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Math Colloquia: Fall 2007

Friday September 07

Title Faculty and Student Opportunities with the NSA
Speaker Dr Michelle D. Wagner, Director, Mathematical Sciences Program, Applied Research Mathematician
National Security Agency

The National Security Agency (NSA) offers a wide variety of opportunities for students and faculty members in the mathematical sciences to become involved in research. One avenue for faculty support is the Mathematical Sciences Grant Program, which funds unclassified research at colleges and universities across the U.S. in five select subdisciplines. Other avenues for involvement include a Faculty Sabbatical Program, and summer intern programs for undergraduate and graduate students in the mathematical sciences. The talk will provide details about these programs, and time will be allotted for questions at the end.

Biosketch: I’m from Asheville, NC and received the B.S.Ed. in Mathematics at Western Carolina University in 1992, and the M.S. in Applied Mathematics at Western Carolina University in 1994. In 1999 I completed the Ph.D. in Mathematics at Emory University with a concentration in combinatorics and graph theory, and with a special fondness for probabilistic algorithms, derandomization techniques, and teaching. To satisfy my desire to both teach and do research, I accepted a tenure-track position at the University of Wisconsin-La Crosse in 1999 and spent 2 enjoyable years there. Life circumstances brought me back to the east coast where I decided to try my hand as an Applied Research Mathematician at the NSA – it’s now 6 years and counting.

Friday September 14

Title A diffeomorphic mean curvature flow for the processing of anatomical surfaces
Speaker Dr John Zweck
Mathematics and Statistics

We introduce a flow on the group of diffeomorphisms of Euclidean space that is obtained by transforming the variational formulation of the classical mean curvature flow of surfaces into an optimization problem on a group of diffeomorphisms. This diffeomorphic mean curvature flow can reduce the area of a surface without inducing changes in topology or creating singularities, which can be useful in both computational anatomy and computer graphics. We discuss some analytical properties of the flow and present numerical examples.

This work is joint with Sirong Zhang, Laurent Younes, and Tilak Ratnanather.

Friday September 21

Title Adaptive Constraint Reduction for Training Support
Speaker Professor Dianne P. O’Leary
Department of Computer Science and Institute for Advanced Computer Studies (UMIACS)
University of Maryland at College Park

Data classification is a fundamental task in science and engineering. For example, given data gathered about a patient’s tumor, we might need to decide whether the tumor is malignant or benign. Ideally, we would like to determine a mathematical function whose evaluation would indicate the classification of the tumor. Linear discriminant analysis provides one such function, but functions more general than a separating hyperplane are needed in many applications.

Support vector machines (SVMs) provide a means to classify data into two groups (positive and negative) using criteria more descriptive than separating hyperplanes. SVMs are {\em trained} using a large set of positive and negative examples. Classifiers such as neural networks are trained by an iterative process of presenting examples and adjusting network weights until convergence. In contrast, the training of an SVM is acccomplished by solving a single quadratic programming problem whose size is determined by the number of examples and the number of parameters in the classifier. This simple training regime is a major advantage of the SVM framework.

These quadratic programming problems can, however, be quite large. In this work we use two approaches to improve computational efficiency. First, we apply an adaptive constraint reduction method in an interior point method for solving the quadratic programming problem. This has the effect of allowing later iterations to focus on only a few of the example datapoints. Second, we cluster the data and initially train on a small number of examples drawn from each cluster. This has the effect of reducing computation time in the early iterations.

We discuss our algorithm and its convergence theory and illustrate its performance on a variety of examples.

(Joint work with Jin Jung and Andre Tits)

Friday September 28

Title Efficient Non-Negativity Preserving High-Order Implicit Time-Stepping for Reaction-Diffusion Equations
Speaker Matthias Gobbert
Department of Mathematics and Statistics

Many problems give rise to mathematical models in the form of systems of transient reaction-diffusion equations coupled through non-linear reaction terms. Examples of applications with particular interesting physical effects that pose challenges for a numerical solution include a problem in one spatial dimension with moving internal layers and a model for the flow of calcium ions in three dimensions with a highly non-smooth source term. I will describe the techniques used for the efficient numerical simulation of these problems and how to deal with several classical challenges effectively, for instance the preservation of non-negativity of the solution while maintaining mass conservation with a fully implicit high-order time-stepping method. Parts of this work were done in collaboration with Thomas I. Seidman (UMBC), Michael Muscedere (UMBC), and Raymond J. Spiteri (University of Saskatchewan).

Friday October 05

Title Nonconforming methods for Maxwell source problems and eigenproblems
Speaker Fengyan Li
Department of Mathematical Sciences
Rensselaer Polytechnic Institute

Partially motivated by the observation that the curl-curl operator behaves differently when it is applied to the divergence-free field and the gradient field in the Hodge decomposition of a function, we introduce the reduced time-harmonic Maxwell (RTHM) equations whose solution is the divergence-free component of the solution to the time-harmonic Maxwell equations. Three schemes are formulated for numerical solving the RTHM system. Two of them use the classical nonconforming finite element approximations, and the other is based on the interior penalty type discontinuous Galerkin methods. To weakly impose the divergence-free condition satisfied by the solutions, the schemes either work with the locally divergence-free trial spaces, or contain a weighted divergence term in the formula. With the properly chosen graded meshes, the optimal error estimates are established which are confirmed by the numerical experiments. The similar numerical schemes and the error estimate results are extended for solving the reduced curl-curl problems.

The discrete operators in these schemes naturally define three Maxwell eigensolvers which are free of spurious eigenmodes and are free of penalty parameters. The analysis for these solvers is closely related to the reduced curl-curl problems and their numerical approximations. Not like many other Maxwell eigensolvers based on the full curl-curl problems, the compactness of the involved operator and the uniform error estimates for the source problems greatly simplify the analysis of our proposed eigensolvers.

Biosketch: Dr. Fengyan Li is currently an assistant professor in the Department of Mathematical Sciences at Rensselaer Polytechnic Institute. She received her B.S. and M.S. in computational Mathematics from Peking University in 1997 and 2000, and her Ph.D. in Applied Mathematics from Brown University in 2004. After that Dr. Li spent two years as a postdoctoral Fellow at University of South Carolina.

Dr. Li’s research interests are mainly in numerical analysis and scientific computing. Her recent research includes the development of the local structure preserving discontinuous Galerkin (DG) methods for solving time-dependent Maxwell equations, MHD equations, Hamilton-Jacobi equations and certain second order elliptic equations, as well as the development of nonconforming finite element methods for solving Maxwell source problems and eigenproblems. Dr. Li has also been working on the design of the DG-based fast sweeping methods for efficiently solving static Hamilton-Jacobi equations.

Friday October 12

Title Force flux and the Peridynamic stress tensor
Speaker Dr Rich Lehoucq
Computational Mathematics and Algorithms Department
Sandia National Laboratories

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differental equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go throughthe point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations. This is joint work with Stewart Silling.

Friday October 19

Title A multilevel computational approach for solving coupled fluid-structure interactions for biological and bio-inspired applications
Speaker Padmanabhan Seshaiyer
Department of Mathematical Sciences
George Mason University

In the last decade, there have been dramatic advances in our knowledge of solving coupled processes characterized by fluid-structure interaction. The efficient solution of modeling the complex nonlinear interaction of a fluid with a structure has remained a challenging problem in computational mathematics. Such applications often involve complex dynamic interactions of multiple physical processes which present a significant challenge, both in representing the physics involved and in handling the resulting coupled behavior. In this talk, a multilevel solution methodology for fluid-structure interactions will be described. Computational results that validate the performance of the technique will also be presented for various benchmark applications.

Friday October 26

(Distinguished Speaker Series)

Title Scalable Solver Infrastructure for Computational Science & Engineering
Speaker David Keyes, Fu Foundation Professor of Applied Mathematics
Columbia University

Multiscale, multirate scientific and engineering applications based on systems of partial differential equations possess resolution requirements that demand execution on the highest-capability computers available, which will soon reach the petascale. While the variety of applications is enormous, their needs for mathematical software infrastructure are surprisingly coincident. Implicit methods for transient and equilibrium problems lead after discretization to large, ill-conditioned algebraic systems. The chief to bottleneck to scalability is often the solver. At their current scalability limits, many applications spend a vast majority of their operations in solvers, due to solver algorithmic complexity that is superlinear in the problem size, whereas other phases scale linearly. Furthermore, the solver may be the phase of the simulation with the poorest parallel scalability, due to intrinsic global dependencies. The Towards Optimal Petascale Simulations (TOPS, project focuses on ameliorating this bottleneck while providing a multilevel programming interface that allows users to advance from initial concerns of correctness and robustness to ultimate concerns of efficiency and performance portability by experimenting with a variety of solvers.

We begin with an overview of the diverse petascale hardware roadmaps at the laboratories served by the TOPS project, with such applications as electromagnetism, magnetohydrodynamics, and quantum chromodynamics. We then describe the algorithmic and software roadmap of TOPS, which includes such well-known packages as Hypre, PETSc, SUNDIALS, SuperLU, and Trilinos.

Bio sketch: David E. Keyes is the Fu Foundation Professor of Applied Mathematics in the Department of Applied Physics and Applied Mathematics at Columbia University. He has authored or co-authored over 100 publications in computational science and engineering, numerical analysis, and computer science, and has delivered over 300 invited presentations at universities, laboratories, and industrial research centers. With backgrounds in engineering, applied mathematics, and computer science, Keyes works at the algorithmic interface between parallel computing and the numerical analysis of partial differential equations, across a spectrum of aerodynamic, geophysical, and chemically reacting flows. Newton-Krylov-Schwarz parallel implicit methods, introduced in a 1993 paper he co-authored, are now widely used throughout engineering and computational physics, and have been scaled to thousands of parallel processors. Keyes has received numerous awards for his teaching and research, most recently the Sidney Fernbach Award 2007. A SIAM Visiting Lecturer since 1992, Keyes became the Vice President-at-Large of SIAM in January 2006. At Columbia, he is faculty advisor to the local student chapter of SIAM.

Friday November 02

No talk today

Friday November 09

Title Extremal ellipsoids of convex bodies and their symmetry properties
Speaker Osman Guler
Department of Mathematics and Statistics

A convex body K is a compact convex set in R which has an interior. It is known that K has a unique circumscribed ellipsoid MVCE(K) and a unique inscribed ellipsoid MVIE(K). The former contains K and has minimum volume, and the latter is contained in K and has maximum volume. The two ellipsoids crop up in many different fields under different disguises.

In this talk, we first develop the basic properties the the ellipsoids using semi-infinite programming. The uniqueness of the ellipsoids have the important property that they inherit the symmetry poperties of K. The seond part of the talk will be about the symmetry properties convex bodies and ellipsoids.

Friday November 16

Title Representation and Statistical Estimation of Deformable Template Models for Shape Recognition and Computational Anatomy
Speaker Stephanie Allassonniere
Johns Hopkins University

This presentation takes place within the framework of image analysis with a statistical point of view. Image analysis has been widely studied this past decade in particular the geometrical side of it which consists in developing ways of matching elements extracted from images.

All these matching methods require the choice of some important elements as the template and the metric put on the deformation space. We thus propose a statistical model which allows us to learn all these quantities as parameters of the model through some derivatives of the EM algorithm: one deterministic, using a mode approximation of the posterior density, the second stochastic (SAEM), coupled with the use of some MCMC methods. We prove that under the model, the parameter estimator is consistent and the stochastic algorithm is convergent. The model is then generalised to a mixture of deformable template models to perform a clustering of the dataset which also enables to consider the model as a classifier.

Monday November 19

Visiting professor at University of Illinois at Urbana-Champaign

Title Billiards and Farey series
Speaker Radu Gologan
University “Politehnica” Bucharest and Institute of Mathematics Bucharest,

One of the problems that has stirred a lot of interest in statistical physics, ergodic theory and number theory over the last decades , is the so called Lorenz gas model, or Sinai billiard. This is a billiard system on the two dimensional torus with one or more circular regions removed. This model was intensively studied from the point of view of dynamical systems. We will present some exact results on the statistical behaviour of such models using the analytic number theory machinery: Kloosterman sums and Farey series. This is a work started by the author jointly with Florin Boca and Alexandru Zaharescu.

Friday November 23


Friday November 30

Title Prediction Problems and the Numerical Calculation of Random Integrals Arising in the Theory of Gaussian Markov Random Fields
Speaker Loren D. Pitt, Emeritus Professor of Mathematics, Statistics, and Education
University of Virginia

The prediction of the values of a Gaussian Markov random field inside a domain based on knowledge of the field outside that domain is given by the solution of a linear boundary value problem with random boundary data. When the boundary of the domain is sufficiently smooth this is a classical problem but typically some of the boundary data (normal derivatives) only exists in the generalized sense as random Schwartz distributions. In any practical setting this leads to the problem of estimating these distributions from a finite set of observations, which gives rise to a large set of both applied and theoretical questions.

Biosketch: Loren got his PhD from Princeton in mathematics in 1967, and he was one of William Feller’s last doctoral students. He has a strong background in analysis and probability theory and even has an MS in biometry to add another dimension. He has worked on analysis problems using the insight of probabilistic techniques and problems in probability using analytic techniques—a reflection of his Princeton training.

Loren was at Rockefeller University and since then has been at the University of Virginia, where he was chair of the Mathematics Department for three or four years. He was on the editorial board of the Journal of Multivariate Analysis for 16 years and has served as Editor of two Birkhauser monograph series for over twenty years.

In addition, Loren has been heavily involved in improving the mathematics education of high school teachers in Virginia and for some years was Director of the Center for Liberal Arts at Virginia, while continuing his reasearch work and his supervision of doctoral students.

Friday December 07

No talk today