Math Colloquia: Fall 2004

Friday September 10

Title Refining Rate Constant Estimates in the BIAcore
Speaker David A. Edwards
Department of Mathematical Sciences
University of Delaware

The BIAcore is an ingenious device that allows the measurement of rate constants for binding processes without disturbing the system. However, accurate mathematical models are needed to interpret the raw data correctly. This talk will discuss the latest enhancements to these models, including the effects of decay in the measuring wave and flow inside the reacting receptor layer. By using asymptotics and perturbation methods, simple expressions may be obtained which are valid for a wide range of experimental parameters. These solutions, which provide corrections to the rate constants measured in the BIAcore, are interpreted physically.

Friday September 17

Title Optimization of Risk Measures
Speaker Alexander Shapiro
Industrial and Systems Engineering
Georgia Tech

Mean-risk approach to optimization under uncertainty is going back to the pioneering work of Markowitz (1952) and is routinely used in portfolio selections. More recently an axiomatic approach to the mean-risk analysis was suggested by Artzner et al (1999), which started an intensive development of a mathematical theory of risk measures. In this talk we discuss the mean-risk methodology from an optimization point of view. We show that the min-max, utility and mean-risk approaches to stochastic optimization, in a sense, are equivalent to each other. We also formulate mean-risk measures in a multi-stage setting and derive dynamic programming type equations.

Friday September 24

Title Curvature of piecewise-linear surfaces
Speaker David Cohen-Steiner
Computer Science
Duke University

In many applications, such as surface segmentation, anisotropic surface remeshing or non-photorealistic rendering, a key step is to estimate the curvature of a smooth surface knowing only a piecewise-linear approximation of it. A popular method for solving this problem is quadric fitting, where the estimated curvature is the one of the quadric that best fits the surface locally. In this talk, I will present a different approach to curvature estimation. Using the theory of normal cycles from geometric measure theory, I will show how curvature tensors can be defined for a large class of surfaces, including smooth and polyhedral ones. The curvature tensor of a piecewise-linear approximation of a smooth surface can then be used as an estimator of the one of the smooth surface. By analyzing the difference between the curvature of the surface and the one of its approximation, one can show that this estimator indeed converges to the curvature of the underlying smooth surface when the resolution increases. If time permits, I will also discuss the special case of restricted Delaunay triangulations, as well as an application of this work to the anisotropic remeshing of triangulated surfaces.

Friday October 01

No talk today

Friday October 08

Title Optimization Models in Quantitative Finance
Speaker Reha Tuntuncu
Mathematical Sciences
Carnegie Mellon University

Optimization models and methods play an increasingly important role in financial decision making. Many problems in quantitative finance including asset allocation, risk management, derivative pricing, and model fitting are now routinely and efficiently solved using modern optimization techniques. In this talk, we will survey some examples of such problems we studied in our recent work. In particular, we will describe our models for robust asset allocation, determination of robust profit opportunities, estimation of risk-neutral densities using option prices, and returns-based estimation of sector allocations.

Friday October 15

Title Large-Scale Nonlinear Optimization in Circuit Tuning
Speaker Andreas Waechter
IBM T.J. Watson Research Center

Circuit tuning is an essential step in the design of digital circuits. The central task is to find optimal sizes of transistors in order to minimize signal delay or area requirement. This problem can be formulated as a large-scale nonlinearly constrained nonlinear optimization problem, where function evaluations are obtained by simulation of gates (small subcircuits). This approach has been implemented in the IBM-internal circuit tuning tool EinsTuner, which has been used for the design of every custom digital circuit designed by IBM for several years.

The numerical optimization engine is IPOPT, which follows a primal-dual interior point approach and uses a line search filter method to ensure global convergence. Details on the optimization algorithm and numerical results will be presented.

Friday October 22

Title SDP approximations for copositive and completely positive matrices (or, Pólya meets De Finetti)
Speaker Pablo A. Parrilo
Electrical Engineering and Computer Science

The recognition and verification of matrix copositivity is a well-known computationally hard problem, with many applications in continuous and combinatorial optimization. In this talk we discuss a hierarchy of approximations for a real matrix to be copositive, based on semidefinite programming (SDP). These conditions are obtained through the use of a sum of squares decomposition for multivariable forms. The completeness of the hierarchies is shown to be equivalent to classical results for homogeneous forms and exchangeable random variables due to Pólya and De Finetti, respectively. We will discuss their relationship, the application of the results to some well-known families of copositive forms, as well as a “quantum” version of the problem.

Bio Sketch: Pablo A. Parrilo is Associate Professor in the Department of Electrical Engineering and Computer Science at MIT, and a member of the Laboratory for Information and Decision Systems (LIDS). His current research interests include control and identification of uncertain complex systems, robustness analysis and synthesis, and the development and application of computational tools based on convex optimization and algorithmic algebra to practically relevant problems in engineering, economics, and physics.

From October 2001 through September 2004, Pablo was Assistant Professor of Analysis and Control Systems at the Automatic Control Laboratory of the Swiss Federal Institute of Technology (ETH Zurich). He received an Electronics Engineering degree from the University of Buenos Aires in 1994, a PhD in Control and Dynamical Systems from the California Institute of Technology in 2000, and held short-term visiting appointments at UC Santa Barbara, Lund Institute of Technology, and UC Berkeley.

Friday October 29

Title An Optimal Design of the M/M/C/K Queue for Call Centers
Speaker William A. Massey
Department of Operations Research and Financial Engineering
Princeton University

Motivated by the performance analysis of call centers, we develop an optimal design analysis for the M/M/C/K queueing system in steady state. The number of servers C corresponds to the number of agents and C+K, where K equals the number of additional waiting spaces, corresponds to the number of telephone lines. Our goal is to find the optimal (C,K) for this multi-server, loss and delay queueing system by holding two key service metrics below their given target values, for a predetermined level of offered load traffic. This is joint work with Rodney Wallace of IBM.

Biosketch: Professor Massey is the Edwin S. Wilsey Professor in the Department of Operations Research and Financial Engineering, a member of the Applied and Computational Mathematics Program, and an associate member of the Department of Mathematics at Princeton University. From 1981 until 2001, he was a researcher in the Mathematical Sciences Research Center at Bell Laboratories of Lucent Technologies. His research interests include queueing theory, applied probability, as well as performance and pricing models for telecommunication systems. He received his undergraduate degree in Mathematics from Princeton University in 1977 and his Ph.D. in Mathematics from Stanford University in 1981.

Friday November 05

Title Eavesdropping on Synaptic Traffic
Speaker Steve Cox
Computational and Applied Mathematics
Rice University

Nerve cells communicate to one another across synapses. The receiver encodes this message as a change in local, in space and time, conductance. This change engenders a postsynaptic change in potential that actively diffuses through the dendritic tree and eventually may lead to the firing of a nervous impulse which may in turn lead to a long term change in the aforementioned synaptic conductance. To quantify this synaptic plasticity we propose a non invasive cocktail of optical imaging via voltage sensitive dyes and numerical determination of synapse location and conductance time course. In this talk we will focus on the mathematical and numerical study of the sideways Hodgkin-Huxley system that permits one to eavesdrop on synapses.

Wednesday November 10

Joint Math/Stat Colloquium

Title Sample Size and Power of Randomized Designs
Speaker Feifang Hu
Department of Statistics
University of Virginia

Randomized designs are often used in clinical trials. In the literature, the power and sample size are usually obtained by ignoring the randomness of the allocation in randomized designs. However, when using a randomized design, the power is a random variable for a fixed sample size $n$. In this talk, we focus on the power function (random) and the sample size of two-arm (drug versus control) randomized clinical trials. We first give an example where a target power can not be achieved with high probability when the requisite sample size (based on the formula in the literature) is used. Then we obtain the power function for any given sample size and study the properties of this power function. Based on the power function, a formula of sample size is derived for randomized designs. This formula is applied to several randomization procedures. We also discuss our finding that response adaptive designs can be used to reduce the requisite sample size. Some simulation studies are reported.

Friday November 12

Title Nonlinear Light Propagation in Photonic Lattices
Speaker Jianke Yang
Department of Mathematics and Statistics
University of Vermont

Recently, light propagation in an optically-induced photonic lattice is stirring a lot of interest due to their novel physics,light-routing applications, as well as connections to photonic crystals and Bose-Einstein condensation.

In this talk, I will report our recent theoretical and experimental results on nonlinear localized states of light in a two-dimensional photonic lattice. These localized states include the fundamental lattice solitons, vortex solitons, dipole solitons, and quadrupole solitons. These solitons exhibit interesting geometric structures, and they arise due to the presence of the photonic lattice. The stability analysis for these novel solitons will also be presented.

Experimentally, we have observed these lattice solitons in photorefractive crystals, and the experimental results are in good agreement with the theoretical predictions.

This work was done in collaboration with Prof. Z. Chen in San Francisco State University.

Friday November 19

Title Homogenization of Porous Media Equations with an Application to Cancellous Bone
Speaker Robert P. Gilbert, Unidel Foundation Professor
Department of Mathematics
University of Delaware

The equations for the acoustic modeling of a poroelastic material will be derived and an application given to the case of cancellous bone. Good acoustic models of cancellous bone are useful for ultrasound interrogation to determine whether the patient has osteoporosis.

Friday November 26


Friday December 03

No talk today

Friday December 10

Title On a multidimensional model for the dynamic combustion of compresssible reacting flow
Speaker Konstantina Trivisa
Department of Mathematics
University of Maryland College Park

We consider a multidimensional model for the dynamic combustion of compressible reacting fluids formulated by the Navier Stokes equations in Euler coordinates. For the chemical model we consider a one way irreversible chemical reaction governed by the Arrhenius kinetics. The existence of globally defined weak solutions of the Navier-Stokes equations for compressible reacting fluids is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl and P.L. Lions.