Math Colloquia: Fall 2011

Friday September 2

No talk today

Friday September 9

Title Multigrid preconditioners for linear systems arising in PDE constrained optimization
Speaker Andrei Draganescu

We will discuss the problem of finding optimal order multigrid preconditioners for linear systems involved in the solution process of large-scale, distributed optimal control problems constrained by partial differential equations. Multigrid methods have long been associated with large-scale linear systems, the paradigm being that the solution process can be significantly accelerated by using multiple resolutions of the same problem. However, the exact embodiment of the multigrid paradigm depends strongly on the class of problems considered, with multigrid methods for differential equations (elliptic, parabolic, flow problems) being significantly different from methods for PDE constrained optimization problems, where the linear systems often resemble integral equations. In this talk we will present a number of model problems for which we were able to construct optimal order multigrid preconditioners, as well as problems where we have been less successful. The test-problems include (a) linear and semi-linear elliptic constrained problems, (b) optimal control problems constrained by Stokes flow (both (a) and (b) without control-constraints), and (c) control-constrained problems with linear-elliptic PDE constraints.

Friday September 16

Title A Low-Order Model of Biological Neural Networks
Speaker James Lo

A Low-Order Model (LOM) of biological neural networks, which is also a new paradigm of learning machines, will be proposed. LOM is a network of biologically plausible models of dendritic nodes/trees, spiking/nonspiking neurons, unsupervised/supervised covariance/accumulation learning mechanisms, feedback connections, and a scheme for maximal generalization. These component models were motivated and necessitated by making LOM learn and retrieve easily; and cluster, detect and recognize multiple/hierarchical corrupted, distorted and occluded temporal and spatial patterns. On one hand, biological plausibility of LOM makes a strong case that LOM is the common cortical algorithm long hypothesized by neuroscientists. On the other hand, with many features and capabilities desirable of a learning machine, LOM is expected to be a powerful engine for intelligent systems.

Friday September 23

No talk today

Friday September 30

Title Stability of Switched Homogeneous Systems: Spectral Radius and Generating Function Approaches
Speaker Jinglai Shen

This talk presents stability analysis of discrete-time switched homogeneous systems on cones under arbitrary and optimal switching rules. Several interrelated approaches, such as the joint spectral radius approach and the generating function approach, are exploited to derive necessary and sufficient stability conditions and to develop efficient algorithms for stability tests. Specifically, the generalized joint spectral radius and the generalized joint lower spectral radius are introduced to characterize the radii of domains of strong and weak attraction. Furthermore, strong and weak generating functions and their radii of convergence are employed to derive stability conditions; their analytic properties, numerically effective approximations and convergence analysis are established. An example is given to demonstrate the proposed generating function approach.

Friday October 7

Title Higher-Order Tensor Factorizations
Speaker Carla D. Moravitz Martin
James Madison University

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Usually, one is interested in decomposing the tensor in a way analogous to matrix decompositions. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.

In this talk, we will cover the existing factorizations and explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Most of the current tensor decompositions are based on multilinear extensions of the matrix singular value decomposition (SVD). We also present a new SVD-like factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors and identify the corresponding characteristics. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications. We conclude with an application in image deblurring.

Friday October 14

Title Price Discrepancy and Optimal Trading of Derivatives
Speaker Tim Siu-Tang Leung
Columbia University

In incomplete markets, where not all risks can be hedged, different risk-neutral or risk-averse pricing models may yield a range of no-arbitrage prices for derivatives. The investor’s model price may disagree with the market price. This leads us to investigate the optimal timing to buy and/or sell a derivative security under price discrepancy. We show that the structure of the resulting optimal stopping problem depends on the interaction between the respective market price of risk and the option payoff. Also, we introduce the delayed purchase premium to characterize the investor’s optimal strategies. Explicit solutions and numerical examples are given for some representative classes of Markovian models, including diffusion stochastic volatility models and intensity credit risk models.

Friday October 21

Title Talk canceled; moved to December 2
Speaker Christian Gromoll
University of Virginia


Friday October 28

Title Two Perspectives on Mathematical Biology
Speaker Kathleen Hoffman

Mathematical biology has received significant attention recently, and has been called `the new physics’. In this talk, we will explore two perspectives in mathematical biology. The first perspective focuses on a mathematical problem with applications to biology and will be illustrated with an example from elastic rod models of DNA. The second perspective is data-driven modeling of biological systems and will be illustrated with an example that models neuromechanical locomotion. We will discuss the impact of each perspective in terms of the impact on the mathematical literature and the biological system.

Friday November 4

Title Geometric Control on Nonlinear Manifolds for Complex Aerospace Systems
Speaker Taeyoung Lee
The George Washington University

Many interesting dynamic systems in science and engineering evolve on a nonlinear, or curved, space that cannot be globally identified with a linear, Euclidean space. Such nonlinear spaces are referred to as manifolds, and they appear in various dynamic systems, varying from a simple pendulum to complex multibody systems. However, the geometric structures of a nonlinear manifold have not been extensively studied in control system engineering. The traditional nonlinear control systems based on local coordinates of a nonlinear manifold may exhibit singularities and complexities, and therefore, it is difficult to obtain global stability properties. This talk summarizes new geometric approaches for computational dynamics, optimization, feedback control, and estimation of mechanical systems evolving on a nonlinear manifold. By constructing a control system directly on a manifold in a coordinate-free fashion, we can control non-trivial, aggressive motion of complex dynamic systems globally over a long time period. The desirable properties of geometric approaches are demonstrated by several aerospace systems, such as formation reconfiguration of satellites, tethered spacecraft, and a quadrotor UAV.

Friday November 11

Title Random graphs based on messaging activity counting processes
Speaker Nam Lee
Johns Hopkins University

In this talk, we consider a community of n individuals where for each pair ij of individuals, i and j are exchanging messages that can be colored with one of K tags. For analyzing such a corpus of social interaction event records, we introduce and motivate a model for a network of point processes.

Friday November 18

Title Non-remotality for closed and bounded convex sets
Speaker T. S. S. R. K. Rao
Indian Statistical Institute

We consider closed and bounded convex sets that are not remotal. Using extremal arguments we show that how such sets arise in infinite dimensional dual spaces. Using integral representations in a joint work with M. Martin from Granada (Spain) we could show that every infinite dimensional Banach space has a closed and bounded convex set that is not remotal. We also consider uniqueness of vectors of maximal length.

Friday November 25


Friday December 2

Title Least-Attained-Service queues and the M_1 topology
Speaker Christian Gromoll
University of Virginia

I’ll describe a recent result on fluid limits for Least-Attained-Service queues. Interestingly, this policy does not have a fluid limit when the standard Skorohod J_1 topology is used on path space. The limit requires using the weaker, and more uncommon, M_1 topology. I’ll describe some of the steps required to work with this topology on the space of cadlag measure-valued paths.