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Math Colloquia: Fall 2012

Friday September 7

Title Analysis and approximation of a constrained optimization problem representing a rod with hard self-contact
Speaker Thomas Seidman

Minimizing the integrated pointwise potential of a stationary elastic rod dates back to Euler, but only recent work includes analysis of the self-contact problem of the rod with itself. In the case of an elastic rod with hard contact, the minimization is complicated by the requirement that the rod is treated as a solid tube without permitting self-penetration. This is a nonconvex inequality constraint and requires construction of the basic normal cone at a minimizer to compute contact forces. One may view the constraint as the imposition of an infinitude of nonlinear scalar constraints. We formulate the variational formulation of the elastic rod with hard contact in this framework and prove existence and regularity of minimizers. We further approximate both the functional and the minimizers of this problem by a sequence of elastic rods with soft contact, which are modeled of by unconstrained optimization problems. We prove convergence of both the energy and the minimizers to the appropriate hard contact energy and minimizers. This is ongoing work with K. Hoffman.

Friday September 14

Title A Finite Element-Discontinuous Galerkin method for coupled flow-transport problems
Speaker Stefan Kopecz
University of Kassel, Germany

The simulation of cavitation in the context of micro foams can be modeled as a combination of a flow model with a second model for transport. The flow equations generalize the incompressible Navier-Stokes equations in terms of variable viscosity and density, plus a pressure and void fraction dependent divergence constraint. The transport of the void fraction is given by an advection equation. A numerical method for the solution of these equations will be presented. It utilizes the Finite Element (FE) method for the solution of the flow equations and a Discontinuous Galerkin (DG) method for the transport. Main focus is on a nonlinear projection scheme for the flow equations, the satisfaction of a maxium principle within the DG method and a post processing strategy to overcome the incompatibilty of the FE and DG discretizations.

Friday September 21

Title Physical transformations between quantum states
Speaker Chi-Kwong Li
College of William and Mary

Given two sets of quantum states $\{A_1, \dots, A_k\}$ and $\{ B_1, \dots, B_k\}$, represented as density matrices, necessary and sucient conditions are obtained for the existence of a physical transformation $T$, realized as a trace-preserving completely positive map, such that $T(A_i) = B_i for $i = 1, \dots, k$. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered. The talk is based on joint work with Zejun Huang, Edward Poon, Yiu-Tung Poon, Nung-Sing Sze

Friday September 28

Title Nonstandard Finite Difference Discretizations of Population Models Satisfying Conservation Laws
Speaker Talitha Washington
Howard University

In this talk, we consider the roles conservation laws can play in providing restrictions on the construction of finite difference discretizations of interacting population systems, modeled by coupled ordinary differential equations. Our analysis is formulated within the nonstandard finite difference (NSFD) methodology of Ronald Mickens. Using a number of well-known population models, we illustrate the details of our procedures by constructing appropriate NSFD discretizations. The relevance of these results to various issues associated with the numerical integration of the original population system differential equations is also presented, especially the role of positivity of the solutions. This is joint work with Ronald Mickens.

Friday October 5

No talk today

Friday October 12

Title Optimal Runge-Kutta smoothers for time dependent partial differential equations
Speaker Philipp Birken
University of Kassel, Germany

We consider implicit time integration schemes for time dependent PDEs. The appearing nonlinear systems are typically solved using either preconditioned Jacobian-Free Newton-Krylov schemes or the FAS variant of multigrid, where the latter are typically designed for steady problems. This leads to suboptimal schemes. With regards to parallel computations, low storage and scalable preconditioners, which can be obtained using multigrid methods with appropriate smoothers. In this talk, we discuss possibilities of finding optimal low storage Runge-Kutta type smoothers that satisfy these requirements.

Friday October 19

Title Boundary Control Approach to Inverse Spectral and Dynamical Problems
Speaker Sergei Avdonin
University of Tennessee at Chattanooga

The Boundary Control (BC) method is based on the deep connection between control theory for partial differential equations and inverse problems of mathematical physics and offers an interesting and powerful alternative to previous identification techniques based on spectral or scattering methods. This approach has several advantages: (i) it is applicable to a wide range of linear lumped and distributed systems and reconstruction situations; (ii) it is, in principle, dimension-independent; (iii) it lends itself to straightforward algorithmic implementations. Being originally proposed for solving the boundary inverse problem for the multidimensional wave equation, the BC method has been successfully applied to all main types of linear equations of mathematical physics. In this talk we discuss connections between the BC method and the classical Gelfand–Levitan and Krein theories, and the recently proposed Simon and Remling approaches. We also demonstrate how our approach can be extended to inverse problems on graphs.

Friday October 26

No talk today

Friday November 2

Title Certified-by-design control synthesis using finite state $\rho/\mu$ approximations
Speaker Danielle Tarraf
Johns Hopkins University

Hybrid systems, pervasive in both engineered and natural systems, pose a broad spectrum of technical hurdles. The problem of synthesizing certified-by-design controllers for these systems is known to be particularly challenging. While several approaches have been investigated, the past decade has witnessed rising interest in finite state approximations as a means of attacking this technical challenge. Plants that are constrained to interact with their controllers via fixed discrete alphabet sets, which are the subject of this talk, can be viewed as a special class of hybrid systems: In addition to the challenging control design problems inherent in hybrid dynamics, the discrete alphabet setting gives rise to non-trivial state estimation problems. We will begin by surveying a set of analysis tools that are tailored towards systems over finite alphabets, as well as synthesis tools for finite state models. A common component of these tools is the use of input-output constraints as a means of describing system properties of interest. We then propose a notion of approximation, referred to as ‘$\rho / \mu$ approximation’, that seeks to approximate systems over finite alphabets by finite memory models, and to quantify the quality of approximation in a manner compatible with the developed analysis and synthesis tools. Finally, we present constructive algorithms for generating such $\rho/\mu$ approximations, and we demonstrate the use of this approach for simple illustrative examples.

Friday November 9

Title Modeling of Microbial Biofilm Communities
Speaker Isaac Klapper
Temple University

Single-celled, microbial organisms are estimated to make up a large fraction of extant biomass. Many of these microbial communities exist in the biofilm form. (A biofilm is a dense aggregation of microorganisms that are embedded in a hydrated polymer matrix of their own secretion.) The distinction between microorganisms in the biofilm state and those in free aqueous suspension (i.e., planktonic) is important. Microorganisms in biofilms function very differently because they are subject to physical, chemical, and biological phenomena that have less impact on conventional planktonic cultures. Multicellular phenomena such as diffusion gradient formation, intercellular communication, differentiation, and extracellular electron transfer operate in biofilms and make them scientifically rich topics of investigation and also inherently complex. Mathematical models are therefore valuable complementary approaches to analyzing and understanding these systems. Resulting models are inherently interdisciplinary; the rich interaction of microbiology, chemistry, and physics requires theory grounded in the mathematics. In this talk, I will discuss a class of biofilm models based on continuum mechanics principles that present a natural platform for combining the relevant biology, chemistry and physics, and will present a few important implications that these models predict.

Friday November 16

No talk today

Monday November 19

Title MCMC for improper target distributions
Speaker Krishna B. Athreya
Iowa State University

MCMC for estimating the means w.r.t a proper target distribution(ie, a probability distribution) are well known. In this talk we consider improper targets and present some MCMC methods for them. We apply this to some Bayesian contexts including Gibbs sampler, importance sampling. We also show how to reduce the improper case to a proper one.(This is joint work with my colleague Vivek Roy of ISU stat dept)

Friday November 23


Friday November 30

Lecture Hall 1 (joint with Biology)

Title The Mathematics of Life — Decisions, Decisions
Speaker James P. Keener
University of Utah

In order to survive, living organisms must constantly make decisions, about what to eat, when and where to move, when to reproduce, when to build, when to destroy, etc. In this talk I will give an overview of the mathematics of decision making, namely the mathematical principles that underlie biological processes of measurements, switches, and signals. The short answer to how decisions are made is that the rate of molecular diffusion contains information that can be transduced by biochemical reactions to give control over behavior. These processes can be given quantitative descriptions using diffusion-reaction equations, and the study of these equations gives valuable insights into how organisms work as well as an opportunity to learn and develop new mathematics. I will illustrate this dual role of quantitative reasoning by way of several specific examples from cell biology.

Friday December 6

Title Capturing Community Behavior in Very Large Networks using MapReduce
Speaker Tamara G. Kolda
Sandia National Laboratories, Livermore, CA

Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to accurately measure and model large-scale networks. We consider especially the role of triangles, which are useful for measuring social cohesion. This talk will focus on two topics: (1) Accurately estimating the number of triangles and clustering coefficients for very large-scale networks, and (2) Generating very large-scale artificial networks that capture the degree distribution and clustering coefficients of observed data. Triangles form a basic pattern of interest for understanding networks because they reflect friend-of-friend relationships. However, counting triangles can be extremely expensive in terms of both computational time and memory requirements. We explain how sampling can be used to accurately estimate the number of triangles and the number of triangles per degree (or degree range) in an undirected graph, as well as the number of each type of directed triangle in a directed graph. Using Hoeffing’s inequality, we can predict exactly how many samples we need for a desired error and confidence level, and this sample size is independent of the size of the graph. We describe a MapReduce implementation and provide examples demonstrating its utility. Once we know how to measure triangles, we can calculate the clustering coefficients and use this data as input to a generative model. Ideally, a good model should be able to reproduce the observations, yet few models are able to do so. We hypothesize that any graph with a heavy-tailed degree distribution and large clustering coefficients must contain a scale-free collection of dense ER subgraphs. From this, we propose the Block Two-Level Erdos-Renyi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks. Moreover, the BTER model can scale to very large networks, and we describe our MapReduce implementation and recent experimental results. This is joint work with Ali Pinar, Todd Plantenga, C. Seshadhri (Sandia National Labs) and Christine Task (Purdue University).