Math Colloquia: Spring 2009

Wednesday February 04

Title Pricing and Hedging in Incomplete Markets
Speaker Dr. Jungmin Choi
Department of Mathematics
Florida State University

Modern financial option analysis was initiated in 1973 by Black and Scholes. The Black-Scholes analysis is used in practice, but many of the underlying assumptions are questionable or even incorrect. New approaches have been introduced to analyze more realistic models and find bounds on prices of options and strategies for managing the risk inherent in options.

In this talk, option pricing problems are considered when we relax the condition of no arbitrage in the Black-Scholes model. Assuming random noise in the interest rate process, the derived pricing equation is in the form of stochastic partial differential equation. Karhunen-Loeve expansion is used to approximate the stochastic term and a combined finite difference/finite element method is used to effect temporal and spatial discretization.

Another assumption in Black-Scholes model is perfect elasticity, which may not be true when there is a “large” agent in the market. Partial hedging problem in financial markets with a large agent is considered, and a Bellman type partial differential equation is derived. An asymptotic analysis leads us to conclude that the expected loss increases when there is a large agent, which means that one would need more capital to hedge away risk in the market with a large agent. This asymptotic analysis is confirmed by performing Monte Carlo simulations.

Monday February 09

Title New Efficient Sparse Space-Time Algorithms in Numerical Weather Prediction
Speaker Dr. Yulong Xing
Courant Institute of Mathematical Sciences
New York University

A major stumbling block in the prediction of weather is the accurate parameterization of moist convection on microscales. A recent multi-scale modeling approach, superparameterization (SP), has yielded promising results and provided a potential solution to this problem. SP is a large-scale modeling system with explicit representation of small-scale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. In this talk, we will present new efficient sparse space-time algorithms of SP which solve the small scale model in a reduced spatially periodic domain with a reduced time interval of integration. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear. The numerical results are compared with the CRM and original SP. It is shown that the new efficient algorithms for SP result in a gain of roughly a factor of 10 in efficiency, and the large scale variables such as horizontal velocity and specific humidity are captured in a statistically accurate way.

Wednesday February 11

Title Performance Analysis of Many-server Queues with Reneging
Speaker Dr. Weining Kang
Department of Mathematical Sciences
Carnegie Mellon University

Motivated by problems of current relevance for call centers, we consider a queuing system with a single pool of N identical servers that process incoming customers who have generally distributed service requirements, and abandon the queue if their waiting time exceeds their so-called patience time. We derive a first-order approximation of this system and study its asymptotic behavior, as the number of servers goes to infinity. We also discuss the implications of our analysis for the design of a call center. The analysis involves a range of mathematical tools, from measure-valued processes and renewal theory to partial differential equations.

Friday February 13

Title Inexact Balancing Domain Decomposition by Constraints Algorithms
Speaker Dr. Xuemin Tu
Department of Mathematics
University of California, Berkeley

Balancing domain decomposition by constraints (BDDC) algorithms are non-overlapping domain decomposition methods for solutions of large sparse linear algebraic systems arising from the discretization of boundary value problems. They are suitable for parallel computation. The coarse problem matrix of BDDC algorithms is generated and factored by a direct solver at the beginning of the computation. It will become a bottleneck when the computer systems with a large number of processors are used. In this talk, an inexact coarse solver for BDDC algorithms is introduced and analyzed. This solver helps remove the bottleneck. At the same time, a good convergence rate is maintained. We will also talk about the extensions of these inexact BDDC algorithms to saddle point problems and problems with mortar finite element discretization.

Monday February 16

Title Perfect simulation of Matern Type III point processes
Speaker Dr. Mark Huber
Department of Mathematics
Duke University

Spatial data are often more widely separated than would be expected if the points were independently placed. Such data can be modeled with repulsive point processes, where the points appear as if they are repelling one another. Various models have been created to deal with this phenomenon. Matern created three procedures that generate repulsive processes. While the third type allows the most flexibility in modeling, Matern was unable to resolve the high dimensional integrations needed to utilize the process for inference. In this talk, I will show how to build an algorithm for using Matern Type III processes that can be used to approximate the likelihood and posterior values for data. First, a Metropolis Markov chain is created using a secondary Poisson process. Next, this chain is used together with bounding chains to obtain perfect draws from the stationary distribution of the chain. Finally, a product estimator is constructed (again using a secondary Poisson process) in order to obtain approximations with provably good error bounds.

Friday February 20

No talk today

Friday February 27

Title A kinetic theory for coupled oscillators
Speaker Dr. Carson Chow
National Institutes of Health

Networks of coupled oscillators have been used to model a wide range of phenomena such as interacting neurons, flashing fireflies, chirping crickets and coupled Josephson junctions. Typically, these networks have been studied analytically in regimes where the number of oscillators are small or in the “mean field” infinite size limit. The dynamics of networks that are large but not infinite is not well understood, although this is where many of the interesting applications lie. I will present a formalism to analyze large but finite-sized networks using an approach borrowed from the kinetic theory of gases and plasmas. The result is an infinite number of coupled equations for the moments of the probability density function for the dynamics (i.e. moment hierarchy) that can be truncated to estimate finite-sized fluctuation and correlation effects. In addition, it can be shown that the moment hierarchy is equivalent to a path integral formulation where diagrammatic methods can be employed to assist in the analytical calculations.

Friday March 06

No talk today

Friday March 13

Title An infinite-dimensional problem in non-smooth optimization
Speaker Dr. Thomas Seidman
Department of Mathematics and Statistics

Joint work with Kathleen Hoffman (cf., her talk 3/9 in the DE Seminar) is formulated as minimizing a convex function on a Hilbert space subject to an interesting constraint: physically, a rod of radius ρ > 0 is not permitted to interpenetrate itself. Here assuming existence of a (local) minimizer, we wish to find the first order optimality conditions and use these to obtain some additional regularity for the minimizer.

Friday March 20

Spring Break

Friday March 27

Title ?
Speaker Dr. Victor Patrangenaru
Florida State University


Friday April 03

Title Proximal Point Algorithms for Large-Scale Ill-Posed and Nonconvex Problems
Speaker Dr. Russell Luke
Department of Mathematical Sciences
University of Delaware

Motivated by several central problems in computational quantum chemistry and image processing, we investigate mathematical foundations of algorithmic strategies for solving large-scale nonlinear equations, most of which can be derived as variational principles for solving nonconvex, ill-posed optimization problems. The applications include: solving the Schroedinger equation for solids using density functional theory; determining the shape and location of acoustic or electromagnetic scatterers from far field data; and the phase retrieval problem in optics and crystallography. We present the most successful computational approaches for solving these problems as instances of a single proximal point algorithm, the foundations of which we investigate in detail. Our investigations of computational foundations of a unifying algorithmic framework yields better strategies for solving the application problems as well as the outlines of new theoretical tools that are needed for these modern applications.

Friday April 10

Title Game Theoretic Sensor Resource Management for High Level Fusion
Speaker Dr. Sang (Peter) Chin
Applied Physics Lab
Johns Hopkins University

We present a hierarchical game-theoretic approach to level 2 and level 3 fusion problems utilizing concepts from statistical modeling, Bayesian analysis, and the theory of uncertain games. Despite the substantial progress in level 1 fusion research over the past couple of decades, far less progress has been made in level 2 or level 3 fusion research, mainly due to the lack of a coherent theory. Given level 1 data such as observations of individual objects on sea, land or littoral regions and prior knowledge of the composition of enemy grouping, we first consider the level 2 problem for determining the enemy identity. We then apply game theory toward the level 3 problem, that is inferring the intent of enemy groups, given the determined identities and prior knowledge of enemy strategies.

Friday April 17

Title The Lorenz manifold: From mathematics to steel
Speaker Dr. Hinke Osinga
Bristol Centre for Applied Nonlinear Mathematics
University of Bristol, UK

The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. The organisation of the dynamics in the Lorenz system and also how the dynamics depends on the system parameters has long been an object of study. This talk addresses the role of the stable and unstable manifolds in organising the dynamics more globally. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold (the Lorenz manifold) and the other two equilibria have two-dimensional unstable manifolds. The intersections of these manifolds in three-dimensional space are heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent study of these manifolds reveals an intriguing combinatoric structure and how it ties in with related structures known for the periodic orbits and homoclinic bifurcations.

The fascination of the Lorenz system goes far beyond mathematics and you will see how the Lorenz manifold was turned into a steel sculpture.

Friday April 24

Title An outsider’s view of entanglement
Speaker Dr. Melvin Currie
National Security Agency

Why in tarnation should the entanglement measure for a pair of quantum bits take on a continuum of values?”

Friday May 01

Title Hybrid inclusions: Modeling and analysis of dynamical systems with continuous-time and discrete-time features
Speaker Dr. Rafal Goebel
Department of Mathematics and Statistics
Loyola University Chicago

Various dynamical systems exhibit behaviors usually attributed to continuous-time systems and behaviors usually attributed to discrete-time systems. For example, currents in a circuit can change continuously, according to Kirchoff’s laws, and instantly, due to switches opening and closing. Similarly, control algorithms designed for continuous-time control systems may involve timers that are reset, modes of operation that switch from one to another, etc. Hybrid inclusions provide a simple framework for modeling and analysis of such systems, through a combination of differential inclusions, difference inclusions, and constraints on the motions resulting from these inclusions.

After motivating examples and an overview of some other mathematical approaches to dynamical systems that combine continuous-time and discrete-time dynamics, the talk will present the framework of and basic results for differential inclusions. The use of generalized time domains for the parameterization of solutions to differential inclusions will be motivated. Elements of asymptotic stability theory for differential inclusions will be presented. In particular, a technique of approximating a hybrid inclusion with a simpler one, far generalizing the concept of linearization, will be described. Then, asymptotic stability for a hybrid inclusion will be deduced from asymptotic stability for the approximation.

Friday May 08

Title Anderson Acceleration for Fixed-Point Iteration
Speaker Dr. Homer Walker
Mathematical Sciences Department
Worcester Polytechnic Institute

Fixed-point iterations occur naturally and are commonly used in a broad variety of computational science and engineering applications. In practice, fixed-point iterates often converge undesirably slowly, if at all, and procedures for accelerating the convergence would be useful. This talk will focus on a particular acceleration method, which originated in work of Anderson (1965). This method has enjoyed considerable success in electronic-structure computations but seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by mathematicians and numerical analysts, Anderson acceleration has received relatively little attention from them, despite there being many significant unanswered mathematical questions. In this talk, I will outline Anderson acceleration, discuss some of its theoretical properties, and demonstrate its performance in several applications.