Applied Mathematics Colloquia Series
Archive

Spring 2005

January

01/28
Robert Ecke
Los Alamos National Laboratory

Adventures in 2D turbulence: topology, transfer and dynamics

Fluid turbulence in two dimensions has some unusual and fascinating properties and is amenable to both physical experiment and numerical simulation on a detailed level. I will show how experiments on 2D turbulence can be used to directly understand how structures in the flow are related to a direct measure of transfer quantities in the inertial ranges of the direct and inverse cascades of 2D turbulence. Further, I will show how measurement of Lagrangian quantities such as mean-square particle separation can be obtained and will present an interpretation of these results compared to theoretical predictions.

February

02/04
Kathleen Dixon
Molecular & Cellular Biology
University of Arizona

Modeling cellular responses to DNA damage

02/11
Anne Gelb
Department of Mathematics
Arizona State University

Multivariate local edge detection method on scattered data

A new local edge detection method is proposed that is effective on multivariate irregular data in any domain. The method is numerically cost efficient and entirely independent of any specific shape or complexity of boundaries. Application of the minmod algorithm to various orders of the method ensures a high rate of convergence away from the discontinuities while reducing the inherent oscillations near the discontinuities. It further enables distinction of jump discontinuities from steep gradients, even in instances where only sparse non-uniform data is available. The method is successfully demonstrated in both one and two dimensions. This method is compared to previously designed edge detection methods based on spectral data. It is well known that edge detection is critical for the reconstruction of an image where the underlying function is piecewise smooth. Many images are obtained from equally spaced data points. In this case, our method can determine regions of smoothness for the underlying function. Application of a high resolution reconstruction method, for example, the Gegenbauer reconstruction method, will then produce a highly resolved reconstruction of the desired image. Several examples are discussed.

02/18
Eric Vanden-Eijnden
Courant Institute
New York University

Rare and not so rare events in complex systems: conformation dynamics, transition pathways, and rates

Many problems in physics, material sciences, chemistry and biology can be abstractly formulated as a system that navigates over a complex energy landscape of high or infinite dimensions. Well-known examples include phase transitions of condensed matter, conformational changes of biopolymers, and chemical reactions. The state of these systems is confined for long periods of time in metastable regions in configuration space and only rarely switches from one region to another. The separation of time scale is a consequence of the disparity between the effective thermal energy and typical energy barrier in these systems, and their dynamics effectively reduces to a Markov chain on the metastables regions. The analysis and computation the transition pathways and rates between the metastable states is a major computational challenge, especially when the energy landscape exhibits multiscale features. I will review recent work done by scientists from several disciplines on probing such energy landscapes. I will then present a new method, the string method, that has proven to be very effective for some truly complex systems in material science and chemistry.

02/25
Linn F. Mollenauer
Bell Laboratories

Transmission distances in dense WDM extended to the noise-imposed limit through conquest of nonlinear penalties with solitons and a novel form of dispersion management

At the height of the recent telecom boom, schemes abounded for ways to fulfill the perceived need for an all-optical, ultra long-haul fiber optic network. Although most died when the boom went bust, at least one, based on dispersion-managed solitons in an all-Raman amplified system, is now beginning to gain significant commercial success. That system, Lucent Technology's "LambdaXtreme", very conservatively claims unrepeatered distances of 4000 km in Wavelength Division Multiplexed transmission of 100 or more channels at 10 Gbit/s each. As impressive as that performance may seem, however, its distance is still limited by a serious non-linear defect, viz., a jitter in pulse arrival times resulting from inter-channel soliton-soliton collisions. In this talk, I shall show how a recently invented technique, based on the use of "periodic-group-delay" devices for an optimal fraction of the dispersion compensation, has enabled experimental demonstration of LambdaXtreme-like, dense WDM transmission to nearly 20,000 km, or essentially the distance imposed by the limiting effects of amplifier spontaneous emission noise alone.

March

03/25
Ramakrishna Ramaswamy
Centre for Systems Biology, School of Natural Sciences
Institute for Advanced Study, Princeton University

The mosaic of the genome: Markov models of segmentation

The metaphor of a mosaic has frequently been applied in describing the manner in which the DNA of an organism - the genome - is organized. Correlations in DNA sequences are of necessity long-ranged, and segmentation methods attempt to understand this correlation, as well as to give methods of partitioning genomic sequences into homogeneous regions. In this talk I will describe statistical methods for segmentation of DNA and discuss a number of Markov (and hidden Markov) models of DNA. Segmentation provides a natural framework for examining evolutionary trends as well as detecting gene transfer processes.

April

04/01
Todd Kapitula
Department of Mathematics
University of New Mexico

The dynamics of matter waves in Bose-Einstein condensates

Especially since the 1997 Nobel Prize winning work of S. Chu, C. Cohen-Tannoudji, and W. Phillips on Bose-Einstein condensates (BECs), there has been a great deal of exciting experimental and theoretical work in the area. Recently, there has been intriguing experimental and theoretical work on the creation and dynamics of vortices in one- and two-species condensates. The governing equations for these systems are Hamiltonian; in fact, they are simply nonlinear Schrodinger equations with nonhomogeneous terms. In addition to questions of existence and stability of these vortices, which are of interest in their own right, the analysis leads to several intriguing fundamental mathematical problems for Hamiltonian systems. These include (a) the connection between integrability and a linear stability analysis, (b) the role in which the energy plays in a stability analysis, and (c) the dynamical effect caused by the wave not being a minimizer for the energy. Answers to some of these problems will be presented, in general and in the context of BECs, as well as interesting open questions.

04/08
Oscar Bruno
Applied & Computational Mathematics
California Institute of Technology

New High-Order, High-Frequency Methods in Computational Electromagnetism

We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers --- even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our algorithms can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects by direct solution of integral equations --- a goal, otherwise achievable today only by supercomputing. A new class of high-order surface representation methods will be discussed, which allows for accurate high-order description of surfaces from a given CAD representation. A class of high-order high-frequency methods which we developed recently, finally, are efficient where our direct methods become costly, thus leading to a general and accurate computational methodology which is applicable and accurate for the whole range of frequencies in the electromagnetic spectrum

04/15
Keith Julien
Department of Mathematics
University of Colorado at Boulder

Rotationally Constrained Rayleigh-Benard Convection

Convection in a rotating layer of fluid has been the subject of a great deal of theoretical and experimental research. This problem is relevant to convectively driven fluid flows in the Earth's atmosphere, ocean and interior and also in the Sun and other stars, where the influence of rotation is generally important. In general numerical simulations of rotationally constrained flows are unable to reach realistic parameter values, e.g., Reynolds $Re$ and Richardson $Ri$ numbers. In particular, low values of $Ro$, defining the extent of rotational constraint, compound the already prohibitive temporal and spatial restrictions present for high-$Re$ simulations by engendering high frequency inertial waves and the development of thin (Ekman) boundary layers.

Recent work in the development of reduced partial differential equations (pde's) that filter fast waves and relax the need to resolve boundary layers has been extended to construct a hierarchy of reduced pde's that span the stably- and unstably-stratified limits. By varying the aspect ratio for spatial anisotropy characterizing horizontal and vertical scales, rapidly rotating convection and stably-stratified quasi-geostrophic motions can be described within the same framework.

In this talk, the asymptotic pde's relevant for rotating Rayleigh-Benard convection are explored. Special classes of fully nonlinear exact solutions are identified and discussed. Direct numerical solutions that correctly capture the regular vortex columnar and irregular geostrophic turbulence regime of recent laboratory experiments are also presented and discussed.

04/22
Govind Menon
Department of Applied Mathematics
Brown University

Dynamic scaling in Smoluchowski's coagulation equation

Smoluchowski's coagulation equation is a fundamental mean-field model of clustering. It has been used to study (among other things) the formation of clouds of smoke, dust and haze, the coagulation of colloids, clustering and lines of descent in population genealogy, and the mergers of financial institutions.

Suitably rescaled distributions of cluster sizes are often observed to approach self-similar form. This is known as dynamic scaling. I will describe rigorous results on dynamic scaling for three ``solvable'' cases of Smoluchowski's equation. A general theme is the fundamental utility of classical (1930s) methods in probability theory (limit theorems, infinite divisibility, Levy processes) for a complete understanding of such scaling dynamical systems. A striking connection with the structure of shocks in Burgers equation with random initial data due to Jean Bertoin will also be described.

This is joint work with Bob Pego (Carnegie Mellon University)