DE Seminars: Fall 2005

Monday September 19

Title Efficient Numerical Schemes for Stochastic Chemical Kinetics: Part II
Speaker Muruhan Rathinam

This will be a continuation of the talk given on Sept 9th at the Applied Math Colloquium. After a quick review of the material covered in the previous talk, I shall go into the details of a new explicit leaping scheme which is showing promise for stiff systems.

Monday October 3

Title Singular perturbations and multiple time scales
Speaker Thomas Seidman

Evolution systems with multiple time scales typically appear as singular perturbation problems involving either a large parameter (on the slow scale) or a small parameter (on the fast scale. We will look at a few case studies to get an idea of some relevant phenomena and approaches. [This will be primarily a tutorial talk.]

Monday October 10

Title Global bifurcations and canard solu tions for the forced van der Pol system
Speaker Kathleen Hoffman

Singularly perturbed problems are ubiquitous throughtout science and engineering, yet the dynamics of their solutions are not well understood. The forced van der Pol oscillator serves as an a case study for the dynamical behavior that can occur is systems with two slow variables and one fast variable. I will present results that characterize this dynamical behavior of the forced van der Pol oscillator and will then extrapolate the results to more general systems.

Monday October 17

Title Worms, Snakes, and Eels: PDE’s for Slender Incompressible Bodies
Speaker Stuart Antman

A material body is incompressible if every deformation of it locally preserves its volume, in particular, if the Jacobian determinant of every continuously differentiable deformation of it is identically 1. (Rubber and much living tissue (which is composed mostly of water) are examples of incompressible materials.) Since the nonlinear PDEs of evolution for such 3-dimensional bodies have largely resisted analysis, it is useful to have effective theories for slender bodies (like worms, snakes, and eels) governed by equations with but one independent spatial variable. This lecture shows that the actual construction of one such very attractive theory requires the solutions of a sequence of first-order PDEs (by the method of characteristics). Although the resulting equations are more complicated than those for bodies not subject to the constraint of incompressibility, they admit some tricky a priori bounds and they have novel regularity properties not enjoyed by the latter. The governing equations for an elastic body can be characterized by Hamilton’s Principle. The ODEs governing travelling waves for these equations can also be characterized by Hamilton’s Principle, but the kinetic and potential energies for these ODEs do not correspond to those of the PDEs. These ODEs, which have a nonstandard structure, admit, under favorable assumptions, periodic travelling waves with wave speeds that are are supersonic with respect to some modes of motion and subsonic with respect to others.

Monday October 24

Title The Newmark Integration Method for Simulation of Multibody Systems”
Speaker Bogdan Gavrea

When simulating the behavior of a mechanical system, the time evolution of the generalized coordinates used to represent the configuration of the model is computed as the solution of a combined set of ordinary differential and algebraic equations (DAEs). The most well-known and time-honored algorithms are the direct discretization approach, and the state-space reduction approach respectively. In the latter, the problem is reduced to a minimal set of potentially new generalized coordinates in which the problem assumes the form of a pure second order set of Ordinary Differential equations (ODE). This approach is very accurate, but computationally intensive, especially when dealing with large mechanical systems. The direct discretization approach is less accurate but significantly faster.

In the context of direct discretization methods, approaches based on Backward Differentiation Formulas (BDF) have been the traditional choice for more than 20 years. We propose a new approach in which BDF methods are replaced by the Newmark formulas. Local error analysis, and global convergence results for the linear, constant coefficient DAE will be presented.

Monday November 7

Title Problems in Biocomplexity from an Experimentalists Perspective
Speaker Theresa Good

Over the past decade, there has been an explosion in technology for use in biomedical research, from tools for genomics, and proteomics, to single cell imaging and microfluidic arrays, and tools for biological mass spectrometry. However, as we become increasingly more sophisticated in our ability to collect data from biological systems, our ability to manage the experimental data or assemble the data into a meaningful picture of what controls the biological system has lagged behind our ability to collect the data.

In this talk, I will address some of the ways that my research group, as experimentalists, have attempted to use computational tools in an attempt to understand the data we collect from biological systems. The examples I will talk about include the development of mathematical models to explain the regulatory networks that control pyrimidine biosynthesis in E. coli, the use of optimization tools to minimize side effects in HIV chemotherapy, and the use of mathematic models and optimization tools to aid in experimental design, mechanism understanding, and therapy development for problems in Alzheimer?s and T cell diseases. As an experimentalist, our focus is not on the development of computational tools, but on using computational tools in order to understand data, or predict how to best obtain experimental data.

Monday November 14

Title Continuous-in-time Error Estimate for Operator-Based Upscaling
Speaker Tetyana Vdovina

Modeling of wave propagation in a heterogeneous medium requires input data that lives on many different spatial and temporal scales. Operator-based upscaling allows us to capture the effect of the fine scales on a coarse domain without solving the full fine-scale problems. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First we solve small independent problems for approximate fine-scale information internal to each coarse block. Then we use these subgrid solutions to define an upscaled operator on the coarse grid. The fine-grid velocity field is used throughout the process (i.e., no averaging of input fields is required). Due to the homogeneous Neumann boundary conditions imposed on each coarse block, the subgrid problems decouple, which leads to the natural parallelization of the first stage of the method. Two variable velocity numerical experiments illustrate that operator-based upscaling captures the essential fine-scale information (even details contained within a single coarse grid block) and models wave propagation quite accurately at considerably less expense than full finite differences.

In this talk we will focus on deriving a priori L-infinity-in-time L2-in-space error estimate for the continuous-in-time two-scale mixed finite element problem. Pressure is approximated on the subgrid scale to order h (fine grid space step) and acceleration is approximated on the coarse scale to order H (coarse grid space step) if RT0 spaces are used on both the subgrid and coarse scales.

Monday November 28

Title Temperature Difference Between the Body Core and the Arterial Blood Supplied to the Brain in Selective Brain Cooling in Humans
Speaker Liang Zhu
Department of Mechanical Engineering

In this study a theoretical model is developed to evaluate the temperature difference between the body core and the arterial blood supplied to the brain. Several factors including the local blood perfusion rate, blood vessel bifurcations in the neck, and blood vessel pairs on both sides of the neck are considered in the model. The theoretical simulation is used to estimate the potential for cooling the blood in the carotid artery on its way to the brain by heat exchange with its countercurrent jugular vein and by radial heat conduction loss to the cool neck surface. It is shown that the cooling of the arterial blood can be as much as 0.8C lower than that body core temperature, which is in agreement with the previous experimental measurements of the temperature difference between the tympanic and body core temperatures during the exercise. The temperature difference between the body core and the arterial blood supplied to the brain could be larger than 1.2C if the neck surface is actively cooled to 0C. This research could provide indirect evidence of the existence of the selective brain cooling (SBC) in humans during hyperthermia. The results of this study can also be used to evaluate the feasibility of lowering brain temperature effectively by placing ice pad on the neck surface during the heat stress.

Monday December 5

Title MATLAB’s ode15s Function and the Efficient and Effective Solution of Time-Dependent Reaction-Diffusion Equations Using It
Speaker Matthias Gobbert

I will consider how to numerically solve a system of time-dependent reaction-diffusion equations coupled by non-linear reactions. The focus will be on how to accomplish this effectively and efficiently in Matlab. Using the method of lines, each partial differential equation is approximated by a system of ordinary differential equations (ODEs). Since such a system of ODEs obtained by a method of lines is necessarily stiff, we have to use an implicit solver designed for stiff problems, such as the one in Matlab’s ode15s function. Its discretization of the system of ODEs results in a non-linear system of equations that is solved by (a variant of) the Newton method, which requires a linear solve in every iteration. My goals for this talk are

  • to show in some detail how Matlab’s ode15s works, in particular, how it integrates the non-linear solver, and
  • to demonstrate how to make the code efficient in Matlab and how to program this effectively.

This talk assumes some familiarity with numerical methods and with Matlab, but it is specifically designed to be open to graduate students. While I will not talk about research results per se, I will make connections to on-going work in passing and point out opportunities.

Monday December 12

Title Geometric Control of Nonlinear Dynamical Systems
Speaker James Dibble
Department of Mathematics and Statistics

In this talk, the basics of geometric control theory will be discussed, including the means by which the differential equations governing dynamical systems can be viewed as vector fields on differentiable manifolds. In control systems, the ability to manipulate those differential equations, and hence alter those vector fields, leads to questions about which states the system can reach given certain initial conditions. In particular, the use of Lie brackets to determine controllability and (strong) accessibility of control-affine systems will be explained, as will results concerning connected analytic systems.