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Math Colloquia: Fall 2008

Friday September 05

Title Parallel Computing for Long-Time Simulations of a Model for Calcium Waves in a Heart Cell
Speaker Dr. Matthias Gobbert
Department of Mathematics and Statistics

The release of calcium ions in a heart cell can lead to self-organized waves of increasing concentration, which plays a role in controlling the beating of the heart. This process is modeled mathematically by a system of transient reaction-diffusion equations. Parallel computing for long-time simulations on high resolution meshes for this problem requires the combination of an appropriate numerical algorithm, its efficient implementation, and crucial hardware components to achieve excellent performance. We will show performance results from the new cluster hpc in the UMBC High Performance Computing Facility (HPCF) and discuss critical choices for hardware and usage of this machine. The long-time simulation results demonstrate that the simulations capture the crucial physical effect of self-organizing waves of calcium. However, they also highlight shortcomings of the underlying model. Current research with our new colleague Brad Peercy is directed at improving the model.

Friday September 12

Title Recent developments in the structure theory of JBW* triples
Speaker Dr. C. Martin Edwards
The Queen’s College
Oxford University

Every complex Banach space possesses a (partially defined) triple product. A JBW*-triple is a dual complex Banach space for which the triple product is globally defined. Examples are von Neumann algebras, Jordan W*-algebras, spin triples, and ternary operator algebras. They have also been proposed as models for physical systems. The talk will include an elementary introduction to JBW*-triples, their structure, and their applications, and will close with a description of some recent results in this area.

Friday September 19

11:00–noon, room MP 401, Joint Applied Mathematics & Statistics Colloquium

Title Classification in High Dimension with Nonparametric Empirical Bayes Estimation
Speaker Dr. Junyong Park
Department of Mathematics and Statistics


Friday September 26

Title The Spherical Radon Transform in Thermoacoustic Tomography – Physical Realities
Speaker Dr. Sarah Patch
University of Wisconsin, Milwaukee


Friday October 03

No talk today

Friday October 10

(Joint with Optimization Seminar)

Title Hopf Bifurcations and Stabilization of Axial Flow Engine Compressors
Speaker Dr. MingQing Xiao
Department of Mathematics
Southern Illinois University

In recent years, control of compression systems has become a topic of much research interest to control engineers. One of the major challenges in the design and operation of compression systems is handling the instabilities that arise in the unsteady fluid structural dynamics. This is because when a turbomachine, such as a jet engine, operates near its optimal operating point, the flow can become unstable. Two kinds of instability phenomena, rotating stall and surge, are of major concern in compression systems, as they can lead to undesirable reduction in performance and even damage to engine components during operations.

In this talk, I will present some of our recent results in controlling compression systems. I will first introduce the full-order compression system model, the so-called Moore-Greitzer model, and show that it is not (topologically) equivalent to its linearized version near the point where the pressure rise closes to its maximum. I will then show that the Moore-Greitzer model features a center manifold near this maximum pressure rise, which makes it possible to translate the study of the behavior of the local flow in the compressor into a study of the flow of two scalar ODE’s on the center manifold. Using the normal form of a nonlinear system obtained through integral averaging, I will introduce a nonlinear state feedback controller, which accomplishes the tasks of preventing the closed-loop system from entering either rotating stall or surge and causing the closed-loop pressure rise coefficient to approach its maximum with the elimination of hysteresis. I will close by presenting numerical simulations of open-loop and closed-loop models, to illustrate the analysis and the results.

(Research supported in part by NSF DMS-0605181)

Friday October 17

Title Neuronal dynamics: Effects of synaptic plasticity
Speaker Dr. Amitabha Bose
Department of Mathematical Sciences
New Jersey Institute of Technology

Neurons are cells in the nervous system that can produce signals to encode and transmit information related to sensory input, motor control or behavioral tasks. A signal is typically in the form of a large amplitude voltage change across the cell membrane called an action potential. A synapse is a point of communication between two neurons. Synaptic plasticity refers to the fact that the strength of the synapse can be frequency dependent. Understanding the dynamics of a neuron’s electrical activity is of central importance to understanding the function of neurons. In this talk, I will discuss how the network architecture (feed-forward or feedback) dictates how synaptic plasticity is used within a neuronal network. I will focus on two specific examples involving multistability of periodic solutions in feedback networks and of phase invariance in feedforward networks. In both cases, I will derive low-dimensional maps that determine network dynamics and show how plasticity affects these maps. The main mathematical techniques involve dynamical systems and geometric singular perturbation theory.

Friday October 24

No talk today

Friday October 31

Title Modeling Imperfect Financial Markets
Speaker Dr. Harbir Lamba
Department of Mathematical Sciences
George Mason University

The standard (efficient market) models of economics and mathematical finance make severe assumptions about the rationality, motivation and decision-making mechanisms of participants. However, their statistical properties differ significantly from those of real markets. I shall describe an alternative class of models within which many of the standard assumptions can be systematically weakened (i.e. the efficient-market paradigm exists as a special case within this framework).

Briefly, the current `strategy’ of a particular agent is defined by a pair of dynamic thresholds straddling the current price. When the price crosses either of these thresholds the agent switches investment position and a new pair of thresholds is generated. Such models are capable of robustly reproducing the most important stylized facts of financial markets, such as the apparent power-law distribution of the largest price-changes. Furthermore, the threshold dynamics can mimic different sources of investor motivation, running the gamut from purely rational information processing and inductive learning, through rational (but often undesirable!) behavior induced by perverse incentives and moral hazards, to irrational psychological effects.

The model can be reformulated as a system of particles moving on a two-dimensional domain that switch state and are reinjected whenever a boundary is crossed. This abstraction helps clarify the question of which phenomena, when introduced into an otherwise efficient market, can cause significant asset mispricing. Finally we demonstrate a connection between these threshold models and the Olami-Feder-Christensen description of earthquakes.

No investment advice will be offered.

Friday November 07

Joint with Optimization Seminar

Title Clustering linear and nonlinear manifolds
Speaker Dr. Rene Vidal
Department of Biomedical Engineering and the Center for Imaging Science
Johns Hopkins University

Over the past few years, various techniques have been developed for learning a low-dimensional representation of data lying in a nonlinear manifold embedded in a high-dimensional space. Unfortunately, most of these techniques are limited to the analysis of a single submanifold of a Euclidean space and suffer from degeneracies when applied to linear manifolds (subspaces). The simultaneous segmentation and estimation of a collection of submanifolds from sample data points is a challenging problem that is often thought of as “chicken-and-egg”. Therefore, this problem is usually solved in two stages (1) data clustering and (2) model fitting, or else iteratively using, e.g. the Expectation Maximization (EM) algorithm.

The first part of this talk will show that for a wide class of segmentation problems (mixtures of subspaces, mixtures of fundamental matrices/trifocal tensors, mixtures of linear dynamical models), the “chicken-and-egg” dilemma can be tackled using an algebraic geometric technique called Generalized Principal Component Analysis (GPCA). In fact, it is possible to eliminate the data segmentation step algebraically and then use all the data to recover all the models without first segmenting the data. The solution can be obtained using linear algebraic techniques, and is a natural extension of classical PCA from one to multiple subspaces.

The second part of this talk will present a novel algorithm for clustering data sampled from multiple submanifolds of a Riemannian manifold, e.g. the space of probability density functions. The algorithm, called Locally Linear Manifold Clustering (LLMC) is based on clustering a low-dimensional representation of the data learned using generalizations of local nonlinear dimensionality reduction algorithms from Euclidean to Riemannian spaces.

The talk will also present a few motivating applications of manifold clustering to hybrid system identification, and computer vision problems such as texture clustering, segmentation of rigid body motions, segmentation of dynamic textures, segmentation of diffusion MRI.

Friday November 14

Title A Poisson structure and Hamiltonian systems on the set of probability measures
Speaker Dr. Wilfrid Gangbo
Georgia Tech


Friday November 21

Title Efficient Preconditioning Strategies for the Incompressible Navier-Stokes Equations
Speaker Dr. Howard Elman
Computer Science Department and Institute for Advanced Computer Studies (UMIACS)
University of Maryland at College Park

We give an overview of new efficient preconditioning methods for solving the algebraic systems of equations that arise from discretization and linearization of the incompressible Navier-Stokes equations. We present several methods that take advantage of the block structure of the underlying problem that, when coupled with Krylov subspace methods, yield asymptotic convergence rates that are independent of discretization mesh size and (for evolutionary problems) time step size, and are at worst mildly dependent on Reynolds number. In addition, we show how boundary conditions affect the performance of these preconditioners. In particular, improved methods can be derived by careful treatment of boundary conditions that improve the quality of certain commutators used in the derivation. We demonstrate that with appropriate treatment of boundary conditions, transient periods of slow convergence are eliminated, leading to significantly faster performance for problems with refined meshes.

Friday November 28


Friday December 05

Title Scalable algorithms for analysis of metabolic and gene networks
Speaker Dr. Calin Belta
Mechanical Engineering, Systems Engineering and Bioinformatics
Boston University

Bacteria continuously respond to environmental changes through a complicated mechanism consisting of an interplay between metabolic and gene networks. Through hundreds of chemical reactions, the metabolic network converts nutrients from the environment to elements necessary for growth and survival of the cell. Most of these chemical reactions are regulated by enzymes produced by a gene network. In turn, the expression of the genes from the gene network is regulated by its own products and metabolites produced by the metabolic network.

In the first part of the talk, I will show how a quasi-steady state assumption on the dynamics of the metabolic network, combined with tools from convex analysis and in vivo survivability data, can lead to predictions such as essentiality of metabolites, nutrients, and chemical reactions and lead to insights into robustness of metabolism to gene mutations. In the second part of the talk, I will focus on gene networks. I will show how a particular approach to modeling, together with discrete abstractions and model checking, can be used to tune and analyze gene networks from qualitative specifications given as arbitrary temporal and logic statements over species concentrations.