Seminar on Nonlinear Waves
Lefschetz Center for Dynamical Systems Seminar
Center for Fluid Mechanics Seminar
Abstract: Improved computational algorithms and hardware have recently afforded the direct simulation of fluid transport in the post-Stokes hydrodynamic regime through porous media with realistic pore geometries. Via the implementation of a second-order-accurate Lattice Boltzmann Method (LBM) on parallel computers, a systematic investigation of the effect of packing disorder and Reynolds number in pressure-driven steady flow through two-dimensional analogues of disordered packed beds has been completed. The numerical results, obtained on statistically significant ensembles of packing realizations, provide a detailed probe into pore-scale hydrodynamics and allow the assessment of old (or the formulation of new) macroscopic transport laws. The simulation of hydrodynamic dispersion on the basis of the detailed pore-scale hydrodynamic fields has also been completed via LBM (for unsteady mass transport) and finite difference methods (for steady-state). The predictions of macroscopic (effective) transport properties agree for the first time with available measurements of the transverse effective diffusivity. Overall, LBM is perceived as a new member of the coarse-grained (or mesoscopic) theoretical view of condensed matter, and as a tool that can accommodate complex interfaces and complex physics. Future work involves the systematic validation of laminar flow LBM results through a single realization of three-dimensional "randomly"-packed beds via Magnetic Resonance Imaging velocimetry.
Special PDE Seminar
Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: This talk will focus on two problems: A new approach to thinking about algebraic 2D curves (and 3D surfaces) and fitting them to unorganized data; a new "complete" set of Euclidean invariants and a "Single-Computation" approach to object position estimation that uses all the information in the coefficients of the algebraic curve representation.
Two difficulties in the past with using algebraic curves and surfaces (sometimes called implicit polynomial curves and surfaces) have been huge amounts of computation and poor stability in fitting these to data, and incomplete sets of invariants for position-independent recognition and lack of understanding of the behavior of these invariants. We introduce a fitting approach that is linear least-squares with ridge regression. The resulting fitting is very fast and highly stable, resulting in fits which represent the data very well and fitted coefficients which are highly stable and thus well suited to position-invariant recognition or position estimation based on the fitted coefficients. We then introduce a complex-function representation for the implicit polynomial curve representation, and from this derive invariants for shape recognition that contain all postion-invariant shape information in the coefficients, and give a geometric interpretation for these invariants. They are stable. Then an object position estimation approach is given that is single computation, rather than iterative, and contains all the information in the coefficients. This technology appears to be effective for shape recognition in the presence of noisy data and significant occlusion (through use of algebraic curves fit to patches of data) and for very large databases of shapes.
Brown Analysis Seminar
Department of Mathematics Colloquium
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