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DE Seminars: Spring 2004

Monday February 2

Title A distributed parameter identification problem from neuronal cable theory
Speaker Jonathan Bell

Dendritic and axonal processes of nerve cells have membranes with ion channels of various kinds. These channels play a major role in characterizing the types of excitable responses expected of the cell type. The ion channel densities are usually represented as constant parameters in neural models. However, through microelectrode measurements and selective ion staining techniques, it is known that ion channels are spatially non-uniformly distributed. To gain a better understanding of neuronal behavior, spatially distributed channel densities should be incorporated into the models. But they need to be recovered from real nerve cells, which is a difficult task due to the small size of these processes. I will discuss an approach to recovering a single spatially non-uniform ion density through use of temporal data that can be gotten from recording microelectrode measurements at the ends of a neural fiber segment. The numerical approach seems to work both for linear and nonlinear models. As time permits I will also mention some issues concerning dynamical changes in these densities and the need for new types of modeling.

Monday February 9

Title Observability and blowup for a thermoelastic system
Speaker Thomas Seidman

We consider a system of partial differential equations coupling a heat equation: $u_t=\Delta u -a\Delta w_t$ with time-dependent elasticity: $w_{tt}+ [\Delta]^2 w =a\Delta u$ (taking $u=w=\Delta w =0$ as boundary conditions). A fairly standard problem of distributed parameter system theory is to ask whether knowledge of the heat flux $f$ at the boundary over a time interval suffices to determine the state $Z = [u, w, \Delta w]$ at a terminal time $T$ in the absence of information about initial data. We use results of nonharmonic analysis to show that the map: $f \mapsto Z(T)$ is well-defined and bounded with a suitable blowup rate for the norm of this map as $T \to 0$.

Monday February 16

Title Corner singularities and boundary layers
Speaker Bruce Kellogg

We shall consider several simple second order elliptic boundary value problems with a small parameter in the highest derivatives, and posed on a polygon in the plane. Thus, the solution displays both corner singularities at the vertices and boundary layers. We shall review recent work of Martin Stynes and myself on this problem.

Monday February 23

Title Adaptive Monte Carlo Algorithm for Stochastic Differential Equations
Speaker Kyoung-Sook Moon

I will present an adaptive Monte Carlo algorithm for weak approximations of stochastic differential. The goal is to compute an expected value of a given function depending on a stochastic process. Based on a posteriori error expansion, the adaptive algorithm is proven to stop with asymptotically optimal number of steps and the approximation error are bounded by the specified error tolerance as the tolerance tends to zero. Finally, I will show numerical results from computations of barrier options in financial mathematics. These results show that the adaptive algorithm achieve the time discretization error of order N^{-1} with N adaptive time steps, while the error is of order N^{-1/ 2} for a method with N uniform time steps.

Monday March 1

Title Stability and Convergence of a Spectral Galerkin Method for the Linear Boltzmann Equation
Speaker Samuel G. Webster

Chemical vapor deposition (CVD) is one production step in the manufacturing of computer chips. CVD involves gas flow at low pressures to deposit conductive material onto the surface of a silicon wafer. This gas flow is modeled by the Boltzmann transport equation (BTE) of rarefied gas dynamics. Possessing no analytic time-dependent solution, the BTE is solved numerically. Our method begins with a discretization of the velocity space through a spectral Galerkin approach which leads to a system of partial differential equations. Stability and convergence results will be presented for this part of the method. The resulting system of hyperbolic partial differential equations is then solved by the discontinuous Galerkin method. I will present results for physical quantities and parallel performance studies.

Monday March 8

Title A Lotka-Volterra Three Species Food Chain
Speaker Joseph P. Previte
Penn State Erie

We investigate a generalization of the Lotka-Volterra equations to a three species food chain. This simplified model yields a more complicated dynamical system than models involving logistic-type equations and lends itself to a study or discussion in a modelling or differential equations course. This work was accomplished at the 2000 REU in mathematical biology at Penn State Erie.

Monday April 5

Title Numerical Studies of a Reaction-Diffusion System with a Fast Reaction
Speaker Ana Maria Soane

A chemical process involving three reactive species in two reactions is modeled by a non-linear system of three reaction-diffusion equations, in one spatial dimension. Because one reaction is several orders of magnitudes faster than the second reaction, this model exhibits both a transient layer in time and an interior layer in space, which makes the problem challenging numerically. Asymptotic analysis exists for the steady-state system. A numerical method was developed that was able to capture the relevant behavior of the time-dependent system. Results both for the steady-state problem and for the transient problem will be presented.

Monday April 19

Title Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems
Speaker Leigh J. Little
SUNY Brockport

A general-purpose block LU preconditioner for saddle point problems is presented. The main difference between the approach presented here and that of other studies is that an explicit, accurate approximation of the Schur complement matrix is efficiently computed. This is used to compute a preconditioner to the Schur complement matrix, which in turn defines a preconditioner for the global system. A number of different variants are developed and results are reported for a few linear systems arising from CFD applications.

Monday April 26

Title Existence Result for the Stationary Problem of the Cahn-Hilliard Equation: Rigorous Numerics
Speaker Valeriy R. Korostyshevskiy

We consider the stationary problem for the modified Cahn-Hilliard equation. In particular, we are interested in proving existence of homoclinic orbits. It is accomplished by means of spectral methods and Banach contraction principle. Together they form a rigorous computer-assisted proof.

Monday May 3

Title Stochastic Chemical Kinetics
Speaker Daniel T. Gillespie
Daniel T. Gillespie Consulting

The time evolution of a well-stirred chemically reacting system is traditionally modeled by a set of coupled ordinary differential equations called the reaction rate equation (RRE). The resulting picture of continuous deterministic evolution is, however, valid only for infinitely large systems. That condition is usually well approximated in laboratory test tube systems. But in biological systems formed by single living cells, the small population numbers of some reactant species can result in dynamical behavior that is noticeably discrete rather than continuous, and stochastic rather than deterministic. In that case, a more physically accurate mathematical modeling is obtained by using the machinery of Markov process theory, specifically, the chemical master equation (CME) and the stochastic simulation algorithm (SSA). This talk will review the theoretical foundations of stochastic chemical kinetics, and then discuss some recent efforts to (1) approximate the SSA by a faster simulation procedure, and (2) establish the formal connection between the CME/SSA description and the RRE description.

Monday May 10

Title Mathematical Image Analysis
Speaker Tomasz Macura

Digital images are typically represented as a lattice of quantized values called pixels. This representation facilitates computer operations and has become the de-facto standard for acquiring, storing, and displaying images. However, the application of mathematical operators, e.g. non zero order derivatives, to images represented in this way is ill-posed in the notion of J. Hadamard. Due to J.J. Koenderink’s method of multi-scale image analysis, there exists an alternative image representation for which such mathematical operations are well posed. This talk will focus on the explanation and derivation of J.J. Koenderink’s method. Several examples will be given that illustrate the importance and use of Konederink’s method in Image Analysis.