Lefschetz Center for Dynamical Systems Seminar
Abstract: It is observed that in the presence of small-scale eddies the transport of large-scale vector quantities is accompanied with depleted, and in some cases even "negative", diffusion. This phenomenon is the result of the weak nonlinear interaction between large and small scales in incompressible fluid. This interaction is described by the eddy viscosity, a four-tensor that relates the large-scale deviatoric stress to the large-scale rate of strain. The goal of this talk is to analyze the eddy viscosity for large Reynolds numbers by means of Homogenization of the Navier-Stokes equations in the case of two-dimensional cellular flows. In this talk I plan to discuss some new analytical and numerical results for eddy viscosity, such as the saddle-point variational principles for a class of non-symmetric, nonlocal operators and the dynamical systems approach to the weak convergence of solutions for Hadamard ill-posed equations.
Special Information Theory Seminar (Amended)
Purdue University, Statistics Department and Visiting Assistant Professor of Reasearch, Brown University, Division of Applied Mathematics | |
Abstract:
After some background on data compression, in the first part of this talk we will introduce a correspondence between "lossy" data compression algorithms and probability measures defined on an appropriate probability space ("lossy,"
in that some distortion is allowed between the original data and the decompressed data). Conceptually, designing good codes reduces to finding good probability measures. This correspondence allows us to restate the data compression problem as a problem in convex optimization, and to characterize precisely the best
achievable performance of arbitrary algorithms. We give coding theorems emphasizing:
(a) Arbitrary source models;
(b) Non-asymptotic bounds; and
(c) Results with probability one (rather than on the average).
Then we turn to the simplest class of random data sources, those that generate independent and identically distributed data. Here it is possible to give an explicit expression for the (asymptotically) optimal compression performance, which reveals an interesting new phenomenon: There is a sharp dichotomy in the speed at which optimality can be reached: For some sources this speed is of order O(log n) and for some it is of order O (square-root of n) [where "n" is the length of the data being compressed]. We give a complete characterization of when each case occurs: Roughly speaking, the speed is almost always O(square-root of n) (but not always!).
Special Information Theory Seminar (Amended)
University of British Columbia, Canada | |
Abstract: Stepping-Stone models originally arose as mathematical models for population genetics. They can be thought informally as systems of interacting Fisher-Wright processes or Fleming-Viot processes. Evans proposed a class of models with continuous site spaces. Such models are defined via moment duality.
In this talk we will go over several results on continuous-sites Stepping-Stone models. We will also point out their connection with multi-type voter models.
Scientific Computing Seminar
Abstract: It was found that the rate of convergence of finite element approximations exceeds the optimal global rate at some exceptional global points. This phenomenon is called ``superconvergence'' and these special points are called ``natural superconvergence point.'' Most earlier works on superconvergence were concentrated on tensor-product rectangular elements and lower order (linear and quadratic) triangular elements. In this work, a systematic method is introduced, analyzed and used to find all gradient superconvergent points of arbitrary rectangular finite elements. The results justify some computer findings. The method is then generalized to three dimensional elements to predict gradient superconvergence points which have not been reported in the literature.
PDE Seminar
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