daily optimization around the constraints of the career, their home, and their families. This seminar will present briefly the content of one applied math career in the context of a national lab, and also discuss how the work-family balance can be achieved in this setting.
Discrete network approximation for highly-packed composites
with irregular geometry in three dimensions
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.
Multifidelity optimization using asynchronous parallel pattern search and space mapping
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.
New perspective for simulating incompressible fluid flows with free boundary
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.
Rigorous numerical computations in complex dynamical systems
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) = (x2+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.
Women mathematicians: We can do more than teach
How many times has someone asked you what your degree is in and when you respond, "Math," they ask, "Oh, do you teach?" While teaching is a noble profession, it is not for everyone. There are other career options for women in the mathematical sciences. I will describe career options that I stumbled upon while job searching during the last phases as a graduate student in applied mathematics, the path I chose, and a brief sampling of some of the research in which I am currently involved.
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.
[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.
On efficient high-order schemes for acoustic waveform simulation
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.
Becoming an applied mathematician - From mathematical logic to airplanes
I will describe briefly some the projects I worked on during my Boeing career, emphasizing the role of a mathematician in a manufacturing environment. In this context I will discuss the advantages and drawbacks of working in industry and offer some practical advice to mathematicians at the beginning of their careers.
Maeve L. McCarthy (Mathematics & Statistics, Murray State University) http://campus.murraystate.edu/academic/faculty/maeve.mccarthy/
Numerical analysis of the Exponential Euler method and its suitability for dynamic clamp experiments
Numerical analysis of the Exponential Euler method and its suitability for dynamic clamp experiments posterabstract: 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.
A convergence analysis of generalized iterative methods in finite-dimensional lattice-normed spaces
This poster introduces a lattice-normed space approach to study convergence of iterative methods for solving systems of nonlinear operator equations. 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.
Myunghyun Oh (Department of Mathematics, The Ohio State University ) http://www.math.ohio-state.edu/~myoh/
Evans function for periodic waves in infinite cylindrical domain
An 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.
The effect of gravity modulation on the onset of filtrational convection
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.
Stochastic modeling of macroevolution
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.
A story about Yangians
We describe the connection between some remarkable matrices with non-commutative coefficients and the quantum groups Yangians.
A symbolic dynamical system for reconstructing repetitive DNA
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.
A mathematical model for cell movement in tumor induced angiogenesis
Abstract. 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.
Building a career at the NSA
If you are looking for an environment where you can bring your mathematical background and talents to bear on problems that really make a difference, then the National Security Agency (NSA) could be the place for you. In this talk I will describe our training programs for new mathematicians, the many ways in which mathematics comes into play at the NSA, some of the opportunities for advancement throughout a career at the NSA, and the unique opportunities for female mathematicians at the Agency.
Mathematics of risk management
In 1996, I made the transition to Wall Street from academia, where I had been an assistant professor of mathematics for almost 10 years. Based on my experiences, I will give an overview of some of the jobs available to mathematicians on the sell-side and buy-side of the street: from investment banking to hedge fund management. I will then briefly discuss the mathematical skills needed to work in quantitative finance today, and introduce the basic mathematical framework of quantitative finance. I will present some of the numerical and analytical research problems I have worked on in stochastic volatility modeling and credit derivatives, and potential research directions in these areas.
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