Math Colloquia: Fall 2013

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).


Friday September 13

Title Introduction to the Boundary Control Method and its Application to Inverse Problems
Speaker Jonathan Bell

The boundary control method is an approach to inverse problems based on the relationship between control and systems theory. I will first give some motivation for studying certain inverse problems, then reduce the problem to a “simple” case. Then I will develop aspects of the boundary control method in a way that leads to an algorithmic approach for estimating a certain distributed parameter. I will wrap up with comments about other problems I am, or would like to attack. This project is joint with S. Avdonin, U. Alaska, Fairbanks.

Friday September 13



Friday September 20

Title Flow Fields at all Speeds: Analysis, Numerics and Simulation
Speaker Andreas Meister
University of Kassel

We will present a comprehensive study of a finite volume method for inviscid and viscous flow fields at high and low speeds. Thereby, the results of a formal asymptotic low Mach number analysis are used to extend the validity of the numerical method from the simulation of compressible flow fields at transonic as well as supersonic speed to the low Mach number regime. After a brief repetition of an asymptotic analysis in the continuous context we present a finite volume approximation of the governing equations. To overcome the failure of the usual numerical method in the low Mach number regime we combine the numerical flux function with a preconditioned formulation. Both, a wide variety of trans-, super-, hyper-, and subsonic realistic test cases and a formal discrete asymptotic single scale analysis are employed in order to prove the validity of the derived numerical method from hypersonic to low Mach number fluid flow.

Friday September 27

Title Mathematical Problems from the Call Center Industry
Speaker Hans Engler
Georgetown University

In 2011, the speaker spent seven months working full-time for a startup company that produces optimized routing software for telephone call centers. He will talk about some of the mathematical problems that he encountered during this time. These include the modeling of Markov chains with huge state spaces, large scale combinatorial optimization problems that can be traced back to work in Banach space theory by A. Grothendieck, and techniques for detecting and classifying heavy tailed distributions in empirical data. A case will be made that the current revolution in data analysis (“Big Data”), which has garnered a lot of attention and created a fair amount of hype, presents exciting challenges not just for statisticians and computer scientists, but also for mathematicians.

Friday October 4

Title Multilevel Algorithms for Symmetric Saddle Point Systems and the Saddle Point Least Square Method
Speaker Constantin Bacuta
University of Delaware

We present a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on the availability of families of stable finite element pairs and on the existence of fast and accurate solvers for symmetric positive definite systems. On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non-increasing number of iterations on each level. Numerical results supporting the efficiency of the method are presented for the Stokes system. We define the “saddle point least-squares” method for solving first order systems of PDEs and relate it with the Bramble-Pasciak’s least-squares approach. We apply our cascadic method to the saddle point least-squares discretization of time harmonic Maxwell equations. Our multilevel iterative solver does not involve edge elements or spaces of bubble functions.

Friday October 11

UBM Special Colloquium joint with Biology

Title Next Generation Phylogenetics
Speaker David Weisrock
University of Kentucky

Research in our lab centers on using genetics to resolve the geographic boundaries of species in nature, reconstruct the relationships among these lineages, and address the mechanisms that have led to their formation. At the population level our research seeks to understand the adaptive and non-adaptive factors that influence gene flow and drive speciation. At a more macroevolutionary level we use phylogenetic approaches to resolve the evolutionary history of species assemblages and the rates at which they form. The advent of genomics has been especially valuable for our research and we increasingly focus our efforts on analytical approaches that permit the summarization of population-genetic and phylogenetic information from across the genome.

Friday October 18

Title Efficient Monte Carlo Counterparty Credit Risk Pricing and Measurement
Speaker Bo Zhang
IBM T. J. Watson Research Center

Counterparty credit risk (CCR), a key driver of the 2007-08 credit crisis, has become one of the main focuses of the major global and U.S. regulatory standards. Financial institutions invest large amounts of resources employing Monte Carlo simulation to measure and price their counterparty credit risk. We develop efficient Monte Carlo CCR frameworks by focusing on the most widely used and regulatory-driven CCR measures: expected positive exposure, credit value adjustment (CVA), and effective expected positive exposure. Our numerical examples illustrate that our proposed efficient Monte Carlo estimators outperform the existing crude estimators of these CCR measures substantially in terms of mean square error. We also demonstrate that the two widely used sampling methods, the so-called Path Dependent Simulation and Direct Jump to Simulation date, are not equivalent in that they lead to Monte Carlo CCR estimators which are drastically different in terms of their mean square error.

Friday October 25

Title Inexact Fixed Point schemes and termination criteria
Speaker Philipp Birken
Kassel University, Germany

We consider fixed point schemes, where the function evaluation corresponds to inexact solves of linear or nonlinear equation systems. This allows to decide how accurate these systems should be solved to obtain a target error in the fixed point iteration, as well as to decide on a good termination criterion. The first applicaton is the Picard iteration which is widely used for the incompressible Navier-Stokes equations. Here we can show that this iteration converges regardless of how accurate we solve the subsystems, provided a relative termination criterion is employed. As a second example, we consider thermal fluid structure interaction problems where heat is exchanged via a coupling interface. This appears in many applications, for example in the cooling process in steel forging. A so called partitioned coupling approach naturally leads to a fixed point iteration, which is then analyzed using the derived methodology.

Friday November 1

Title Cellular Dynamics: Intracellular Signals and Extracellular Environments
Speaker Bradford Peercy

The molecular movements within a cell act to control its processes from the millisecond to many minute timescale. As well, cells can interact with each other and their environment to create emergent behavior beyond that of individual cells. In this talk I will relate our efforts to describe mathematically and analyze cellular processes, many in a spatial context, critical to a variety of health issues. Applications vary from subcellular calcium dynamics in cardiac cells (arrhythmias) and neurons (learning and memory) to genetic regulation in skeletal muscle (muscle atrophy) and pancreatic beta cells (diabetes). We also consider how the extracellular environment, specifically the geometry of the domain, can impact the intracellular signals in two additional systems of Vitamin D regulation (immune function) and cell migration determination (cancer metastasis). While the applications are widely varying in the physiology, we apply a similar series of reductions, approximations, and simulations in modeling often complex spatio-temporal phenomena to gain insight and even predict important biophysical results.

Friday November 8

Title Random holomorphic fields on complex manifolds
Speaker Renjie Feng
University of Maryland, College Park

Zeros of random polynomials and analytic functions have been studied for a long time. By the maximal principal, there are only constant holomorphic functions on compact complex manifolds, thus random holomorphic sections of line bundle are natural generalization of random polynomials to the complex manifolds. In the talk, I will first explain several classical results on random polynomials and random holomorphic sections, such as the distribution of zeros, two or k-point correlation and their recaling limits, universality of zeros, hole probability,the distribution of critical points. Then I will explain my recent results with S. Zelditch on the distribution of critical values and sup norm of random holomorphic sections under Gaussian and spherical ensembles. Many classical results on probability and geometry are crucial in the proof, such as the full expansion of Bergman kernel on diagonal, Dudley and Sudakov’s entropy bounds. The talk should be very elementary and accessible to large audience.

Friday November 15

Title Biphasic Acoustic Behavior of a Non-periodic Porous Medium
Speaker Robert Gilbert
University of Delaware

We study the problem of derivation of an effective model of acoustic wave propagation in a two-phase, non-periodic medium modeling a fine mixture of linear elastic solid and a viscous Newtonian fluid. Bone tissue is an important example of a composite material that can be modeled in this fashion. We extend known homogenization results for periodic geometries to the case of a stationary random, scale-separated microstructure. The ratio ε between a typical size of microstructural inhomogeneity and the macroscopic length scale is a small parameter of the problem. We employ stochastic two-scale convergence in the mean to pass to the limit ε → 0 in the governing equations. The effective model describes a biphasic viscoelastic material with long time history dependence. Homogenized system describes macroscopically anisotropic media and appears to be more general than the Biot system; however, numerical realizations show that the non-Biot coefficients are much smaller than the usual Biot coefficients. Hence, we obtain a numerical scheme for accurately determining the Biot coefficients.

Friday November 22

No talk today

Friday November 29


Friday December 6

Title Rays, Waves and Rainbows!
Speaker John Adam
Old Dominion University

Rainbows are exquisitely beautiful both optically and mathematically. This talk is an attempt to summarize some of the direct and indirect connections that exist between classical ray and scattering, wave and potential scattering theory. The primary bow is the lowest-order bow that can occur by scattering from a spherical drop with constant refractive index n, but zero-order (or direct transmission) bows can exist when the sphere is radially inhomogeneous. The refractive index profile automatically defines a scattering potential, but with a significant difference compared to the standard quantum mechanical form: the potential is k-dependent. A consequence of this is that there are no bound states for this system. The correspondences between the resonant modes in scattering by a potential of the ‘well-barrier’ type and the behavior of electromagnetic ‘rays’ in a transparent (or dielectric) sphere are discussed. The poles and saddle points of the associated scattering matrix have quite profound connections to electromagnetic tunneling, resonances and ‘rainbows’ arising within and from the sphere. The rainbow can also be regarded as a fold catastrophe.