This talk investigates the upscaling methods to the following
parabolic equation
$$
\partial_t c+\nabla\cdot(\mathbf{u}
c)-\nabla\cdot(\mathbf{D}\nabla c)=f(x,t)
$$
which stems from the application of solute transport in porous
media. Because of the high oscillating permeability of the porous
media, the Darcy velocity $\mathbf{u}$ hence the dispersion tensor
$\mathbf{D}$ has many scales with high contrast. Thus, how to
calculate the macro-scale equivalent coefficients of the above
equation becomes the target of this talk. Two kinds of upscaling
formulations are discussed in this work. The two different
equivalent coefficients computing formulations are based on the
solutions of two different cell (local) problems, which one utilizes
the elliptic operator with terms of all orders while the other only
uses the second order term. Error estimates between the equivalent
coefficients and the homogenized coefficients are given under the
assumption that the oscillating coefficients are periodic (which is
not required by our methods). Numerical experiments are carried out
for the periodic coefficients to demonstrate the accuracy of the
proposed methods. Moreover, we apply the two upscaling methods to
solve the solute transport in a porous medium with a random
log-normal relative permeability. The results show the efficiency
and accuracy of the proposed methods.
Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and
Angeli) is their decomposition in terms of so-called monotone systems, which can be thought of as systems with exclusively positive feedback.
In this talk I discuss some general properties of monotone dynamical systems, including recent results regarding their generic convergence
towards an equilibrium. Then I will discuss the use of monotone systems to model biochemical behaviors such as switches and oscillations under
time delays.
We study particle systems corresponding to highly connected queuing
networks, like Internet. We examine the validity of the so called Poisson
Hypothesis (PH), which predicts that such particle system, if started
from a reasonable initial state, relaxes to its equilibrium in time
independent of the size of the network. We show that this is indeed the
case in many situations.
However, there are networks for which the relaxation process slows down.
This behavior reflects the fact that the corresponding infinite system
undergoes a phase transition. Such transition can happen only when the
load per server exceeds some critical value, while in the low load
situation the PH behavior holds. Thus, the load plays here the same role
as the inverse temperature in statistical mechanics.
I will discuss some results on the vanishing and non-vanishing of
critical values of L-functions and their derivatives, both experimental and
theoretical. I will present an example of a computational "elliptic Stark
point" in a cyclic quintic extension of the rationals.
The control of the density and location of dislocations (line defects)
in heteroepitaxial thin film is very important in designing
semiconductor-based electronic devices. We have developed a level set
method based, three dimensional dislocation dynamics simulation method
to describe the motion of dislocations in thin films. The dislocation
location is given by the intersection of the zero level sets of a pair
of level set functions. This representation does not require
discretization and tracking of the dislocation, and handles topological
changes automatically. The simulation method incorporates the elastic
interactions of the dislocations and the stress fields throughout the
film and substrate. Using the above approach, various dislocation
motion and interactions within a heteroepitaxial thin film are simulated
and analyzed.
I will discuss asymptotic-based algorithms for the study of
the electronic structure of materials, in the context of density
functional theory. I will illustrate the ideas using both the Kohn-Sham
and orbital-free formulations.
This is joint work with Weinan E (Princeton University), and Jianfeng Lu
(Princeton University).
In this talk semismooth Newton methods for solving
nonlinear non-smooth equations in Banach spaces are discussed.
In varaitional optimization problems function for which we
desire to find a root is typically Lipschitz continuous but not $C^1$
regular. A globalization theory is presented and applications to
complementarity problems and variational inequalities are discussed.
Classical potential theory converts linear constant-coefficient elliptic
problems in complex domains into integral equations on interfaces, and
generates robust, efficient numerical methods. The conversion is
usually carried out for a particular situation such as the Poisson
equation in dimension 2, and the efficiency of the resulting methods
then depends on detailed analysis of the appropriate special functions.
We present a general conversion scheme which leads naturally to a fast
general algorithm: arbitrary elliptic problems in arbitrary dimension
are converted to first-order systems, a periodic fundamental solution is
mollified for convergence, and the mollification is locally corrected
via Ewald summation. Local linear algebra and the elementary theory of
distributions yield a simple boundary integral equation. With the aid
of a new nonequidistant fast Fourier transform for piecewise polynomial
functions, the resulting numerical methods provide highly accurate
solutions to general elliptic systems in complex domains.
Some reaction-diffusion equations admit traveling wave
solutions, which are simple models of a chemical reaction spreading with
constant speed. Even in a heterogeneous medium, solutions to the initial
value problem may develop fronts propagating with a well-defined
asymptotic speed. I will describe recent progress in understanding how
fronts propagate in heterogeneous media. In particular, I will discuss
properties of generalized traveling waves for one-dimensional
reaction-diffusion equations with variable excitation. I also will
discuss multi-dimensional fronts in stationary ergodic random media.
We will discuss several projects with the general goal of
designing second-order accurate numerical methods for
the motion of a viscous fluid with a moving interface of
zero thickness which exerts a force in response to its
stretching. The interfacial force results in jumps in
the fluid quantities at the interface. In recent work with
Anita Layton we have found that the problem of the Navier-Stokes
equations with an elastic interface can be simplified by decomposing
the velocity at each time into a ``Stokes'' part, determined
by the (equilibrium) Stokes equations, with the interfacial
force, and a ''regular'' remainder which can be calculated
on a rectangular grid without special treatment at the interface.
For the Stokes part we use the immersed interface method; for the
regular part we use the semi-Lagrangian method. Smaller time
steps can be used to advance the interface with Stokes flow,
using boundary integrals, if needed, to handle the boundary force.
This decomposition exhibits second-order accuracy in
simple test problems. Analytical issues of accuracy and some related
error estimates for the immersed interface method will be
described. We allow for the possibility of more general
boundary motion in work with John Strain for the case of Stokes flow,
in which we use Strain's semi-Lagrangian contouring method
to move the interface. We represent the velocity, on or off
the interface, as a singular integral, and calculate it using
Ewald splitting. The smooth or regularized part is computed as a
Fourier series, while the local part is approximated analytically.