Lefschetz Center for Dynamical Systems Seminar
Abstract: We model a dilute gas Bose--Einstein condensate trapped in a standing light wave by the cubic nonlinear Schr\"odinger equation with an elliptic function potential. New families of stationary solutions are presented and their stability is examined using analytic and numerical methods. Jacobi elliptic $Dn(x, k)$ solutions are found to be stable for defocusing, whereas Jacobi elliptic $Cn(x, k)$ solutions are found to be stable for focusing. The linearized stability calculations allow us to generate a set of criteria concerning the stability and instability of the various families of solutions. Our results imply that for defocusing (repulsive BEC), a large number of condensed atoms is sufficient to form a stable, periodic condensate. For focusing (attractive BEC), solutions with nodes are necessary.
Brown University Center for Statistical Sciences Seminar
Sponsored by the Mollie B. Mandeville Lectureship And The C. V. Starr Foundation Lectureships Fund |
Abstract: Item response theory (IRT; e.g. van der Linden and Hambleton, 1997) is a method of modeling data from social surveys and educational and psychological tests, dating back at least to Lord (1952) and Rasch (1960). IRT enables us to study the characteristics of test or survey items across multiple respondent populations, and to study respondents' propensities to answer postively across different sets of items (these propensities are often interpreted as psychometric abilities or attitudes, for example). Although it arose to answer questions about educational tests that traditional factor analysis could not easily answer, IRT would be recognized by many statisticians today as a mixed-effects or hierarchical Bayes logistic regression methodology for multivariate discrete response data.
In this talk I will briefly introduce IRT models in educational testing and then describe the hierarchical rater model (HRM) of Patz, Junker and Johnson (2000), which extends IRT to model bias and variability in human raters (judges) of essays, art portfolios, and other student work that cannot be encoded as machine scorable multiple choice responses. I will show how the HRM can be used to make inferences about examinees, test items and raters, in a statewide mathematics exam given in the State of Florida.
(*) Joint work with Richard Patz, CTB/McGraw-Hill, Matt Johnson, Carnegie Mellon University and Educational Testing Service, and Lou Mariano, Carnegie Mellon University.
** Reception following seminar at 167 Angell Street, 2nd floor conference room.
Center for Fluid Mechanics Seminar
Abstract: During the last decade microflows has become an important scientific field with applications in the area of chemical, biological, and biomedical analyzes. The increasing interest in using microflows for sorting and analyzing cells or particles has resulted in the need of methods for computing the transport dynamics of these objects.
In particulate microflows two main problems arise. Firstly, the size of the particles are only one or two orders of magnitude smaller than smallest length scale of the carrying flow. Secondly, in many microsystems a considerable amount of particles are present (leading to a high volume fraction). So far no equation of motion for a particle in a fluid flow has been able to overcome these two problems. Consequently, another approach is needed for computing the particle dynamics in fluidic microsystems.
A method called force coupling is introduced and developed. The method is not limited to small particles or to few particles (low volume fraction). It uses knowledge about the forces from the particles on the fluid flow to solve the equations of motion for the fluid. The motion of the particle is determined from the fluid velocity field as a volume average of the fluid velocity in the neighborhood of the particle.
The results from the force coupling method is verified by comparing with experimental data for a single sphere, two spheres, and three spheres rising in an inclined or vertical channel. Comparison of the trajectories and particle velocities show good agreement. Special attention is also given to the pressure driven Poiseuille flow, since microflows are typically of this kind.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Algorithms for probability propagation in graphs have been independently discovered and analyzed in a variety of fields including artificial intelligence, speech recognition and error-correcting codes. In these various fields it was shown that the algorithms converge to the globally correct posterior probabilities for singly connected graphs. Recently, however, a number of researchers have empirically demonstrated good performance of these same algorithms on graphs with loops. Perhaps the most dramatic instance is the near Shannon limit performance of "Turbo Codes" whose decoding algorithms is equivalent to probability propagation on a loopy graph.
I will describe analytical results for probability propagation in graphs with loops --- including formulas relating the correct probabilities with the probabilities calculated using propagation algorithms and a variational interpertation of the propagation fixed-points. In particular, for a category of loopy graphs the decision based on probability propagation can be proven to be optimal (although the probabilities will be incorrect). I will illustrate the analysis using examples from image understanding and error-correcting codes.
* Joint work with Bill Freeman (MERL) and Jonathan Yedidia (MERL).
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: I will discuss a new joint project at Carnegie Mellon and the University of Pittsburgh Medical Center aimed at developing models, algorithms, and software for simulating blood flow at the cellular level. This capability will permit us to model the behavior of critical regions in artificial heart devices, as well as strengthen our insight into strokes, atherosclerosis and thrombosis, sickle cell disease, and the development of blood additives to improve transport and treat hypertension.
Blood flow belongs to a class of flow problems with dynamic interfaces: it is a mixture of interacting gel-filled solid cells and fluid plasma. Current blood flow models are macroscopic, treating the mixture as a homogeneous continuum. Microstructural models resolve individual cell deformations and their interactions with the surrounding fluid plasma. The major challenge faced in simulating such flows is resolving the interfacial motion. Lagrangian methods are ideally suited for such problems, since interfaces are naturally represented and propagated. However, the material description of motion results in dynamic meshes, which become hopelessly distorted unless they are regularly regenerated. Lagrangian methods are particularly challenging on parallel computers, because scalable dynamic meshing algorithms remain elusive.
I will present a parallel dynamic mesh Lagrangian method for flows with dynamic interfaces. We take an aggressive approach to dynamic meshing by triangulating the propagating grid points at every timestep using a parallel Delaunay algorithm recently developed by Blelloch, Hardwick, Miller, and Talmor. Contrary to conventional wisdom, we show that the costs of the geometric components (triangulation, coarsening, refinement, and partitioning) can be made small relative to the flow solver. For example, in a simulation of 10 interacting viscous cells with 500,000 unknowns on 64 processors of a Cray T3E, dynamic meshing consumes less than 5% of a time step. Moreover, our experiments on up to 64 processors show that the computational geometry scales about as well as the flow solver. Therefore, we anticipate that overall scalability on larger problems will be as good as that of the flow solver.
This work is joint with Guy Blelloch, Ivan Malcevic, Gary Miller, and Noel Walkington of CMU, and Jim Antaki of UPMC.
PDE Seminar
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