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DE Seminars: Fall 2006

Monday September 18

Title The Fox and the Hound: a convolution game
Speaker T. I. Seidman

We will see why certain kinds of state-constrained optimal control problems can be effectively reformulated as convolution games. Some partial results are available, but this remains work in progress, joint with Boris Mordukhovitch.

Monday September 25

Title How Does One Assess the Quality of a Trajectory-based Phase-space Sampler for Molecular Dynamics Simulations?
Speaker Andrei Draganescu

Molecular dynamics (MD) is a computer simulation technique where the time evolution of a set of interacting atoms is followed by integrating their equations of motion. Our goal is to explain the numerical analysis behind the recipes of MD for practitioners and researchers in numerical analysis and computational mechanics. The vast majority of these practitioners and researchers work with continuum mechanics. In contrast, an atomistic method such as MD is both culturally and intellectually distinct. The recent interest in multi-scale analysis, in particular, Atomistic-to-Continuum coupling necessitates a sophisticated understanding of MD, its goals, limitations and computation. In contrast to continuum methods where accurate trajectories are of interest, MD is rarely concerned with accurate trajectories. In point of fact, accurate trajectories are not, in general, possible. Instead, quantities of interest are statistical averages computed during the sampling of phase space. Primarily, in this talk we explain why MD works from a mathematical perspective. Rather than focusing on geometric time integration issues, we will explore the relevant features of a time integrator that result in an acceptable sampling of phase space. Much of our discussion revolves around explaining in a mathematical fashion what the MD community of users and researchers has learned via dint of hard work and careful physical reasoning. This represents joint work with Rich Lehoucq at the Sandia National Labs.

Monday October 2

Title Operator Upscaling for the Wave Equation
Speaker Tetyana Vdovina

Wave propagation in heterogeneous media results in models involving multiple scales. Operator upscaling solves the problem on a coarse grid, but still retains subgrid information and fine-scale input data. The method applied to the constant density, variable sound velocity acoustic wave equation consists of two stages. First, the problem is solved for the subgrid component defined locally within each coarse block. Then, the subgrid solutions are used to augment the coarse-grid problem. Due to the homogeneous Neumann boundary conditions imposed on each coarse block, the subgrid problems decouple, which leads to the efficient parallel algorithm. Two variable velocity numerical experiments illustrate that operator upscaling captures the essential fine-scale information. We develop convergence analysis for the method using energy techniques. We show that in the H1 norm the upscaled solution converges linearly on the coarse scale, and pressure (which is not upscaled in this implementation) converges linearly on the fine scale if Raviart-Thomas zero spaces are used on both the subgrid and coarse scales. The fully discrete scheme is shown to be second-order in time. We also discuss the application of operator upscaling to the three-dimensional elastic wave equation.

Monday October 30

Title Hadamard Wellposedness for a Class of Non-Linear Shallow Shell Problems
Speaker Catherine Lebiedzik
Wayne State University

This talk is concerned with the nonlinear shallow shell model introduced in 1966 by W .T. Koiter. We consider a new version of this model which is based upon the intrinsic shell modeling techniques introduced by Michel Delfour and Jean-Paul Zol\’esio. We show existence and uniqueness of both regular and weak solutions to the dynamical model and that the solutions are continuous with respect to the initial data. While existence and uniqueness of regular solutions to nonlinear dynamic shell equations has been known, full Hadamard wellposedness of weak solutions, as discussed, is a new result which solves an old open problem in the field.

Monday November 13

Title Dynamics and Stability of Rotating Rigid-body Systems: some geometric and algebraic perspectives
Speaker Jinglai Shen

This talk addresses dynamics and stability of rigid-body attitude systems, which is motivated by a physical testbed in aerospace engineering. The talk consists of two parts: the first part focuses on variational principle of the rotating rigid body and the heavy top systems, and the second part on the asymptotic stability of the two systems using linear velocity dampings. We show in the first part how the SO(3) geometry helps to develop equations of motion via the reduced Lagrangian. In the second part, we demonstrate the connection between the asymptotic stability and nonlinear observability via LaSalle’s invariance principle; we further show finite verification of observability, and thus asymptotic stability, by making use of some algebraic techniques.

Monday November 20

Title Exponential Attractors for phase field equations with Dynamic Boundary Conditions
Speaker Ciprian G. Gal
Morgan State University

In the present article, we consider a model of phase separation based on the Allen-Cahn equation with nonconstant temperature, with dynamic boundary conditions for both the order parameter as well as the temperature function. We will show how to derive all the boundary conditions (including dynamic) for both the unknown functions. Using a fixed point argument, we obtain the existence and uniqueness of a global solution to our problem. The longtime behavior of the solution is investigated proving the existence of an exponential attractor (and thus, of a global attractor) with finite fractal dimension in a suitable Sobolev space. This is joint work with Maurizio Grasselli, Italy.

Monday November 27

Title Giant squid, hidden canard: The 3D geometry of the Hodgkin-Huxley model
Speaker Jonathan Rubin
University of Pittsburgh

The classical Hodgkin-Huxley (HH) model for the action potential of a space-clamped squid giant axon represents a rich source of interesting dynamics. It has been demonstrated that changes in the timescales associated with the gating variables in the HH model can lead to a significant slowing of the firing rate, in some cases including chaotic dynamics. In this joint work with M. Wechselberger, we explain these phenomena via a thorough geometric analysis, including application of recently developed theory on canards, and associated mixed-mode oscillations, in 3D systems with two slow variables.

Monday December 4

Title Biology, the “degenerate” world: A call for more quantitative analysis
Speaker Sergei P. Atamas

Certain biological systems, e.g. immune system, brain cortex, and evolution, are represented by the repertoires of structurally similar, yet functionally diverse elements that are responsive to the external and internal signals. These repertoires are selective, i.e. the signals are recognized by and cause selective numeric expansion of some but not all repertoire elements (recognizers). The nature of signal recognition in these repertoires is degenerate, meaning that one kind of signals can be recognized by more than one type of recognizers and conversely, one recognizer can recognize signals of more than one kind. For example, in the immune system, T cell antigen receptors and antibodies demonstrate relative, but not absolute specificity toward antigens. The degenerate nature of signal recognition, in combination with the subsequent selection of responders in such .degenerate repertoires., results in unusual features that so far have been observed only in biology. These systems 1) are highly successful in adaptation to novel environments, they .learn through associating novelty with previous signal patterns;. 2) are extremely reliable, and easily withstand a substantial loss of elements; and 3) capable of complex self-organizing pattern formation. Despite significant progress in studying the behavior of specific types of degenerate repertoires, particularly in immunology, the conceptual understanding of these systems remains undeveloped. This presentation is an attempt to start .building a bridge. between biologists and mathematicians and thus facilitate new collaborations in this intrinsically interdisciplinary field.

Monday December 11

Title Elliptic boundary value problems in non-smooth domains: An application of weighted Sobolev spaces
Speaker Ana Maria Soane

We study a variational formulation of the Poisson problem in H2, rather than the traditional H1 setting. We show the existence of solution in smooth domains. Then we introduce weighted Sobolev spaces to handle the corner singularities in non-smooth domains. We describe the basic functional analytic setting for weighted spaces and prove a Poincare-type inequality in that context. We present several numerical calculations based on such variational problems in weighted spaces. This investigation is motivated by our work on H2 formulation of the Navier-Stokes equations.