Math Colloquia: Spring 2013

Wednesday January 30

Title Modeling the Inflammatory Response: Wound Healing and Mechanical Ventilation
Speaker Angela Reynolds
Virginia Commonwealth University

The acute inflammatory response, triggered by a variety of biological or physical stresses on an organism, is a delicate system of checks and balances that, although aimed at promoting healing and restoring homeostasis, can result in undesired and occasionally lethal physiological responses. In this talk, I will focus on the role of the inflammatory response in wound healing and mechanical ventilation.

In order to promote wound healing in diabetic patients, we developed an ordinary differential equations model, which tracks fibroblasts, collagen, inflammation and pathogens. The model was validated by comparison to the normal time course of wound healing where appropriate activity for the inflammatory, proliferative, and remodeling phases were recorded and compared to collagen accumulation experiments. The model was then used to investigate the impact of local oxygen levels on wound healing. This model is the first step to creating a patient-specific treatment model.

Additionally, we will look at a multi-scale model for mechanical ventilation focusing on the interplay between gas exchange and inflammatory response. This portion of the multi-scale project is a partial differential equations model of gas exchange with inflammatory stress on a small physiological unit of the lung, referred to as a respiratory unit (RU). Linking multiple RUs with various ventilation/perfusion ratios and taking into account pulmonary venous blood remixing yields our lung-scale model. Using the lung-scale model, we explored the predicted effects of inflammation on ventilation/perfusion distribution and the resulting pulmonary venous partial pressure oxygen level during systemic inflammatory stresses. This model translates changes in inflammation levels to changes in peripheral PO2, which can be measured in real time from patients.

Friday February 1

Title Results on Valid Inequalities for Quadratic Programming with Indicator Variables
Speaker Hongbo Dong
University of Wisconsin-Madison

We study some fundamental convex hulls in quadratic programming with binary indicator variables. Problems of this type arise in many applications, including portfolio selection, sparse least-squares, discrete time filter design, etc. Valid inequalities for our convex hull can be obtained by extending inequalities for a related set without binary variables (QPB), that was studied by Burer and Letchford. After closing a theoretical gap about QPB, we characterize the strength of different classes of lifted QPB inequalities. We show that one class, lifted-posdiag-QPB inequalities, capture no new information from the binary indicators. However, we demonstrate the importance of the other class, called lifted-concave-QPB inequalities, in two ways. First, all lifted-concave-QPB inequalities define the relevant convex hull for the case of convex quadratic programming with indicators. Second, we show that all perspective constraints are a special case of lifted-concave-QPB inequalities, and we further show that adding the perspective constraints to a semidefinite programming relaxation of convex quadratic programs with binary indicators results in a relaxation equivalent to the recent optimal diagonal splitting approach of Zheng et al. Finally, we show the separation problem for lifted-concave-QPB inequalities is tractable if the number of binary variables involved in the inequality is small. We also present a new reformulation mechanism using 2-variable lifted-concave-QPB cuts, which significantly improves branch-and-bound solvers in some instances.

Monday February 4

Title The Cancer identity and roles of macrophages
Speaker Duan Chen
MBI, The Ohio State University

Hypoxia, acidosis, and strong reducing capacity are distinguish properties of solid tumors from healthy tissues. These parameters change along cancer development and are among the most critical parameters for optimization of anti-cancer therapies and screening of anti-cancer drugs. Multi-scale models are established to explore explanations, correlations and impacts of the three properties in tumor growth with corresponding chemotherapies, from the aspects of tumor-immune system interactions at the tissue level, chemical interactions inside individual cancer cells and proton transport through membrane proteins. High-performance numerical algorithms are developed to handle computational challenges in solving the PDE systems and the models are validated by the comparisons between numerical simulations and experimental data. Additionally, the models can be summarized as different types of free boundary problems and the well-posedness of their solutions are analyzed.

Wednesday February 6

Title Stochastic modeling of biochemical networks
Speaker Hye-Won Kang
MBI, The Ohio State University

Stochastic effects may play an important role in mathematical modeling of biological and chemical processes in case the copy number of some component involved in the system is small. In this talk, we consider stochastic modeling of biochemical networks with several examples. First, we look at a mathematical model of lung cancer involving microRNAs. MicroRNAs take charge of cellular differentiation, apoptosis, and growth, and they serve as biomarkers in cancer. The model involves an important component of the large signaling pathway in lung cancer. The background noise due to the unknown part of the pathway is modeled in terms of stochastic differential equations.

Next, multiscale approximations of stochastic chemical reaction networks are suggested. Evolution of the network is modeled in terms of a continuous-time Markov jump process. Chemical reaction networks are generally large in size and they involve various scales in species numbers and reaction rate constants. The multiscale approximation method is introduced to reduce the network complexity and to derive limiting models with simple structure. Last, we consider stochastic reaction-diffusion systems to model pattern formation in developmental biology. Spatially distributed signals called morphogens influence gene expression that determines phenotype identity of cells. A stochastic model for boundary determination between different cell types is suggested using signaling schemes for patterning.

Friday February 8

Title Clique and cluster identification using convex optimization
Speaker Brendan Ames
IMA, University of Minnesota

Identifying clusters of similar objects in data plays a significant role in a wide range of applications such as information retrieval, pattern recognition, computational biology, and image processing. A classical approach to clustering is to model a given data set as a graph whose nodes correspond to items in the data and edges indicate similarity between the corresponding endpoints, and then try to decompose the graph into dense subgraphs. Unfortunately, finding such a decomposition usually requires solving an intractable combinatorial optimization problem. In this talk, I will discuss a convex relaxation approach to identifying such dense subgraphs with surprising recovery properties: if the data is randomly sampled from a distribution of clusterable data, then the correct partition of the data can be recovered from the optimal solution of the convex relaxation with high probability.

Monday February 11

Title Discrete total variation flows without regularization and applications
Speaker Abner J. Salgado
University of Maryland, College Park

We propose and analyze an algorithm for the solution of the L2-subgradient flow of the total variation (TV) functional. The algorithm involves no regularization, thus the numerical solution preserves the main features that motivates the use of this type of energy both in imaging and materials sciences. We derive L2 error estimates under minimal regularity assumptions, and introduce a TV-diminishing interpolation operator which yields improved error bounds. We also propose an iterative scheme for the solution of the ensuing discrete problems and analyze it. This methodology extends to the TV functional augmented with a strictly convex functional, such as a p-Laplacian term. We discuss several numerical experiments which illustrate the power and potentials of the method, and explore a model arising in materials science.

Wednesday February 13

Title Markov Modulated Stochastic Networks in Heavy Traffic
Speaker Xin Liu
IMA, University of Minnesota

Stochastic networks arise as models in various areas including computer systems, telecommunications, manufacturing, finance, and service industry. The networks are often too complex to be analyzed directly and thus one seeks suitable approximate models. One class of such approximations are diffusion models that can be rigorously justified when networks are operating in heavy traffic, i.e., when the network capacity is roughly balanced with network load. In this talk, some recent study on Markov modulated stochastic networks in heavy traffic will be presented. We first consider generalized Jackson networks with Markov modulated arrival and service rates and routing structure, and develop suitable reduced models using techniques from diffusion approximations and heavy traffic theory. We then develop a comprehensive stability theory for such Markov modulated stochastic networks and their diffusion limit. At last, we study optimal control problems for Markov modulated multiclass single-server queueing systems in heavy traffic, and establish an asymptotically optimal policy which is an “average” cμ rule.

Friday February 15

Title A study of the Modied Buckley-Leverett equation: An interplay of analytical and numerical methods
Speaker Ying Wang
University of Minnesota


The Buckley-Leverett (BL) equation is a transport equation used to model two-phase flow in porous media.One application is secondary recovery by water-drive in oil reservoir simulation. The modified Buckley-Leverett (MBL) equation differs from the classical BL equation by including a balanced combination of diffusive and dispersive terms. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profiles for certain Riemann problems as suggested by the experimental observations.

In this talk, I will first show that for the MBL equation, the solution of the finite interval [0,L] boundary value problem converges to that of the half-line [0;+∞) problem exponentially as L → +∞. This result provides a justication for the use of the finite interval in numerical studies for the half line problem. Furthermore, we numerically verify that the convergence rate is consistent with our theoretical estimate.

I will then illustrate how we extend the central schemes originally designed for hyperbolic conservation laws to solve the MBL equation, which is of pseudo-parabolic type. This extension can also be applied to other conservation law solvers. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks, which is consistent with the experimental observations.

Wednesday February 20

Title Gevrey class techniques for dissipative equations: from turbulenceto flocking
Speaker Animikh Biswas
University of North Carolina-Charlotte

A basic tool, in both numerical and theoretical study of an infinite dimensional dynamical system, is to approximate it by a suitable finite dimensional system that “shadows” its dynamics. This may take the form of Galerkin projections or finite elements in computational fluid dynamics, or other suitable interpolant (such as “determining modes” or readings from spatially dispersed weather stations) in data assimilation and weather forecasting. It turns out that in certain dissipative systems, these approximates converge exponentially fast. This is related to “gain of analyticity” of solutions over time and “uniform space analyticity radius” of the attractor. In fluid dynamics, this radius is related to the Kolmogorov length scale and it demarcates the length scale below which the viscous effect dominates the (non-linear) inertial effect. Foias and Temam introduced an effective approach to estimate space analyticity radius for the Navier-Stokes equations via a class of (exponential) norms called Gevrey norms. We will discuss how Gevrey functional classes (and their generalizations)can be used to study the dynamics for a wide class of equations with applications ranging from turbulence theory to flocking and social networks. In particular, we will show that Gevrey class techniques can be used to obtain higher order decay rate of solutions. We will also discuss some recent connections to intermittency and (anomalous) energy cascade in 3D turbulent flows. Our result establishes a rigorous connection between the appearance of complex singularities and intermittency which had earlier been conjectured by several authors (for instance Frisch and Morph).

Friday February 22

Title Stable Discretizations and Fast Solvers for Simulating non-Newtonian Fluids Models and Their Applications
Speaker Young-Ju Lee
Rutgers University

Unlike Newtonian fluids, the viscosity of non-Newtonian fluids is variable based on applied stress or stress history. There are abundant daily life examples of such non-Newtonian fluids, just to cite a few, cornstarch dissolved in water, ketchup, toothpaste, paint and shampoo, from which it is often possible to understand physics in an exciting hands-on way. In this talk, we introduce stable discretizations and fast solvers that can handle rate-type non-Newtonian fluid equations in a unified fashion and demonstrate their performance and validity by using both benchmark problems and mathematical theory. We then apply these methodologies to investigate and attempt to identify a mathematical model for the unusual phenomenon observed in motion of the sphere falling through a wormlike micellar fluid in 2003; a sphere falling in a wormlike micellar fluid undergoes nontransient and continual oscillations. By tackling the Johnson-Segalman models in the parameter regimes that have been unexplored for the flow past a sphere, we obtain the self-sustaining, continual, (ir)regular and periodic oscillations. If time permits, we shall discuss some open issues as well as potential solutions in mathematical modelings.

Friday March 1

No talk today

Friday March 8

No talk today

Friday March 15

Title Lyapunov-like transformations on proper cones
Speaker Muddappa Gowda

Given a proper cone K in a real finite dimensional Hilbert space H, a linear transformation L on H is said to be Lyapunov-like on K if

x ∈ K, y ∈ K∗ , and x, y = 0 ⇒ L(x), y = 0,

where K∗ denotes the dual of K. The Lyapunov transformation L_A (X) = AX+XA^T on the semidefinite cone is an example. This introductory/survey talk deals with the relevance and properties of Lyapunov-like transformations in dynamical systems, complementarity theory, and conic optimization. By relating Lyapunov-like transformations to the automorphism group of the underlying cone and its Lie algebra, we address the issues of (i) characteriza- tion of such transformations on polyhedral/symmetric/completely positive cones and (ii) computation of the so-called Lyapunov rank of a proper cone.

Friday March 22

Spring Break

Friday March 29

Title Team-Based Learning in Mathematics
Speaker Kalman Nanes

This talk documents an adaptation of Team Based Learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, for use in the collegiate mathematics classroom. We discuss the standard components of TBL and the necessary changes to those components for the needs of our classroom. We also run an empirically controlled analysis of the effects of TBL on the student learning experience in the first term of TBL use in Linear Algebra (MATH 221).

Friday April 5

Title An optimal block iterative method and preconditioner for banded matrices with applications to PDEs on irregular domains
Speaker Daniel Szyld
Temple University

Classical Schwarz methods and preconditioners subdivide the domain of a partial differential equation into subdomains and use Dirichlet or Neumann transmission conditions at the artificial interfaces. Optimizable Schwarz methods use Robin (or higher order) transmission conditions instead, and the Robin parameter can be optimized so that the resulting iterative method has an optimal convergence rate. The usual technique used to find the optimal parameter is Fourier analysis; but this is only applicable to certain domains, for example, a rectangle.

In this talk, we present a completely algebraic view of Optimizable Schwarz methods, including an algebraic approach to find the optimal operator or a sparse approximation thereof. This approach allows us to apply this method to any banded or block banded linear system of equations, and in particular to discretizations of partial differential equations in two and three dimensions on irregular domains. This algebraic Optimizable Schwarz method is in fact a version of block Jacobi with overlap, where certain entries in the matrix are modified.

With the computable optimal modifications, we prove that the Optimizable Schwarz method converges in two iterations for the case of two subdomains. Similarly, we prove that when we use an Optimizable Schwarz preconditioner with this optimal modification, the underlying Krylov subspace minimal residual method (e.g., GMRES) converges in two iterations. Very fast convergence is attained even when the optimal operator is approximated by a sparse transmission matrix. Numerical examples illustrating these results are presented.

This is joint work with Martin Gander (Geneva) and Sebastien Loisel (Heriot-Watt U.)

Wednesday April 10

Title Dirichlet mixtures, the Dirichlet process, and the topography of amino acid multinomial space
Speaker Stephen Altschul
National Center for BioInformatics, National Library of Medicine, NIH

The Dirichlet Process is used to model probability distributions that are mixtures of an unknown number of components. Amino acid frequencies at homologous positions within related proteins have been fruitfully modeled by Dirichlet mixtures, and we have used the Dirichlet Process to derive such mixtures with an unbounded number of components. This application of the method requires several technical innovations. The resulting Dirichlet mixtures model multiple alignment data substantially better than do previously derived ones. They consist of over 500 components, in contrast to fewer than 40 previously, and provide a novel perspective on the structure of proteins. Individual protein positions should be seen not as falling into one of several categories, but rather as arrayed near probability ridges winding through amino-acid multinomial space.

Friday April 12

Title Mathematical modeling, analysis and simulation of coupled flow-structure interaction with applications to biological, bio-inspired and engineering systems
Speaker Padmanabhan Seshaiyer
George Mason University

Over the last decade, there have been dramatic advances in mathematical modeling, analysis and simulation techniques to understand fundamental mechanisms underlying fluid-structure interaction (FSI). To model a strong FSI interaction one must efficiently couple mathematical models with proper kinematics and dynamic coupling conditions. This work will present the results from projects that evolved from multidisciplinary applications of differential equations for fluid-structure interaction problems in biological, bio-inspired and engineering systems. Specifically, numerical methods for efficient computation of nonlinear interaction for coupled differential equation models that arise from such multiphysics applications will be presented. Some theoretical results that validate the reliability and robustness of the proposed computational methodology will also be presented. We will also discuss how such projects can provide opportunities for students at all levels to employ transformative research in multidisciplinary areas.

Friday April 19

No talk today

Friday April 26

Undergraduate Senior Thesis Presentations

Title A Force-Based Biophysical Model of Border Cell Migration: Unraveling the mechanism of collective cell migration
Speaker David Stonko (advisors: Peercy-Math/Stat, Starz-Gaiano-Biol Sci)


Title Ideals and automorphisms of the semidefinite cone under the Schur product
Speaker Babhru Joshi (advisor: Gowda-Math/Stat)


Title A Mathematical Model of Melanopsin Activation
Speaker Drew Thatcher (advisors: Hoffman-Math/Stat, Robinson-Biol Sci)


Friday May 3

Title A dynamical system modeling a supply chain: stability and instability
Speaker Tom Seidman

We model a demand controlled inventory/resupply system using a signal kanban policy for re-ordering a batch from a supplier (processor/manufacturing station,…), which then takes both supplies and time for this. This defines a dynamical system response to external demand and, assuming a natural capacity condition, we ask whether this always will be stable (requiring only bounded buffers). Stability is shown under an acyclicity condition on the assignments of suppliers to buffers and an example (somewhat based on the Kumar-Seidman instability example for source control supply) showing the necessity of such a condition.