Background Image

Seminars & Colloquia

In this Section
Kushani DeSilva
December 15,2014
Department of Mathematics, Clarkson University
Bayesian Approach to Particle Velocimetry Using Splines in Tension 


Abstract:  Particle Velocimetry is very popular and widely used in applications related to fluid, gas and aero dynamics, plasmas and object tracking.  It helps in
 retrieving quantitative information such as velocity or acceleration.  In particular,Particle Image Velocimetry (PIV) is a widely used technique in capturing velocity of particles from high-speed image records.  In post-processing steps, the 2D recordings are considered either as intensity fields or individual particles  are identified and traced over time.  Cross-Correlation based on Fast Fourier Transform is used in extracting velocity, when the 2D recording are considered as intensity fields.  When individual particles are identified and traced over time, often velocities are derived by finite-differencing or fitting a smoothing regression function (e.g. cubic splines) with a subsequent analytic derivate.  However, these traditional techniques derive the velocity with respect to the position of the particle in space and therefore encounter a fundamental flaw.  Computing the derivative to obtain velocity is ill-conditioned because small uncertainties in position estimates are magnified and thus may result in very poor results.  Here, we propose a new approach using a Bayesian method to infer the velocity of a particle.  The new method is built using splines in tension and more important it is formulated in the space of velocity, and thus taking the derivative in order to derive the velocity is avoided.  We suggest that the new method will perform better than the traditional methods of deriving particles velocities, because the spline model is formulated in the velocity space (continuous) and therefore automatically includes the physical constraints of finite acceleration.  Also we suggest that it is important that the model is formulated in a space where the quantity to be modeled (velocity) is continuous, instead of being placed in the data space.
Kelum D. Gajamannage
December 2,2014
Department of Mathematics Clarkson University
Nonlinear Dimensionality Reduction and Manifold's Learning of Collective Motion
Abstract:  Animal groups, such as schools of fish and flocks of birds often exhibit collective motion driven by mutual interactions.  We have developed a mathematical perspective that the definitive feature of such collective motions is the presence of low-dimensionality.  Computationally identifying this feature, Nonlinear Dimensionality Reduction (NDR) techniques may be executed on the manifold representing the collective motion.  We propose a specialized approach for these problems that topologically summarizes the high-dimensional data.  Furthermore, transitions of behavior, of speed, coordination as well as structure of a multi-agent systems may be defined as singularities of the embedding manifolds, where traversing particle swarms under-go switching of behavior describing phase transitions.
Ye Li
October 10,2014
Department of Mathematics, Clarkson University
Discrete Epidemic Model with Mobile Human Population
Abstract:  We introduce an SIR model coupled to social mobility model (SMM), which we discretize by a forward Euler Method, and a mixed type Euler method (structured with both forward and backward Euler elements).  We calculate basic reproduction number R using a next generation matrix method.  In our model, when R < 1, there will be a disease-free equilibrium (DFE), and implies DFE will be locally asymptotically stable, while R > 1 implied DFE is unstable.