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DE Seminars: Spring 2009

Monday February 2

Title Dynamical Analysis of a Model for a Skeletal Muscle Fiber with Myotonia or Periodic Paralysis
Speaker Jonathan Bell
Department of Mathematics and Statistics

Medical research has indicated that abnormalities of skeletal muscles, myotonia and periodic paralysis, are caused by alteration in the voltage-gated sodium channels. This assumption led to studies of channel behavior based on the dynamics of membrane potentials. To simulate such behavior, a two-compartment Hodgkin-Huxley type model has been modified by reformulation of the sodium current term and a model reduction due to disparity of time scales. A geometric perturbation analysis is carried out on the reduced model system. The conditions on the system parameters under which the model exhibits dynamic behavior that is consistent with clinical observations shall be derived. We are able to detect slow-fast limit cycles which generate bursts of action potentials characteristic of the clinical case where active and non-active phases are observed to alternate in a pulsatile fashion, such as that in patients with Hyperkalemic periodic paralysis. Relying on the observation that the state variables possess drastically diversified dynamics, we are able to explain the differences between the action potential dynamics of a normal subject and those of myotonia or periodic paralysis cases.

Monday February 9

Title New Efficient Sparse Space-Time Algorithms in Numerical Weather Prediction
Speaker Yulong Xing
Courant Institute of Mathematical Sciences
New York University

A major stumbling block in the prediction of weather is the accurate parameterization of moist convection on microscales. A recent multi-scale modeling approach, superparameterization (SP), has yielded promising results and provided a potential solution to this problem. SP is a large-scale modeling system with explicit representation of small-scale processes provided by a cloud-resolving model (CRM) embedded in each column of a large-scale model. In this talk, we will present new efficient sparse space-time algorithms of SP which solve the small scale model in a reduced spatially periodic domain with a reduced time interval of integration. The new algorithms have been applied to a stringent two-dimensional test suite involving moist convection interacting with shear. The numerical results are compared with the CRM and original SP. It is shown that the new efficient algorithms for SP result in a gain of roughly a factor of 10 in efficiency, and the large scale variables such as horizontal velocity and specific humidity are captured in a statistically accurate way.

Monday March 9

Title Stability and Existence Results for Elastic Rods Models with Self Contact
Speaker Kathleen Hoffman
Department of Mathematics and Statistics

In the classic theory of elastic rods, two non-adjacent points along the rod may upon contact occupy the same physical space. In this talk, I will develop an elastic rod model with a pairwise repulsive potential such that if two non-adjacent points along the rod are close in physical space, there is an energy barrier that prevents contact. For adjacent pairs, the repulsive potential is negligible and the elastic rod is described by a classical elastic rod model. The framework for this model is developed to prove the existence and stability of minimizers.

This is joint work with Rob Manning (Haverford College) and Tom Seidman (UMBC).

Monday March 23

Title Improvement of numerically integrated satellite orbits by inclusion of temporal variations in Earth’s gravitational field from GRACE
Speaker Peter Hinkey
Department of Mathematics and Statistics

The numerical integration of satellite orbits is accomplished using several mathematical models and equations. The improvement of satellite orbit prediction using improved gravitational modeling will be the focus of this paper. Using NASA Goddard.s software package Geodyn, precise orbit determination is performed by using specific commands regarding a spacecraft.s position, and the forces acting on the spacecraft. Using a Least Squares adjustment process, discrepancies between the observations and the predicted ranges are used to improve the a priori orbital model. The success of this process is demonstrated with statistics based on observations from ground stations. This process can also support the estimation of Earth-related parameters, including station positions, Earth orientation , the gravity model itself, Earth tides, and ocean tides when data from several spacecraft and over an extended period of time are used.

Monday April 6

Title COMSOL Multiphysics Features in Scripting and Batch Processing
Speaker Matthia Gobbert
Department of Mathematics and Statistics

COMSOL Multiphysics is an extremely powerful and versatile finite element package for the solution of partial differential equations. Some of its key features include its CAD capabilities for the creation of complicated 2-D or 3-D domains and its sophisticated meshing capabilities. These highly visible features are accessible through the Java-based graphical user interface (GUI). Beginning users will start to learn the software by using this GUI, and this is suitable for the immediate solution of a problem. However, for the solution of larger problems with correspondingly larger memory requirements and longer run times, other features of COMSOL become crucial. In these cases, it is often useful to run COMSOL on a different machine than the desktop computer, such as a remote machine with larger memory or faster processors like in the UMBC High Performance Computing Facility (HPCF; Moreover, to ensure reproducibility of one’s research results, or to perform parameter studies, the use of the GUI is not ideal. These problems are addressed by COMSOL’s capabilities for scripting and batch processing. These features are not easy to use, and in fact, it is even quite confusing what features all exist and what they mean; this is a result of the extreme versatility and power of COMSOL. My talk will attempt to structure and explain the options for the use of COMSOL that exist and show them at work, including specifically how to use m-files in conjunction with MATLAB as scripting tool and COMSOL’s own binary format for batch processing. To put in perspective, COMSOL also has additional parallel computing capabilities on multi-core and multi-processor systems, though these are not covered by this talk. Help for using COMSOL and all of its features are available via our the Center for Interdisciplinary Research and Consulting (CIRC; or e-mail in the Department of Mathematics and Statistics at UMBC.

Monday April 13

Title A stability study of explicit numerical schemes for a system of differential equations with a large skew-symmetric component
Speaker Katharine Gurski,
Howard University

Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time-step, but require more computations. When the differential equations are dominated by a skew-symmetric component , the problem is not stiffness in the sense that the size of the eigenvalues are unequal, rather that the real eigenvalues are dominated by imaginary eigenvalues. This talk will present and compare several explicit methods including the super-time-stepping method which is a explicit Runge-Kutta method for parabolic partial differential equations and new methods modeled on a predictor-corrector scheme with multiplicative operator splitting. Two of these new explicit methods increase stability without forcing the step size to zero. The talk will include how this method applies to the system of Black-Scholes differential equations and to equations for Hall magnetohydrodynamics with ambipolar diffusion..

Monday April 20

Title DoD Funding/Collaboration Opportunities Presentation
Speaker Heather White
DoD Research Specialist

I will discuss potential collaborations and funding opportunities available through the United States Army Medical Research and Materiel Command (USAMRMC) at Ft. Detrick.

Monday April 27

Title Group Dynamics in Phototaxis
Speaker Doron Levy

Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients. Optimizing the environmental conditions is conducted by moving in a particular direction, a motion known as ”chemotaxis” or ”phototaxis.” The various patterns of motion appear to be a complex function of cell density, surface properties and genotype. In this talk we will present a hierarchy of new models for phototaxis that were constructed based on experimental observations: a stochastic model, a particle system, and a system of partial differential equations. Our main theorem proves that the resulting system of PDEs is indeed the limit of the particle system. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute) and Tiago Requeijo (Math, Stanford).

Monday May 4

Title Fast reconstruction methods for diffuse imaging with many sources
Speaker Gunay Dogan

In this work, we address a 2d tomography problem, where we try to reconstruct the absorption coefficient of an elliptic partial differential equation (PDE) from boundary measurements induced by a large number of sources. Our motivation for this problem is diffuse optical tomography, a new imaging modality that can be used for breast tumor detection and functional brain imaging. We pose our model problem on a square geometry where the light sources and measurements are located regularly on opposite sides of the domain. This corresponds to a popular data acquisition pattern. The problem in this form requires solving a large nonlinear inverse problem, where the forward problem is given by multiple elliptic PDEs, and is thus computationally intensive. To address this, we propose to solve a linearized version of the problem based on the Born approximation and show that substantial gains can be made in computation. By revealing the special structure of the problem, we design fast methods to assemble the coefficient matrix for the linearized problem. We also propose fast matrix-vector product routines that can be used to solve the linear system with iterative methods or sparse singular value decomposition. Finally we introduce a fast inversion algorithm that produces the solution of the inverse problem by solving a sequence of small systems. We demonstrate the effectiveness of our method with several examples. This is joint work with George Biros.