| Air Quality Modeling Project
 Amundson,
Fitzgibbon, Glowinski, He, and Kuznetsov are currently working on that project. The EPA funded
research on air quality modeling is aimed at developing
a performance-portable multi-scale air quality modeling
system to simulate various chemical and physical
processes that are important for understanding atmospheric
trace gas transformations and distributions. The
photochemical model consists of a set of coupled
partial differential equations, one for each chemical
species. The input of these equations is complete
local weather data and concentrations of chemical
precursor molecules, ideally determined by real-time
monitoring. Many of the monitored species undergo
chemical reactions, decreasing the concentration
of one species, and increasing the concentration
of another species to potentially harmful levels.
The most advanced applied mathematical techniques
and massively parallel supercomputer simulations
are required to predict the future concentration
of potentially harmful species. The methods currently
employed depend on the computational science of the
late 1970's and early 1980's. They do not utilize
the dramatic breakthroughs of the last twenty years
in the science of computation. Effective use of modern
technology, especially large-scale distributed parallel
computing and state of the art computational methodology,
will allow us to increase both the efficiency and
speed of computation.
Air Quality Modeling is already organized around
research, training, and vertical integration. Current
participants include postdocs (Basak, Lubertino,
Yoo), graduate students (Martynenko, Myers, Smith,
Wang), and undergraduate students (Ghere, Lewis).
The graduate students currently work on a variety
of problems involving novel operator splitting techniques
and stiff ODE solvers, as well as studies of various
aspects of parallel and high performance computing.
They present their work as PhD and MS thesis projects
in the Computational Sciences program. This program
allows students to obtain a multi-disciplinary education
across traditional departmental lines. Through capstone
senior research projects, the undergraduate students
participate in the development of support tools for
the modular visualization environment and statistical
model evaluation. The group holds biweekly meetings.
Students, along with postdocs and faculty, participate
in three end-of-semester review meetings (Fall, Spring,
and Summer) at which they present talks and summaries
of research projects.
This project has two primary goals: a performance-enhanced
multi-scale air quality model system,
better able to exploit modern mathematical algorithms
and microprocessor-based parallel computers, and
an interdisciplinary program on air quality modeling.
Physiological
Fluid Dynamics Project
 Canic is working on an interdisciplinary project in the endovascular
treatment of abdominal aortic aneurysms (AAA) that
uses expertise in the fields of cardiovascular interventions
(Krajcer, M.D., Interventional Cardiologist at St. Luke's
Episcopal Hospital and the Texas Heart Institute),
mechanical engineering (Ravi-Chandar, UT Austin Center
for Mechanics of Solids, Structures and Materials),
computer science Mirkovic, and applied mathematics
Canic. Endovascular prostheses, called stents, are
used in the treatment of AAA. There are two principal
objectives: to carry out a quantitative analysis
of the (hemodynamics) equations that model the performance
of stents, and to aid physicians in the choice of
a stent.
A first generation hemodynamics model was developed
to study blood flow through axisymmetric elastic
tubes with elastic properties that change discontinuously.
Solving this model requires a new mathematical theory
for solutions of quasilinear hyperbolic equations
with discontinuous coefficients. The next steps are
the development of a three-dimensional model, its
implementation on advanced parallel computer architecture,
and its simulation.
Both education and research have been integrated
into this project from its inception. Three undergraduate
students (Pritts, Burns in Math; Roy in CS) and a
postdoc (Kim) have worked on modeling, numerical
simulation, and the development of a theory to study
the first generation equations. In Summer 2001, two
REU students also worked on the numerical
analysis and simulation of the simplified equations.
Both students wrote papers that will be archived
as Mathematics Department preprints. Two graduate
students (Burns and Sharma) will begin their PhD
thesis work on this project in Fall 2001, and a postdoc
(Vassilevski) will begin work on the 3D hemodynamics
model in Spring 2001.
Wavelets and Visual Computing
Project
Papadakis, and Paulsen are working on that project that will bring together
researchers and students in the fields of functional,
harmonic, and wavelet analysis with researchers in
the fields of visual computing and signal processing.
Students can integrate their mathematical research
with projects on visual computing and signal processing
under development in UH's Center for Bioimaging and
Biocomputation.
For example, Papadakis is involved in developing
multiresolution deformable models and non-separable
multidimensional multiresolution designs in collaboration
with faculty in computer science (Kakadiaris), electrical
and computer engineering (Karayiannis), and chemistry
(Kouri). These designs are applied to low bit rate
video compression, high resolution video transmissions,
compression and progressive transmission of images
of surfaces, geoscientific data analysis and edge
detection, and texture segmentation. Paulsen and
a PhD student (Holmes) are involved in more abstract
research on frame theory, aimed at classifying and
generating frames with special properties that should
eventually have application to data compression.
Student projects will be drawn from such applications
as the analysis and synthesis
of human motion, biomedical imaging and geoscientific
data analysis. Graduate students and postdocs will
participate in the development of algorithms based
on their research, while undergraduates can perform
simulations and experiments to test the newly developed
codes.
Symmetry and Neuroscience
Project
 Field,
Golubitsky, and Stewart are working on that project. Identical coupled systems of
ODE or {\em coupled cell systems} are discrete-space
continuous-time models that have proved useful as
models of locomotor central pattern generators (animal
gaits) and of the visual cortex (geometric visual
hallucination patterns). The dynamics of cell systems
can be quite complex (oscillation, heteroclinic cycles,
cycling chaos), even when there are only a few cells
and the dynamics within each cell are simple. For
example, the coupling of two one-dimensional cells
can produce oscillation even though neither one by
itself can oscillate. A central question about coupled
cell system dynamics concerns the relative balance
between the internal dynamics of each cell and the
way the cells are coupled. Coupled cell systems also
have symmetry (permutations of the cells that preserve
the coupling), which often organizes much of the
interesting dynamics. The patterns of oscillation
that cell systems can exhibit depend solely on the
architecture of the cell system, as our work on animal
gaits has shown.
Coupled cell systems can be explored by simulation
and theory, and students have written undergraduate,
masters, and PhD theses on the subject. For example,
undergraduates can simulate systems, such as a four-cell
model for biped locomotion (that models differences between walk and run), whereas advanced students
can work on the sophisticated dynamics, symmetry-breaking
bifurcations, and pattern formation
(as in the visual cortex) that appear in these models.
The mathematical classification of neuronal bursting
states has been an important focus of study for the
past 20 years. Using a combination of theory and
numerical simulation, patterns of bursting in coupled
cell systems will be explored.
Einstein Metrics Project
Bao is working on that project. Einstein metrics (the Ricci tensor is a multiple
of the metric tensor) comprise a major focus in differential
geometry. These metrics are more general than those
with constant curvature (the space forms); nevertheless,
explicit examples are scarce and graphical descriptions
of known examples almost totally lacking.
Einstein metrics generalize to Finsler manifolds
(that are equipped with norms instead of inner products).
Broadening the context to Finsler geometry holds
promise because explicit examples of Einstein metrics
are available among a special class known as Randers
spaces: Riemannian spaces for which there are a preferred
direction at each point. This preferred vector field
can arise as the distribution of wind/fluid velocities,
or magnetic polarizations. There is an ongoing effort
to express the Einstein condition for Randers spaces
as a system of nonlinear PDEs coupling the Riemannian
metric to the preferred vector field.
This project can initiate undergraduates, train
graduate students, and supply postdocs with interesting
problems. Undergraduates can use computer software
to picture the geodesics of known examples (Bass).
Graduate students can work on the classification
of Randers spaces which are Einstein (Robles). Postdocs
can focus on the construction of Randers-Einstein
metrics on spaces with interesting topology.
This project has an interdisciplinary flavor. Regard
the preferred vector field as the steady-state flow
of particles in a given medium. Model the Riemannian
metric as the warping of space (occupied by the medium)
due to the presence of elastic obstacles. Alterations
in the flow reshape the elastic obstacle, and vice
versa. This model provides a paradigm for understanding
the coupling between the Riemannian metric and the
preferred vector field (which may be visualized using
large scale computing).
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