Math Colloquia: Fall 2005

Friday September 02

No talk today

Friday September 09

Title Efficient Numerical Schemes for Stochastic Chemical Kinetics
Speaker Muruhan Rathinam
Department of Mathematics and Statistics

In general many chemically reacting molecular systems are modeled by deterministic and continuous models using ODEs or PDEs. However there are certain situations when the continuous and deterministic models are inadequate. Prime examples are intra-cellular gene transcription mechanisms where the fluctuations of certain molecular species present in very low numbers may have a critical effect on the final state of the system.

On a microscopic level molecular reactions may be modeled by continuous time discrete state Markov processes. In principle such a process can be simulated exactly by keeping track of each random jump event. This accurate method however would take prohibitively long computing time for most real problems. In this talk we describe an ongoing project which aims to develop methods, software, and rigorous mathematical theory for simulating these systems in an efficient way by “leaping over” several events at a time. We address certain important issues such as stiffness and preserving nonnegativity of states.

More broadly these schemes would be applicable to other situations such as the dynamics of market microstructures and traffic models where the detailed models are discrete and stochastic.

Friday September 16

Title Finite Element Navier-Stokes Solvers without Inf-Sup Conditions
Speaker Jian-Guo Liu
Department of Mathematics
University of Maryland College Park

We will show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity in an equivalent unconstrained formulation. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. We provide a simple proof of unconditional stability and convergence for discretization schemes that are implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions, particularly for corresponding fully discrete finite-element methods with C1 elements for velocity and C0 elements for pressure. It is important to note that we impose no inf-sup compatibility condition between the finite-element spaces for velocity and pressure. The inf-sup condition (also known as the Ladyzhenskaya-Babuska-Brezzi condition) has long been a central foundation for finite-element methods for all saddle-point problems including the stationary Stokes equation. Its beautiful theory is a masterpiece documented in many finite-element books. In the usual approach, the inf-sup condition serves to force the approximate solution to stay close to the divergence-free space where the Stokes operator is dissipative. However, due to the fully dissipative nature of the new unconstrained formulation, the finite-element spaces for velocity and pressure can be completely unrelated. This is a joint work with Bob Pego and Jie Liu.

Friday September 23

Title Warmstarting Interior-Point Methods for Linear Programming
Speaker Hande Y. Benson
LeBow College of Business
Drexel University

Engineering problems with changing specifications and business problems with changing market prices and demand may require us to solve a series of closely related optimization problems in an efficient manner. One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize after a warmstart. In this talk, we investigate the use of a primal-dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems.

Friday September 30

Title The Minimum Labeling Spanning Tree Problem and Some Variants
Speaker Bruce L. Golden
Robert H. Smith School of Business
University of Maryland College Park

Given a connected, undirected graph G whose edges are labeled (or colored), the minimum labeling spanning tree (MLST) problem seeks a spanning tree on G with the minimum number of distinct labels (or colors). The MLST is motivated by applications in communications network design.

The MLST has been shown to be NP-hard and an effective heuristic has been proposed and analyzed. In addition, metaheuristics (including genetic algorithms) have been developed. In computational tests, the genetic algorithms perform exceptionally well. In this presentation, we summarize much of this work and discuss some variants, if time allows.

Friday October 07

Title Care and feeding of a pet discontinuity
Speaker Thomas I. Seidman
Department of Mathematics and Statistics,

My own pet discontinuity is the `elementary hysteron’ W[.] – switching its output when the input crosses upper and lower threshholds. After an introduction to some general properties of W[.], we consider a system

$$ u_t+V grad u = f_j(t,x,u), j=j(.,x)=W[u(.,x)] $$

in which convection/reaction is coupled (independently at each point) with W, controlling modal switching. The well-posedness argument involves a free boundary problem for the region where j=1 and a related double-obstacle problem. [If time permits, we consider some further coupling, applying this to a bioremediation model.]

Friday October 14

Title Modeling and Analysis of Gene Regulatory Networks
Speaker Mustafa Khammash
Department of Mechanical and Environmental Engineering
University of California Santa Barbara

The heat shock response in bacteria is an essential mechanism for combating the stress resulting from the increase in the ambient cellular temperature. Cells respond to this stress by producing an array of protective proteins referred to as heat shock proteins (HSPs). The production of HSPs is itself regulated directly by alterations in the level, activity, and stability of regulatory proteins sigma-factors. The logic of the heat shock response is implemented through an intricate hierarchy of feedback and feedforward controls that regulate both the amount of the sigma-factor and its functionality.

We present a dynamic model that captures known aspects of the heat shock system. With the aid of this model, we discuss the logic of the response from a control theory perspective, drawing comparisons to synthetic engineering control systems. Questions related to robustness analysis, and model validation will also be discussed and related mathematical challenges will be outlined.

While mathematical modeling of genetic networks like that of the heat shock system often represents gene expression and regulation as deterministic processes, there is now considerable experimental evidence indicating that significant stochastic fluctuations are present in these processes. The investigation of stochastic properties in genetic systems involves the formulation of a correct representation of molecular noise and devising efficient computational algorithms for computing the relevant statistics of the modeled processes. We present some of these techniques and use them to provide compelling examples that illustrate the richness of phenomena that can result from the interaction of dynamics and noise in genetic networks.

Friday October 21

Title Nonlinear Perron-Frobenius Theory
Speaker Roger Nussbaum
Department of Mathematics,
Rutgers University

Classical linear Perron-Frobenius theory concerns eigenvalues and eigenvectors of nxn matrices A with nonnegative matrices. Particularly precise results are available when A is column stochastic (columns sum to one), in which case theorems about eigenvalues translate to statements about asymptotic behaviour of iterates of A. We shall describe some theorems which have been obtained over the past eighteen years and which provide remarkably precise information about the asymptotic behaviour of iterates of nonlinear maps f which generalize the column stochastic matrices. We shall motivate our discussion by describing a simple “sand-shifting” game which a mathematically inclined child might play on the seashore. Interestingly, the sand-shifting game already displays all essential difficulties of the general problem.

Friday October 28

Title Radiative transfer equations and applications
Speaker Guillaume Bal
Department of Applied Physics and Applied Mathematics
Columbia University

Radiative transfer equations have long been used to model the energy density of waves in random media, with applications in light propagating through turbulent atmospheres, underwater acoustics, and elastic wave propagation in the Earth’s crust to name a few. This talk will derive such models from first principles, i.e., from equations for the wave fields in the high frequency regime. A very useful tool in such a derivation is the Wigner transform applied to the wave fields. The correlations of two wave fields propagating in possibly different media will be estimated in the high frequency regime, with applications in the theory of time reversed waves propagating in highly heterogeneous media. Finally recent numerical simulations performed on parallel architectures of two-dimensional acoustic wave propagation in random media on large domains will show the very good accuracy of the macroscopic radiative transfer models to describe the energy density of the acoustic waves. Applications to detection and imaging of buried inclusions in highly cluttered environment will also be mentioned.

Friday November 04

No talk today

Friday November 11

Title A Semiparametric Approach to Time Series Prediction
Speaker Benjamin Kedem
Department of Mathematics
University of Maryland College Park

Given m time series regression models, linear or not, with additive noise components, it is shown how to estimate the predictive probability distribution of all the time series conditional on the observed and covariate data at the time of prediction. This is done by a certain synergy argument, assuming that the distributions of the residual components associated with the regression models are tilted versions of a reference distribution. Point predictors are obtained from the predictive distribution as a byproduct. Applications to US mortality rates prediction and to value at risk (VaR) estimation will be discussed.

The basic underlying idea is an optimization problem based on an assumed connection between two or more distributions. You don’t know the distributions, but once you make the connection you obtain “optimal” distributions.

Friday November 18

Title An improved site percolation threshold universal formula
Speaker John C. Wierman
Applied Mathematics and Statistics
Johns Hopkins University

Site percolation models are infinite random graph models in which each vertex of an underlying graph is retained with probability p independently of all other vertices. The percolation threshold is a critical value of the probability p, above which the random graph has an infinite connected component, and below which all components are finite almost surely. Percolation models arose from applications such as thermal phase transitions, gelation processes, and epidemics, in which there is a sharp phase transition at the percolation threshold.

A challenge since the introduction of percolation theory has been to find a “universal” approximation formula, based on a small number of features of the underlying lattice, for accurately predicting the values of the percolation threshold for all lattice graphs. Several site percolation universal formulas based on dimension and average degree of the lattice have been proposed in the physics literature. These formulas have been proposed on an ad hoc basis, and there has been no systematic evaluation of the performance of the formulas, until one was introduced recently by Wierman and Naor.

We propose a new universal formula for predicting the site percolation threshold of two-dimensional matching lattices. The formula is slightly more accurate for these lattices than the best of the previous formulas, based on a comparison over a class of 38 lattices. The new formula illustrates how the evaluation framework can be used to improve existing universal formulas.

Friday November 25


Friday December 02

Title The STEM Education Landscape in Maryland
Speaker Anne M. Spence
Department of Mechanical Engineering,

The National Bureau of Labor Statistics is predicting a shortage of well-trained engineers, mathematicians and scientists within the next ten years. International testing reveals that children in the United States perform significantly under their international counterparts on mathematics and science tests. “No Child Left Behind” (NCLB) has forced school systems into high stakes testing to prove that every child is learning reading, mathematics, history, and science. In an effort to support NCLB, both the National Science Foundation and the U.S. Department of Education created Math/Science Partnership programs to support teacher development and student achievement. UMBC has moved into the national forefront as a NSF MSP Comprehensive Project. Dr. Freeman Hrabowski has shown his support for these initiatives by saying that UMBC will be the Maryland leader in STEM (Science, Technology, Engineering, and Mathematics) teacher education within the next five years.

Friday December 09

Title Convex duality in nonconvex quadratic optimization
Speaker Marc Teboulle
School of Mathematical Sciences
Tel-Aviv University

Quadratic problems and their relatives, conic optimization problems, are currently one of the most active area of research in optimization. Such problems arise in a wide range of fields including for example, mathematics, engineering sciences, control theory and combinatorial optimization problems. This talk will concentrate on the role of convex duality in the analysis of some instances of quadratic problems and other closely related topics such as: the convexity of the range of quadratic maps, the S-procedure, and semidefinite relaxations of combinatorial optimization problems. The main theme is to show that convex duality is a general methodology that can be successfully used to identify classes of problems where a dual bound is exact; to derive global optimality conditions; to detect hidden convexity in seemingly hard nonconvex quadratic problems; to show that dual approximations problems are somehow providing best computationally tractable bounds, and to formulate some open and challenging related questions.

The talk is intended to a wide audience. We will assume (almost) no prior knowledge in continuous optimization, convex duality, and on the aforementioned topics.