DE Seminars: Spring 2005

Monday February 7

Title Carleman estimates and some applications
Speaker Thomas I. Seidman

In 1935 Carleman introduced an interesting type of estimate to show unique continuation for elliptic PDEs in two variables. This type of estimate has more recently become a major tool for analyses of inverse problems and, still more recently (in the 90’s), for control problems for systems governed by PDEs. We will show here how this type of estimation works for the heat equation to get global information from local observation with some implications for related control problems.

Monday February 14

Title Mathematical Modeling in Pharmacodynamics
Speaker Wojciech Krzyzanski
SUNY at Buffalo

In my talk will present the following topics:
a) Brief history and highlights of pharmacokinetic and pharmacodynamic (PK/PD) modeling.
b) PK/PD modeling and pharmaceutical industry.
c) Selected mathematical problems arising in pharmacodynamics.

In part a) I will briefly introduce pharmacokinetics and pharmacodynamics as health related sciences, highlight their goals and basic methodology. Part b) will refer to the link between PK/PD modeling and drug development process, and impact it has on pharmaceutical industry. In the last section I will discuss few mathematical problems I encountered in my research on pharmacodynamics.

Monday February 21

Title Modeling tumor growth: A computational approach in a continuum framework
Speaker Cosmina S. Hogea
Watson School of Engineering and Applied Science
State University of New York at Binghamton

Tumor growth models can encompass a broad range of microscopic and macroscopic biological and physical phenomena. Continuum based mechanical structure models have been developed to help understand the temporal growth and spatial evolution of tumors. Models of this type may be used to predict the effect that local environmental conditions and methods of intervention (e.g. chemotherapy) have on tumor growth behavior. A broad variety of continuum-based models currently exist in the literature, and more are being developed; yet, little computational modeling has been accomplished to date. Simulating tumor growth typically leads to a complex nonlinear moving boundary problem, where a nonlinear system of PDEs must be solved, with the tumor boundary evolution determined as part of the solution procedure. This requires advanced state-of-the-art numerical techniques. The goal of the present research is to develop a general computational framework for simulating solid tumor evolution using a mechanics based continuum modeling approach and nontraditional moving boundary techniques. Ideally, this framework might be employed for the investigation of various continuum-based tumor growth models, in arbitrary two and three-dimensional geometries. The level set method provides a robust and flexible computational methodology for treating moving boundary problems. Our approach is fundamentally targeted on generality and simplicity of implementation. The feasibility of the proposed methodology is tested on two different types of continuum tumor growth models: a simple Greenspan-type model and a complex multicell model, centered on the tumor-induced angiogenesis phenomenon. The results obtained for the first model show good agreement with published calculations via a boundary-integral method. For the second model, to our knowledge, this is the first attempt to investigate the model behavior in arbitrary two-dimensional geometries; the results obtained prove in good qualitative agreement with experimental observations reported in the medical literature.

Monday February 28

Title Carleman estimates and some applications: Part II
Speaker Thomas I. Seidman

In 1935 Carleman introduced an interesting type of estimate to show unique continuation for elliptic PDEs in two variables. This type of estimate has more recently become a major tool for analyses of inverse problems and, still more recently (in the 90’s), for control problems for systems governed by PDEs. We will show here how this type of estimation works for the heat equation to get global information from local observation with some implications for related control problems.

Monday March 7

Title Applications of Carleman estimates to control theory
Speaker Thomas I. Seidman

This is a continuation of the 2/7 talk on estimates of Carleman type for the heat equation. Here we see how these estimates can be applied to obtain new results for observation and control of problems governed by linear (and quasilinear) parabolic PDEs. For example, we show existence of a boundary patch nullcontrol (Dirichlet boundary data with support in a specified patch) for which the solution of $u_t=\Delta u +f(u)$ with given initial data will vanish at $t=T$.

Monday March 14

Title Bifurcations of Relaxation Oscillations near Folded Saddles
Speaker Kathleen A. Hoffman

Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slow-fast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one parameter families of relaxation oscillations. We will investigate the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, in which relaxation oscillations become homoclinic to a folded saddle.

Monday April 4

Title Numerical Simulation and Analysis of Fiber Optic Compensators
Speaker Susan E. Minkoff and John Zweck

In fiber optic communication, natural irregularities in fiber design result in fast and slow polarization states for the light. As a signal propagates, it spreads out and the encoded message is distorted. To mitigate the effect of polarization mode dispersion, one can place compensating devices along the fiber in an attempt to realign the fast and slow polarization states. We use numerical simulation to compare the performance of different feedback mechanisms that optimize this compensation.

Monday April 11

Title An introduction to neural field theory: bumps and waves
Speaker Jonathan Bell, UMBC

The neocortex is a distinctly layered, densely packed, neural structure. Thus, it is reasonable to model the layers as continuum fields. With the increasing use of multiple electrode recordings and fMRI imaging studies, large-scale traveling waves of activity have become more important to consider. Also, short term memory suggests the presence of localized areas of neuronal activity in the cortex. I will introduce the simplest model case and discuss the existence, shape, and stability of wave fronts, then discuss what I know and don’t know about stationary states. Finally, I’ll introduce some ongoing projects as extensions to the work presented.

Monday April 25

Title Rigid body dynamics applied to apples
Speaker Rouben Rostamian

The subject of this talk was brought to me by Dr. Uri Tasch of the Department of Mechanical Engineering and his collaborators at UMBC and US Department of Agriculture. The goal of this study is to analyze/explain a curious phenomenon that is observed in laboratory experiments at UMBC and USDA. The experiment consists of rolling an apple — the fruit, not the computer — down an inclined track. It is observed that after a short distance, the apple orients itself so that its axis of symmetry becomes horizontal. After that the apple continues rolling down without a change of orientation. This effect appears to be reproducible and is independent of the apple’s initial condition. In this talk I will do the following:

  • Will show a movie clip, made using a high-speed camera, that shows a close-up of the apple’s movement.
  • Will describe the reason for USDA’s interest in this problem.
  • Will describe a mathematical model that attempts to explain the apple’s behavior. To that end, I will give a quick introduction to Lagrangian mechanics with nonholonomic constraints and apply it to derive the equations of motion.
  • Will show numerical solutions of the resulting differential equations and interpret their meaning.

Monday May 2

Title Long-Term Dynamics of the Cahn-Hilliard Equation
Speaker Thomas Wanner
George Mason University

This talk is concerned with the long-term dynamical behavior of the Cahn-Hilliard model, a fourth-order parabolic partial differential equation which describes phase separation phenomena in alloys. I will present results on the structure of the equilibrium set, as well as on the long-term dynamics as described by the global attractor. In addition, it will be shown how these results relate to a phase separation phenomenon called nucleation, which can be observed in the stochastic Cahn-Hilliard model.

Monday May 9

Title Fluid Dynamics of the Hydraulic Jump
Speaker Ana Maria Soane

When a vertical jet of fluid hits a horizontal surface, the fluid spreads out radially in a thin, fast-moving layer until at a certain distance from the jet a sudden increase of depth occurs, which is called a circular hydraulic jump. The common approach in studying the hydraulic jump is to use the boundary-layer approximation to the full Navier-Stokes equations. Our objective is to compute the flow field with the full Navier-Stokes equations, using finite elements, and eventually to analyze the stability of the solution. The main difficulty in the computation of this type of flow is due to the free boundary, i.e., the unknown shape of the exposed surface.In this talk:

  1. I will review the main literature results on the circular/two-dimensional hydraulic jump
  2. Derive the Navier-Stokes equations from basic principles
  3. Present an iterative numerical method for solving this free boundary problem

Monday May 16

Title Modeling Uncertainty in Optical Fiber Communication Systems
Speaker Lawrence Fomundam

An experimental recirculating loop is traditionally used to study straight-line optical fiber communication systems. However, random polarization effects within a loop system are very different from those in a line because of the loop?s periodicity. Experimentally, a device known as a polarization scrambler makes the loop model behave more like a straight line. The polarization scrambler accomplishes this by randomly rotating the polarization state of the light after each round trip of the loop. However, the distribution of these random rotations is biased. In this work, we use a biased model for the polarization scrambler to simulate how the system performance, measured by the bit error ratio, depends on the biasing strength. This work is joint with John Zweck and with Hai Xu in CSEE at UMBC.