Lefschetz Center for Dynamical Systems Seminar
Abstract: We present a description of optical pulse propagation in fibers with randomly varying values of chromatic dispersion. The corresponding mathematical model is the nonlinear Schroedinger equation with a randomly varying dispersion coefficient. Pulse propagation is described by a statistical description of pulse degradation during its propagation along the fiber. Using path integrals along trajectories, we obtain the Fokker-Planck equation for the probability distribution function. We will present different statistical scenarios of pulse evolution in terms of the distribution function which are of practical interest.
Center for Fluid Mechanics Seminar
Abstract: Vortex filament simulations are used to gain new insight into the mechanism leading to vortex breakdown and the formation of time dependent recirculation zone downstream a naturally occurring stagnation point. The structure of the "bubble" and the concomitant self sustained oscillation frequency are analyzed. Vortex line evolution is dissected to investigate the dynamics leading to this intriguing phenomenon.
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: First I present a quick review of the inference problem in stochastic processes, and then a sketch of ``asymptotically stationary'' random fields. Also a testing problem and a consistent estimation question of interest in template construction will be discussed, leading to some new mathematical problems for solutions.
Brown Analysis Seminar
Applied Mathematics Colloquium
Abstract: This talk will briefly overview the development of solution techniques for Maxwell's equations as an analysis tool. It reviews the solution techniques that were prevalent around the beginning of the 20th century. During that period, closed form solutions were sought for simple shape objects such as spheres, cylinders, half-planes, half-spaces, etc. That was the age of simple shapes. As the demand of science and engineering called for solutions to more complex problems, scientists and engineers developed approximate methods to solve Maxwell's equations approximately. This was often the combined use of asymptotic and perturbation methods. Examples of such are the asymptotic high-frequency methods in scattering, the small perturbation method in rough surface scattering, and the match asymptotic methods in boundary value problems, multiscale analysis in random media, and optics. Consequently, a larger class of solutions can be sought by such techniques, empowering the solutions available to Maxwell's equations. This was the age of approximations.
The advent of computer technology spurred interests in solving Maxwell's equations numerically. Much of these works were to find the corresponding matrix equations, which were sufficiently well posed and represented a high fidelity representation of the original Maxwell's equations. This lifted the restriction on solution types solvable for Maxwell's equations, and greatly expanded the scope and versatility of Maxwell's solvers. That was the age of numerical solvers. However, due to the inefficiency in the solution techniques, only small problems can be solved.
The most recent developments of analysis methods for Maxwell's equations are the fast solvers. These fast solvers have been developed for electrostatics, electrodynamics, as well as fast solvers that are continuously valid from static to electrodynamics; for the frequency and time domains, as well as for layered media. These fast solvers use resources that are orders of magnitude smaller than traditional numerical solvers. Their importance in electromagnetic simulations is as important as fast Fourier transforms in signal and image processing. It is quite certain that these solvers will precipitate a revolution in analysis methods in electromagnetics. Problems that required 10 years of computer time to solve in the past can now be solved within a day.
Department of Mathematics Colloquium
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