Applied Mathematics Colloquia Series
Archive

Fall 2003

September

09/12
Matthew Hastings
Los Alamos National Laboratory

Mean-field and Anomalous Behavior on a Small-World Network

Recently, the study of systems on networks, general graphs with some combination of nodes and vertices, has become of interest. I will consider a particular network: the small world network. This network combines long-range and short-range interactions, and has become a standard model in the field. I will show that for a wide range of systems the behavior on this network can be described by mean-field critical behavior, and I will analyze the crossover to this behavior.

Finally, I will finish with the example of the Edwards-Wilkinson equation. Recent work of Toroczkai, Korniss, and others, has shown the relevance of this equation to synchronization in parallel processing. I will apply the results above to this example, and find that in some cases the mean-field behavior applies and in some cases it does not. (Some of this is joint work with B. Kozma and G. Korniss.)

09/19
Orly Alter
Department of Genetics
Stanford University

Modeling Genome-Scale Expression Datasets: From Matrix Algebra to Genetic Networks

October

10/03
Kevin Zumbrun
Department of Mathematics
Indiana University

Stability of Multi-Dimensional Shock Waves

The Navier--Stokes equations of gas- and magnetohydrodynamics are of an unusual hyperbolic--parabolic type combining features of both hyperbolic and parabolic evolution equations, and this is true both at high frequencies, corresponding to short-time behavior/well-posedness of the equations, and low frequencies, corresponding to large-time behavior. The dual nature of the equations comes to the fore in the study of stability of viscous shock waves, or pressure fronts. In this talk, we discuss the interesting blend of analytical techniques that has evolved to treat this problem, centered loosely around the spectral representation (inverse Laplace tranform) formula for C0 semigroups and the spectral determinants arising in normal modes analyses for hyperbolic and parabolic problems, known respectively as Lopatinski and Evans functions. Our results reduce the question of stability to a simple and numerically checkable {Evans function (ODE) condition}, equivalent to spectral stability plus hyperbolic stability of the associated ideal shock (easily checked) plus transverality of the connecting orbit defining the profile (automatic for extreme shocks).ngs" a subroutine

10/10
Michael Berry
Department of Physics
Bristol University

Singularity-dominated strong fluctuations

The fluctuations of a physical quantity can be described by its moments. In many cases, these diverge as an asymptotic parameter becomes large (or small), through the influence of geometric singularities. These large moments are described by power laws whose exponents can be determined from a knowledge of the singularities. Examples are twinkling starlight, the sex life of moths, certain contour integrals, and several properties of spectra in quantum chaology.

10/24
Peter Petropoulos
Department of Mathematics
New Jersey Institute of Technology

Absorbing Layer Boundary Conditions for the Numerical Solution of the Time-Dependent Maxwell Equations in Open Domains

I will review a class of absorbing boundary conditions, based on absorbing layers, for hyperbolic and elliptic partial differential equations posed on open domains. The effect of numerical discretization on their performance will be explored and comparisons to exact absorbing boundary conditions (Dirichlet-to-Neumann maps) will be shown. Finally, energy estimates will be derived for the solution in these layers. Future directions will be briefly discussed.

10/31
Yannis Kevrekidis
Department of Chemical Engineering
Princeton University

Equation-Free Complex Systems Modeling: Enabling Microscopic Simulations to Perform System-Level Tasks

In current modeling , the best available descriptions of a system come at a fine level (atomistic, stochastic, microscopic, individual-based) while the questions asked and the tasks required by the modeler (prediction, parametric analysis, optimization and control) are at a much coarser, averaged, macroscopic level. Traditional modeling approaches start by first deriving macroscopic evolution equations from the microscopic models, and then bringing our arsenal of mathematical and algorithmic tools to bear on these macroscopic descriptions.

Over the last few years, and with several collaborators, we have developed and validated a mathematically inspired, computational enabling technology that allows the modeler to perform macroscopic tasks acting on the microscopic models directly. We call this the ``equation-free" approach, since it circumvents the step of obtaining accurate macroscopic descriptions.

I will argue that the backbone of this approach is the design of (computational) experiments. In traditional numerical analysis, the main code "pings" a subroutine containing the model, and uses the returned information (time derivatives, function evaluations, functional derivatives) to perform computer-assisted analysis. In our approach the same main code "pings" a subroutine that sets up a short ensemble of appropriately initialized computational experiments from which the same quantities are estimated (rather than evaluated). Traditional continuum numerical algorithms can thus be viewed as protocols for experimental design (where "experiment" means a computational experiment set up and performed with a model at a different level of description).

Ultimately, what makes it all possible is the ability to initialize computational experiments at will. Short bursts of appropriately initialized computational experimentation through matrix-free numerical analysis and systems theory tools like variance reduction and estimation- bridges microscopic simulation with macroscopic modeling. Remarkably, if enough control authority exists to initialize laboratory experiments "at will", this computational enabling technology can become a set of experimental protocols for the equation-free exploration of complex system dynamics.

November

11/07
David Cai
Courant Institute of Mathematical Sciences
New York University

Kinetic Theory of Neuronal Networks - Mean-Driven vs Fluctuation-driven Dynamics

A new coarse-grained representation of the dynamics of neuronal networks is introduced and developed in terms of kinetic equations, which, via a novel moment closure, are derived mathematically, directly from the original large-scale integrate-and-fire (I&F) network. This powerful kinetic theory can capture the full dynamic range of neuronal networks ---from the mean-driven limit (a limit as the number of neurons N??, in which the fluctuations vanish) to the fluctuation-dominated limit (such as in small N networks). Comparison with full numerical simulations of the original I&F network establishes that the reduced dynamics is very accurate and numerically efficient over all dynamic ranges. Both analytical insights and numerical scale-up of neuronal computation can be achieved via this kinetic approach. Here the theory is illustrated by a study of the dynamical properties of networks of various architectures, including excitatory and inhibitory neurons of both simple and complex type, which exhibit rich phenomena, e.g., transition to bistability and hysteresis, even in the presence of large fluctuations. The implication for possible connections between the structure of the bifurcations and the behavior of complex cells is discussed. Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for "ring model" architectures which characterize changes in the response of neurons located from near to far from "orientation pinwheel centers".

11/14
Vadim Zharnitsky
Department of Mathematics
University of Illinois, Urbana-Champaign

Ground States in Higher Order Dispersion Managed NLS with Vanishing Residual Dispersion

Microdisk lasers have been invented and developed over the past decade and the billiard model has been successfully used in designing the "optimal" resonator shape. We will show that a new class of billiard tables possessing an isolated caustic, which carries only periodic orbits, can be obtained by studying a certain nonholonomic system, which naturally arises in the billiard problem.

We will also describe some important implications of this construction for the microdisk lasers, including some results of the wave equation simulations. This work is being done in collaboration with Y. Baryshnikov (Lucent) and P. Heider (University of Koln).

11/21
Thomas Spencer
Institute for Advanced Studies
Princeton University

Analysis of a Quasi-Species Model of Evolution

We describe a simple model of evolution which incorporates random mutations and a random fitness profile. This model is a close relative of the quasi-species model. A phase transition occurs as we vary the strength of the randomness in the fitness profile.

December

12/05
Richard Rand
Theoretical and Applied Mechanics
Cornell University

Recent Advances in Parametric Excitation

Parametric Excitation refers to dynamics problems in which the forcing function enters into the governing differential equation as a variable coefficient. The paradigm example is given by Mathieu's equation: x'' + (d + e cost) x = 0. This has application to many engineering systems, the simplest example of which is the vertical forcing of a pendulum. In this lecture, the basics of parametric excitation will be reviewed and some new results will be presented.