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DE Seminars: Fall 2012

Monday October 15

Title The Effect of Bisexuality on the Spread of Incurable Sexually Transmitted Disease
Speaker Evelyn K. Thomas
Howard University

Dynamical systems techniques are used to study the effect bisexual males have on the spread of an incurable sexually transmitted disease within a population consisting of sexually active homosexual, bisexuals, and heterosexual females and males. Our approach is to study decoupled subpopulations of the sexually active populations and determine the influence of other populations on the basic reproductive number and endemic equilibria of these subpopulations. In particular, the severity of the disease in the bisexual male and decoupled sexually mixing bisexual and female subsystems determine the conditions under which the disease will persist in the full model or die out. In addition, we see the severity of the disease in the homosexual male population plays a significant, but secondary role to the disease in the full system. For instance, the disease could be persistent in the homosexually mixing male populations and not dominate the entire system. The bridge between bisexual males and heterosexual females is the main determinant for whether or not the entire system will be engulfed with the disease.

Monday October 22

Title A diffusion/reaction/switching system
Speaker Tom Seidman
Department of Mathematics and Statistics

We consider a model of bioremediation in which the bacteria involved switch discontinuously between `dormant’ and `active’ modes based on threshold levels of a critical nutrient. In contrast to a similar previous model, the nutrient is here transported by diffusion, rather than convection. This is treated as an optimal control problem for insertion of nutrient at the boundary and the results are existence of solutions and of an optimal control.

Monday November 5

Title Analysis of an input-queued switch operating under a maximum weight matching policy
Speaker Weining Kang
Department of Mathematics and Statistics

In this talk we consider an N by N input-queued switch operating under a scheduling policy, called a maximum weight matching policy. To analyze the system performance in the heavy traffic, We establish a second order approximation for the workload process associated with this switch model when all input ports and output ports are heavily loaded. A class of reflected Brownian motions living in polyhedral domains with piecewise constant reflection fields on the boundary will arise in the approximation process. Its stationary distribution in terms of the solution to a second order PDE with boundary condition is discussed.

Monday November 12

Title Optimal control of the stationary Navier-Stokes equations: A multigrid solution
Speaker Ana Maria Soane
Department of Mathematics and Statistics

We consider the distributed optimal control problem associated with the tracking of the velocity of a Navier-Stokes flow in a bounded two-dimensional domain. The goal of our work is to construct multigrid preconditioners to accelerate the solution process. Our approach on previous work on the Stokes control problem was to eliminate the state and adjoint variables from the optimality system and to construct efficient preconditioners for the Schur-complement of the block associated with these variables. In this talk, I will discuss how we extend this work to construct similar preconditioners for the reduced Hessian in the Newton-PCG method to be used for the numerical solution of the Navier-Stokes control problem. The results of some computational experiments are presented.

Tuesday November 20

Title Rigidity of Self-shrinkers of Mean Curvature Flow
Speaker Lu Wang
Johns Hopkins University

Recently, using the desingularization technique, Kapouleas-Kleene-Moller and independently Nguyen has successfully constructed a new family of smooth complete embedded self-shrinkers asymptotic to cones. These are the first non-rotationally symmetric examples after planes, spheres, cylinders and Angenent’s torus.

In this talk, we report some new rigidity (at infinity) theorems of self-shrinkers. The results are two folds with emphasis on the asymptotically cylindrical case. First, we show the uniqueness of smooth properly embedded self-shrinkers asymptotic to any given regular cone in Euclidean space. Second, we discuss the optimal condition on the asymptotics of self-shrinkers, so that the uniqueness of self-shrinkers asymptotic to generalized shrinking cylinders holds true. This gives a partial affirmative answer to the cylinder rigidity conjecture. The point of our theorems is that we do not require completeness of self-shrinkers. One of the main ingredients of the proofs is the (anisotropic) Carleman estimates inspired by the work of Escauriaza-Seregin-Sverak. Among applications, we obtain some non-existence results of self-shrinkers. Namely, except hyperplanes, there do not exist any other smooth complete properly embedded self-shrinkers with ends asymptotic to rotationally symmetric cones.

Monday December 3

Title Model Reduction for Dynamical Systems with Symmetry Constraints
Speaker Mili Shah
Loyola University

Dimension reduction in molecular dynamics simulation is often realized through a principal component analysis based upon a singular value decomposition (SVD) of the trajectory. The left singular vectors of a truncated SVD provide the reduced basis. In many biological molecules, such as HIV1 protease, reflective or rotational symmetry should be present in the molecular configuration. Determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop an SVD that best approximates the given set while respecting the symmetry.

Interesting subproblems arise in the presence of noisy data or in situations where most, but not all of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflection symmetry is an iterative re-weighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).

Monday December 10

Title Optimal control approach for determining best fitting constrained smoothing splines and convergence analysis of related method
Speaker Teresa Lebair
Department of Mathematics and Statistics

There are a plethora of significant phenomena that can be modeled with shaped restricted functions, e.g., monotone or convex functions. A best fitting smoothing spline that adheres to such shape restrictions can be found by minimizing a particular objective functional over an appropriate set of functions. J. Shen and X. Wang demonstrate the existence and uniqueness of best fitting shape restricted smoothing splines in their paper A Constrained Optimal Control Approach to Smoothing Splines. Additionally, they present a method to obtain the best fitting smoothing spline, and discuss the convergence analysis of this method in their paper. Additional work is being done in order to determine how much the sufficient conditions for convergence analysis presented in the paper may be relaxed for monotone and convex smoothing splines.