Professor
Ellad Tadmor, Department of Aerospace
Engineering and Mechanics, University of
Minnesota
A Quasicontinuum for
Multilattice Phase Transforming Materials
The quasicontinuum (QC) method is applied to
materials possessing a
multilattice crystal structure. Cauchy-Born (CB)
kinematics, which accounts for the shifts of the
crystal basis, is used in continuum regions to
relate atomic motions to continuum deformation
gradients. To avoid failures of the CB
kinematics, QC is augmented with a phonon
stability analysis that detects lattice period
extensions and identifies the minimum required
periodic cell size. This augmented approach is
referred to as "Cascading Cauchy-Born
kinematics". Applications of this methodology to
one-dimensional test problems that highlight the
salient features will be presented. Extension to
higher dimensions is straightforward and is
currently being pursued. Some preliminary
results will be shown.
Professor
Alina Chertock, Department of Mathematics,
North Carolina State University
A Positivity Preserving
Central-Upwind Scheme for Chemotaxis and
Haptotaxis Models
In this talk, I will present a new
finite-volume method for a class of chemotaxis
models and for a closely related haptotaxis
model. In its simplest form, the chemotaxis
model is described by a system of nonlinear PDEs:
a convection-diffusion equation for the cell
density coupled with a reaction-diffusion
equation for the chemoattractant concentration.
The first step in the derivation of the new
method is made by adding an
equation for the chemoattractant concentration
gradient to the original system. We then show
that the convective part of the resulting system
is typically of a mixed hyperbolic-elliptic type
and therefore straightforward numerical methods
for the studied system may be unstable.
The new method is based on the application of
the second-order central-upwind scheme,
originally developed for hyperbolic systems of
conservation laws, to the extended system of
PDEs. We show that the proposed central-upwind
scheme is positivity preserving, which is a very
important stability property of the method. The
scheme is applied to a number of two-dimensional
problems including the most commonly used
Keller-Segel chemotaxis model and its modern
extensions as well as to a haptotaxis system
modeling tumor invasion into surrounding healthy
tissue. Our numerical results demonstrate high
accuracy, stability, and robustness of the
proposed scheme.
This is a joint work with A. Kurganov, Tulane
University.
Professor
Fadil Santosa, School of Mathematics and
Institute for Mathematics and its Applications,
University of Minnesota
Modeling and Design of
Optical Resonators
We consider resonance phenomena for the
scalar wave in an inhomogeneous medium.
Resonance can be described as a solution to to
the wave equation that is spatially localized
while its time dependence is (mostly) harmonic
except for decay due to radiation. The
reciprocal of the decay rate is referred to as
the quality of the resonator. We will discuss
the problem of modeling resonators and propose a
method for designing resonators which has high
quality. We will start with an introduction to
resonance and its computation, and describe a
continuation approach that allows one to create
a medium whose resonance has a high quality.
Numerical examples which illustrate our method
will be presented.
Professor Jinchao
Xu, Department of Mathematics, Penn State
University
Numerical Methods for
High Order and Coupled PDE Systems
In this talk, I will present some new numerical
solution techniques for high order and coupled
PDE Systems. I will first discuss some relevant
numerical difficulty and subtlety for this type
of problems, including the potential risk of
reducing a higher order PDEs into a system of
lower order PDEs. I will then report a family of
finite element methods for three types of
problems:
(1) any 2m-th order elliptic boundary value
problems in $R^n$ for any $1\le m\le n$
(2)
a 4-th order PDEs in terms curl operator
(arising from magneto-hydrodynamics modeling) in
$R^3$
(3) Stokes-Darcy-Brinkman systems for
coupled fluids and porpus-media (arising from
fuel-cell and subsurface flow modeling).
If time
allows, I will also discuss algebraic techniques
for solving the discretized systems.
Professor Markus
Püschel, Department of Electrical and
Computer Engineering, Carnegie Mellon University
Linear Transforms:
Theory and Automatic Implementation
Linear transforms, such as the discrete Fourier
transform, discrete cosine transforms, and many
others, are among the most important numerical
kernels used in signal processing and many other
disciplines.
Most transforms possess a surprising number of
fast algorithms, a fact established by the more
than hundred publications on this topic.
However, with few exceptions, the algorithms are
derived through ingenious manipulation of the
transform coefficients—a method that gives no
insight into existence or structure of the
algorithm. In the first part of this talk we
sketch an algebraic theory of transform
algorithms, which solves this problem for many
transforms and also enables the derivation of
many new algorithms, which could not be found
with previous methods. (More info:
http://www.ece.cmu.edu/~smart)
In the second part of the talk we focus on the
efficient implementation of transforms, which is
a difficult problem on fast-changing and
increasingly complex and parallel computing
platforms. We present Spiral, a system that
overcomes this problem by automatically
generating highly optimized code directly from a
problem specification. Optimization includes
vectorization and parallelization for multicore
platform. The performance of the generated code
competes with and sometimes outperforms the best
handwritten libraries. We show that Spiral’s
framework extends beyond transforms and may
serve as a prototype on how to teach computers
to write fast numerical libraries. (More info:
http://www.spiral.net)
Professor
Saswata Hier-Majumder, Department of
Geology, University of Maryland
Cross-scale Modeling in
Magma Dynamics
Thermal and chemical evolution of
planetary bodies involves migration of magmatic
melts through a viscous, rocky matrix. Magma
migration and storage involves a set of coupled
processes operating at different length scales.
On the one hand, efficiency of melt extraction,
thickness of melt-rich layers, distribution of
radioactive elements within the mantle, and
formation of oceanic crust are processes that
are deeply related to magma migration over
length scales of hundreds to thousands of
kilometers. On the other hand, effective
physical properties such as permeability, total
interfacial tension, viscosity and elastic
moduli are deeply influenced by the
microstructure of the partially molten
aggregates. Coupled motion of the viscous rocky
matrix and the magmatic melt over large length
scales is modulated by the physical properties
arising from the microstructure of the rock. A
robust description of magma migration and
planetary evolution thus requires coupled
modeling of microstructure in the grain scale as
well as melt migration modeling in the planetary
scale. In this talk, I will present some recent
results from both scales of modeling, and some
directions for future work.
Dr. Jim
Purser, National Centers for Environmental
Prediction (NCEP) Presentation Slides
Discrete and
Differential Geometry applied to the Efficient
Numerical Synthesis of Spatially Adaptive
Covariances
The assimilation
of spatially distributed meteorological data
into the gridded form needed to initialize the
time integration of a numerical forecasting
model involves several specialized statistical
and numerical procedures. These are designed to
exploit what is known about the distribution of
errors in the pre-existing best estimate, or
‘background’ field of dynamical variables. Of
particular importance is the synthesis of the
errors’ covariances, modeled as spatially
adaptive filtering operators that act upon
appropriate adjoints of the field in question.
Recent developments illustrate the considerable
value to be gained by employing advanced methods
of both discrete and continuous geometry in the
efficient synthesis of the covariance operators.
The discreteness involves the systematic search
for special sets of generalized lines threading
the computational lattice. These line sets are
suitable for the application of low-pass line
filters which, when applied sequentially, will
reproduce the desired degree of local anisotropy
and coherence scale. The normalization of the
resulting synthetic covariances is achieved by
asymptotic methods derived from the tools
provided by non-Euclidean geometry, with the
scale and anisotropy of spatial covariance taken
as an effective Riemannian metric in which the
collective application of the aforementioned
filters is interpreted as the action of a
constant isotropic diffusion. The talk will
emphasize the fundamental role played by
symmetry in successfully orchestrating these
geometrical ideas.
April 2
CANCELLED
Professor Michael J. Shelley, Applied
Mathematics Lab, The Courant Institute CANCELLED
Dynamics and Transport
in Active Suspensions
Fluids with suspended micro-structure --
complex fluids -- arise commonly in micro- and
bio-fluidics, and can have fascinating and novel
dynamical behaviors. I will discuss some
interesting examples of this, but will
concentrate on my recent work on "active
suspensions", motivated by recent experiments of
Goldstein, Kessler, and their collaborators, on
bacterial baths. Using large-scale
particle-based simulations of hydrodynamically
interacting swimmers, as well as a recently
developed kinetic theory, I will investigate how
hydrodynamically mediated interactions lead to
large-scale instability, coherent structures,
and mixing.
Professor
Alexander Gorban, Applied Mathematics,
University of Leicester
Limiters in lattice
Boltzmann methods
The lattice
Boltzmann method has been proposed as a
discretization of Boltzmann's kinetic equation
and is now in wide use in fluid dynamics and
beyond. Instead of fields of moments
(hydrodynamic fields), the lattice Boltzmann
method operates with fields of discrete
distributions. This allows us to construct very
simple limiters that do not depend on slopes or
gradients.
We construct a system of nonequilibrium entropy
limiters for the lattice Boltzmann methods (LBM).
These limiters erase spurious oscillations
without blurring of shocks, and do not affect
smooth solutions. In general, they do the same
work for LBM as flux limiters do for finite
differences, finite volumes and finite elements
methods, but for LBM the main idea behind the
construction of nonequilibrium entropy limiter
schemes is to transform a field of a scalar
quantity - nonequilibrium entropy.
There are two families of limiters: (i) based on
restriction of nonequilibrium entropy
(entropy "trimming") and (ii) based on filtering
of nonequilibrium entropy (entropy filtering).
The physical properties of LBM provide some
additional benefits: the control of entropy
production and accurate estimate of introduced
artificial dissipation are possible.
The constructed limiters are tested on classical
numerical examples: 1D athermal shock tubes with
an initial density ratio 1:2 and the 2D
lid-driven cavity for Reynolds numbers Re
between 2000 and 7500 on a coarse 100*100 grid.
All limiter constructions are applicable both
for entropic and for non-entropic equilibria.
Joint work with R. A. Brownlee and J. Levesley
(Leicester)
Professor James
M. Greenberg, Department of Mathematics,
Carnegie Mellon University
Two-Dimensional
Sloshing Flows for the Shallow-Water Equations
In this talk I’ll discuss “sloshing” 2-Dimensional flows for the “shallow-water” equations.
The model describes the motion of a finite volume of viscous fluid taking place in container whose bottom is
described by a paraboloidal like surface of the form
z = (αx2 +βy2)/2, α> 0, β> 0
,
or more generally, z
= a(x,y) where a->∞ as (x2 +y2) ->∞.
The model includes gravity, Coriolis, and viscous forces.
Professor
Irene Fonseca, Department of Mathematical
Sciences, Carnegie Mellon University
Variational Methods in
Materials and Imaging
Several
questions in applied analysis motivated by
issues in computer vision, physics, materials
sciences and other areas of engineering may be
treated variationally leading to higher order
problems and to models involving lower dimension
density measures. Their study often requires
state-of-the-art techniques, new ideas, and the
introduction of innovative tools in partial
differential equations, geometric measure
theory, and the calculus of variations. In this
talk it will be shown how some of these
questions may be reduced to well understood
first order problems, while in others the higher
order plays a fundamental role.
Applications to phase transitions, to the
equilibrium of foams under the action of
surfactants, imaging, micromagnetics and thin
films will be addressed.
Professor
David Gottlieb, Division of Applied
Mathematics, Brown University
A Modified Optimal
Prediction Method and Application to the
Particle Method
In the
numerical solution of nonlinear PDE's there are
always small scales that can not be resolved.
Often the small scales themselves are not
important but their influence on the large
scales is crucial.
Chorin suggested the modified prediction method
as a tool to statistically model the impact of
the small scales. In this talk a different
variant of the optimal prediction method will be
presented.
Also we introduce and compare several
approximations of this method. We apply the
original and modified optimal prediction methods
to a system of ODEs obtained from a particle
method discretization of a hyperbolic PDE and
demonstrate their performance in a number of
numerical experiments.
This is a joint work with Alina Chertock and
Alex Solomonoff.
Professor Lisa
Fauci, Department of Mathematics, Tulane
University
Interaction of Elastic
Biological Structures with Complex Fluids
The bio-fluid-dynamics of reproduction
provide wonderful examples of fluid-structure
interactions. Peristaltic pumping by wave-like
muscular contractions is a fundamental mechanism
for ovum transport in the oviduct and uterus.
While peristaltic pumping of a Newtonian fluid
is well understood, in many important
applications the fluids have non-Newtonian
responses. Similarly, mammalian spermatozoa
encounter complex, non-Newtonian fluid
environments as they make their way through the
female reproductive tract. The beat form
realized by the flagellum varies tremendously
along this journey. We will present recent
progress on the development of computational
models of pumping and swimming in a complex
fluid. An immersed boundary framework is used,
with the complex fluid represented either by a
continuum Oldroyd-B model, or a Newtonian fluid
overlaid with discrete visco-elastic elements.
