DE Seminars: Spring 2010

Monday March 8

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

Due to mutations in the coding region of ion channel genes, various diseases of muscle cells can arise. One such class of .channelopathies., myotonia, has been studied from a clinical and electrophysiological standpoint, and a mathematical model has been developed to further understand the dynamics of the disease. In our presentation we will discuss reduction of this model to an analytically tractable two-compartment 3D system, and through geometric perturbation theory and simulation, solution behavior corresponding to some rat muscle observations will be explained. In particular, 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, mimicking observations seen in patients with Hyperkalemic periodic paralysis.

This is work with Kamonwan Kocharoen and Yongwimon Lenbury.

Monday March 22

Title Numerical Approximation of Solutions to Nonlinear Inverse Problems Arising in Olfaction Experimentation
Speaker Donald French
Department of Mathematics and Statistics
University of Cincinnati

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. An interdisciplinary research team has been working to determine the distributions of ion channels in frog olfactory cilia. The cilia are long thin processes that extend from the olfactory receptor neurons. The first step in the transduction of an odor into an electrical signal occurs in the membranes of the cilia and is controlled primarily by ion channels.

Mathematical models and simple approximation methods are derived to obtain estimates of the spatial distributions of the ion channels along the length of a cilium.

In this talk we review our results and describe a specialized problem involving a constrained integral equation.

This work is with S. J. Kleene in the University of Cincinnati College of Medicine.

Monday April 19

Title Mathematical Modeling, Analysis and Simulation of a Resonant Optothermoacoustic Sensor
Speaker Noemi Petra
Department of Mathematics and Statistics

Cost-effective sensor systems with the ability to identify trace gases with sensitivities in the parts-per-million (ppm) range are becoming essential tools for environmental monitoring, medical diagnostics, and homeland security. Sensors based on optothermal detection hold great promise because of their simple design, compact size, and potentially low cost. We model an optothermal sensor that incorporates a laser source and a quartz tuning fork (QTF) receiver. To detect the presence of a trace gas, the laser beam modulated with a frequency that matches the QTF resonance frequency, is focused between the tines of the tuning fork where the gas under study is located. This action induces a periodic excitation of the gas molecules which diffuse in space and come into contact with the QTF receiver. As a result, the QTF undergoes periodic thermal expansion and contraction cycles. This change in temperature produces a strain, which gives rise to a stress via elasticity, which is converted into an electric response. Modeling this system numerically requires the computation of thermal stresses induced by the temperature difference and the computation of the generated signal induced by the vibration of the QTF. We use the model to study the dependence of the generated signal on the position of the laser beam with respect to the QTF. We show that the output is largest when the source is focused near the base of the QTF. These simulation results closely match experimental data.

Monday May 3

Title Hermite methods for hyperbolic initial-boundary value problems
Speaker Xi Chen
Department of Mathematics and Statistics
University of New Mexico

We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. For hyperbolic problems, these cell polynomials can be evolved independently over a time step comparable to the macroscopic cell size. This remarkable fact, which directly results from the coalescence of the degrees-of-freedom to the cell center combined with the inherent stability properties of Hermite interpolation, allows extremely efficient time-stepping with minimal storage, minimal cell-to-cell communication, and derivative operators which scale optimally with cell width H and polynomial degree m. Numerical experiments are included to demonstrate the resolution of the methods for large m as well as illustrate the basic theoretical results.

Some of my past research will also be briefly discussed including I. Scattering from a lossless sphere. We study fundamental issues in electromagnetic scattering theory, with an emphasis on pole behaviors of a lossless sphere arising from the singularity expansion method (SEM). II. Modeling supercontinuum generation in fibers with general dispersion characteristics. The generation of broadband supercontinua (SC) in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity.

Monday May 10

Title Parallel Numerical Algorithms of Inverse Problems in Image Processing for a GPU Environment
Speaker Youzuo Lin
School of Mathematical and Statistical Sciences
Arizona State Univesity

Regularization techniques have been widely used in many application domains for the solution of ill-posed inverse problems. In the first part of the talk, I will go over Tikhonov regularization and discuss how to numerically parallelize the algorithm. In the second part of the talk, I will discuss how Graphical Processing Unit (GPU) can help in speeding up numerical solvers like Conjugate Gradient (CG). Applications of image restoration and image reconstruction will be utilized as testing problems.

Firstly, I will consider the parallelization of the least squares regularization problem by using multisplitting which is a method of domain decomposition. The algorithm is composed of global and local iterations. The global iteration requires that solutions are obtained in the local iteration to solve the sub problem system for multiple right-hand sides, where each right-hand side depends on the current global iterative solution. The algorithm is made more efficient by using updating of the initial local Krylov subspaces per problem with minimal restarts. We also study both the corresponding convergence and computational cost. The strength of our algorithm is illustrated by applying it to both image reconstruction and restoration problems.

Secondly, the emerging GPU has shown its enormous computability in high performance computation (HPC). Many efforts has been devoted to the efficient algorithm design of Krylov subspace solvers for large scale linear systems. In this part, I will introduce a well-known numerical method, Lanczos-Galerkin projection originated from the numerical analysis field to the GPU community. We shows that this method can fit well with the single precision accuracy of the current GPUs. One of the benefit gaining from this method is that, it overcomes the large amount of matrix vector (mat-vec) multiplications, which is usually the bottleneck for an efficient GPU based Krylov iterative algorithm. CG is utilized as an example in this work. Numerical results running on GPU are provided to support our proposed algorithm.