TIME: Mondays, 4pm
PLACE: Multipurpose Science & Technology Building, Room 122
ORGANIZER: Natalia Komarova
TRAVEL AND LODGING INFORMATION FOR SPEAKERS
Title: An Invertible Auditory Transform
Abstract: Sound
signal processing is based on spectral analysis normally done with
Fourier type transforms. In this talk, we discuss an invertible
transform with built-in auditory filter characteristics, its
mathematical properties and applications.
Title:
Two-phase flows with soluble surfactant:
local existence of strong solutions
Abstract: The
presence of surfactants, ubiquitous at most gas/liquid interfaces, has
a pronounced effect on the surface tension, hence on the stress
balance at the phase boundary: local variations of the capillary
forces induce transport of momentum along the interface -
so-called Marangoni effects. Surfactants are often soluble in one of
the adjacent bulk phases, in which case there is also exchange of
surfactant between the relevant bulk phase and the interface by
adsorption and desorption. Along the interface surfactant is
transported by convection and diffusion. Further, changes of the
interfacial area due to compression or stretching cause corresponding
changes in surfactant concentration.
We discuss the mathematical model governing the dynamics of such
systems. This leads to the two-phase balances of mass and momentum,
complemented by a species equation for both the interface and the
relevant bulk phases. Within the model, the motions of the surfactant
and of the adjacent bulk fluids are coupled by means of an interfacial
momentum source term that represents Marangoni forces. Employing
Lp-maximal regularity we obtain local (in time) strong well-posedness
of this model for certain initial configurations. The proof is based
on recent Lp-theory for two-phase flows without surfactant.
Joint work with Gieri Simonett (Vanderbilt University, Nashville TN)
and Jan Pruss (Universitat Halle-Wittenberg, Germany)
Title: Gradient Networks: From Transport Efficiency in Scale-free Graphs to Social
Influence Structures
Abstract: It has recently
been recognized that a large number of complex networks are
scale-free, having a power-law degree distribution. Here we propose
that the emergence of many scale-free networks is tied to the
efficiency of transport and flow processing across these
structures. In particular, we show that for large networks on which
flows are influenced or generated by gradients of a scalar distributed
on the nodes, scale-free structures will ensure efficient processing,
while non scale-free structures, such as random graphs, will become
congested. As an application, we then make a connection to a simple
agent-based model of a market and study the effects of the social
network on the evolution of the collective/market.
Title:
Paradigms of Stress Evolution in Growing Tumors
Abstract:
The evolution and spatial distribution of tissue stresses is of fundamental
importance in a number of physiological phenomena. The
experimentally-observed collapse of tumor blood vessels, for example, which
has been attributed to the elevated tissue stresses resulting from confined
proliferation of tumor cells, represents a significant barrier to the
delivery of blood-borne therapeutic agents. Such stresses are residual in
nature, arising in the tissue in the absence of external loads, and result
from the incompatibility of growth strains.
Nevertheless, the underlying phenomenological determinants of residual
stresses, as well as their purpose and implications in both normal
tissue development and various pathological conditions, are poorly
understood since there is currently a paucity of mathematical models
to elucidate these phenomena. In this presentation a number new
mathematical ideas germane to the study of residual stresses in
growing soft tissues will be discussed. Emphasis will be placed on
'solid-multiphase' tissue modeling, which represents a new class of
mathematical models in which the concepts of poroelasticity are
extended to accommodate continuous volumetric growth.
Title:
Statistical Physics, Bounded Rationality and Distributed Control
Abstract: Abstract: A long-running difficulty with conventional game theory is
how to modify it to accommodate bounded rationality. A recurring issue
in statistical physics is how best to approximate joint probability
distributions with decoupled (and therefore far more tractable)
distributions. A major problem in control theory is how to implement
control on (massively) distributed systems, especially in an adaptive
manner, with mixed types of control variables.
This talk shows that the same information-theoretic structure, known
as Probability Collectives (PC), underpins all three issues. This
means that statistical physics, game theory, and distributed control
are fundamentally identical. Accordingly techniques and insights from
one of those fields can be applied to the others. One example of
this, presented here, is the use of the grand canonical ensemble of
statistical physics to elaborate game theory in which the number of
players is not pre-determined, but varies stochastically. Another
example is how to apply steepest descent techniques to
optimize/control systems of discrete variables.
Title:
Light transport in two-layer tissues
Abstract: Light
propagation in tissues is governed by the theory of radiative
transport. The radiative transport equation takes into account
absorption and scattering due to inhomogeneities. A two-layer medium
is a useful model for tissues because it accounts for the differences
in optical properties between the superficial and deep regions of
tissues. We are interested in probing only the superficial layer
because most pre-cancerous tissues develop there. To do this we
introduce an alternate boundary condition that allows for the removal
of the bottom layer from the problem. For the case when the top layer
is thin, we compute an asymptotic solution. We validate our results by
comparing them with numerical solutions.
Title:
Simulating Semiconductor Charge Transport
Abstract:
Computer simulations of charge transport in semiconductor devices
(like diodes and micro-chips) are used by the semiconductor industry
as a tool for reducing the cost of developing new devices and new
process technologies. At the scale of micron or sub-micron, the
semiconductor Boltzmann equation is the most exact model. In order to
alleviate computing load, macroscopic models have been derived,
assuming that the state of the electron gas is described by certain
averaged quantities. These models take similar forms as those in fluid
mechanics, and we may apply CFD techniques to probe this promising
field of academic importance and commercial value.
We shall present some of our recent results. First, by simulating a
hydrodynamic model, we demonstrate the (direct) applicability of CFD
techniques. Secondly, the continuing trend of scaling-down and
speed-up makes the modeling and computing of quantum effect and
transient behavior among the top issues in semiconductor
research. Careful numerical tests helped identifying well-posedness
problem in a quantum hydrodynamic (QHD) model. Viscous QHD model
derived from a Wigner Fokker-Planck equation yields more reliable
numerical results, and demonstrate interesting nonlinear phenomena,
such as negative differential resistance and hysteresis.
Title:
Evolution of the universe
Abstract: We know that the Newtonian N-body problem cannot be solved
in a normal sense. On the other hand, we can find all possible
asymptotic behaviors as time goes to infinity of all possible
solutions for all possible values of N. That is, we can describe the
evolution of Newton's universe. In doing so, I will introduce some
of the history of the problem showing where "chaos" came from, etc.
Title:
Weak Wave Turbulence and its Challengers
Abstract: I will begin my talk with a brief overview of WWT during which
I aim to give an intuitive picture of the phenomenon using the example of
surface water waves. This example will be revisited throughout the talk.
Next I will try to give a welcoming (although selective) introduction to
the calculations of WWT. Equipped with the results of these calculations,
I will discuss the relationship between WWT, power-law spectra (both
Kolmogorov-Zakharov and MMT) and intermittency.
The challengers to WWT are highly nonlinear events, breakdown and the
alternative symmetries of the governing equation. I will make some
remarks which point out the interconnectedness of these phenomena and,
simultaneously, the goals of my research interests.
Title:
Multiscale Asymptotic Analysis of Wave Propagating in Nonlinear Periodic Media
Abstract: New models describing wave propagation in transversely modulated optically
induced waveguide arrays are proposed. In the weakly guided regime, a
discrete nonlinear Schrodinger equation with the addition of bulk
diffraction term and an external ``optical trap'' is derived. In the
defocusing regime the optical trap induces a stable localized mode. In the
limit of strong transverse guidance, the dynamics is governed by a model
which represents the optical analogue of wave action.
Title:
Introduction and recent results for LANS-alpha, the Lagrangian averaged Navier-Stokes alpha model of turbulence
Abstract (click)
Title:
Stochastic models in colon cancer
Abstract: I'll
begin by describing our recent attempts to model the evolution of
crypts in the colon, using methylation patterns as markers [see Yatabe
et al.]. Mutations in colon crypts are thought to play an important
role in pre-tumor progression, and therefore in understanding the time
to cancer. One common complaint about such multistage and multihit
models is that they require unrealistically high mutation rates to
explain the observed incidence of cancer. I'll use our model, together
with classical extreme value theory, to show that we can explain the
SEER incidence data for colon cancer using typical mutation rates
[Calabrese et al.]. A number of corroborative datasets and open
problems will be discussed.
Title:
Mathematical modeling of cancer
Abstract: I will
give an overview of the recent work I have done on stochastic modeling
of cancer. I will first talk about the concept of multistage
carcinogenesis and how we can describe cancer as "bad evolution"
within an organism. I will introduce some simple models and explain
the phenomenon of "stochastic tunneling". Then I will talk about the
role of stem cells in cancer initiation and present some hypotheses
about the cellular origins of colon cancer.
Finally, I will talk about growing cellular colonies and models of
treatment: how does resistance arise and what can we do about it?
Therapies which target specific molecular alterations in cancer cells
have shown promising results. Resistance, however, poses a problem,
especially in advanced disease. An example is the treatment of chronic
myeloid leukemia (CML) blast crisis with Gleevec. I will elucidate the
principles which underlie the emergence of drug resistance in
cancer. The model (a birth-death process on a combinatorial mutation
network) is based on measurable parameters: the turnover rate of tumor
cells, and the rate at which resistant mutants are generated. In the
context of CML, the prediction is that a combination of three drugs
can successfully treat blast crisis.
Title:
American Football, Barberpoles and
Clouds: Pattern Formation in Biased Diffusion of Two Species
Abstract:
Motivated by several physical systems, we study a simple model
of driven, two-species lattice gases. Our system consists of only two
types of NON-interacting (apart from an excluded volume constraint)
particles, diffusing on a periodic lattice and with biased moves in
opposite directions. On a square system, increasing the overall
particle density leads to a transition - from a homogeneous phase with
high particle current to one with spatial structure and minimal
current. For rectangular cases, several structures can appear, with
relative frequency depending on the aspect ratio of the system. Using
a simple continuum theory, we are able to describe much of the novel
transitions. Variations and generalizations, as well as the physical
systems they model, will be discussed.
Title: Success of a continuum lagrangian in twentieth-century solid state physics
Abstract: The
development of quantum mechanics in the 1920s convinced physicists
that future progress in understanding solids would flow from that
discipline, and so continuum mechanics of solids was abandoned by
physicists. Thus, success of the latter in late twentieth century is
regarded as surprising by most.
We present an overview of the construction of a very general Lagrangian
of a closed system of a dielectric crystal interacting with
the electromagnetic field. The Lagrangian is first constructed
for discrete particles, a long-wavelength (continuum) limit is
taken in a manner to preserve all of the eigenmodes. The crystal
can be of any class of symmetry, have any structural complexity,
and have interactions between its various eigenmodes and between
them and the electromagnetic field to any order of nonlinearity.
All eigenmodes are included: electromagnetic, acoustic
and optic modes of vibration, spin, and all polaritonic
combinations of them.
The photoelastic effect was show to have been wrongly formulated for
155 years: the independent variable characterizing the deformation had
been wrong! Thus, the interaction tensor has a more general
symmetry. The accepted relation between the photoelastic effect and
electrostriction was shown to have been wrong for almost as long. The
elastic stiffness tensor was shown to lose its traditional symmetry
when a soft optic mode became involved. All treatments of acoustic
harmonic generation in piezoelectrics were shown to be wrong. The best
derivation of optical activity was shown to have missed a fundamental
contribution having a different dispersion. The Abraham - Minkowski
controversy about the momentum of a light wave in a medium was
resolved. The most general Poynting vector in a medium was
found. Several nonlinear interactions were characterized and
interpreted for the first time.
Title:
Models of Social and Biological Contagion: are Puma shoes some kind of
virus?
Abstract: I will discuss two simple models of contagion relevant to the
desciption of social and biological spreading processes.
The first model aims to unify existing models of the spread of social
influences and infectious diseases. This generalized model of
contagion incorporates individual memory of exposure to a contagious
entity (e.g., a rumor or disease), variable magnitudes of exposure
(dose sizes), and heterogeneity in the susceptibility of individuals.
Through analysis and simulation, we have examined in detail the
mean-field case where individuals may recover from an infection and
then immediately become susceptible again. We identify three basic
classes of contagion models: epidemic threshold, vanishing critical
mass, and critical mass respectively. The conditions for a particular
contagion model to belong to one of the these three classes depend
only on memory length and the probabilities of being infected by one
and two exposures respectively. (For both models, a key quantity is
the fraction of vulnerables, i.e., individuals who are typically
infected by one exposure.) These parameters and their elaborations
are in principle measurable for real contagious influences or
entities, suggesting novel measures for assessing (as well as
strategies for altering) the susceptibility of a population to large
contagion events. We also study the case where individuals attain
permanent immunity once recovered, finding that epidemics inevitably
die out but may be surprisingly persistent when individuals possess
memory. I will also discuss some related work by others.
The second model describes the spreading of social influences on
networks, and is a natural extension of the threshold model due to
Granovetter. For this model on various kinds of random networks,
analytic results are known for when cascades (epidemics) are possible.
In all cases, the density of the network must belong to an
intermediate range referred to as the cascade window. When links are
scarce, not enough individuals are connected for global spreading to
occur, and when links are overly abundant, too few individuals are
vulnerable. In our recent work for this model, we have examined the
role of influentials (a.k.a. opinion leaders or, for a biological
feel, super-spreaders). We examine cascades after they have occurred,
as is invariably done for real cascades. Contrary to much ascribed to
influentials, we find that highly connected nodes are not the chief
determinants of whether or not a cascade will occur. While cascade
initiators are typically more connected than the average individual,
the discrepancy is not pronounced. We further observe that cascades
arise through a multi-step process and that for dense networks, `early
adopters' may in fact be less connected than on average. Also, for
dense networks, cascades rapidly take off after a long and `quiet'
build up period, making them difficult to identify until after they
have been realized. In sum, influentials are limited in their effect
since the condition for a cascade to occur is really a global one;
there must be a sufficient population of vulnerables available, and it
is the most influential of these vulnerables that dictate the spread
of an influence.
Title:
Fibonacci and Plants
Abstract: For
over four hundred years, natural scientists have been intrigued and
mystified by patterns appearing on plants and by the appearance of
Fibonacci sequences when one counts the numbers of arms in the
families of spirals on which the primordia of the plant surfaces
lie. To date, there has been no widely accepted mechanistic
explanation for these observations. I hope that this lecture goes some
way towards providing answers.
Title:
Curvelets and Wave Equations:
Theory and Potential for Scientific Computing
Abstract: This
talk explores the potential of new geometric multiscale ideas in the
area of partial differential equations. We present a recently
developed multiscale system - curvelets - based on parabolic scaling,
in which basis functions are supported in elongated regions obeying
the relation width ~length^2. This system provides optimally sparse
representations of the solution operators for a large class of
symmetric systems of linear hyperbolic differential equations - such
as the wave propagation operator. This has important implications
both for analysis, and for numerical applications, where sparsity
allows for faster algorithms. In the second part of the talk, we
report on preliminary calculations which suggest that it is possible
to derive accurate solutions to a wide range of differential equations
in O(N log N) where N is the number of voxels; this complexity holds
for arbitrary initial conditions. This is joint work with Laurent
Demanet (Caltech)
Title:
Mathematical Model Applications to Disease and Homeland Security
Abstract: The
events of 9/11 in the US changed the way we look at routine activities
such as air and mass-transportation travel. We (as a society) are
somewhat prepared to respond to natural acts (epidemics, earthquakes,
etc.) but have no data or reliable information that would help in the
planning or identification of a set of responses if a deliberate act
(against unsuspecting population) were to take place. I will highlight
some of the challenges that we face and outline the use of
mathematical models in our efforts to meet some of them. I will use
recent SARS and foot and mouth epidemics to ground some of the
ideas. Should we prepare for worst case scenarios? If so, how do we
define worst case scenarios mathematically? I will conclude with the
use of some of these ideas on the potential impact or consequences
associated with the deliberate release of a biological agent in the
mass transportation system of a major metropolitan area.
Mathematical Models and Their Application to the Spread and Control of
Tuberculosis
Tuberculosis high levels of prevalence in the world have been the
norm, particularly in poor and/or developing nations. The impact of
travel and immigration as well as the costs associated with the TB
treatment and the consequences associated with treatment compliance
(antibiotic resistance) will be discussed. The application of
mathematical models in the evaluation of epidemiological and
sociological factors associated with TB dynamics and its control at
the population level will be highlighted.
Title:
Abstract:
Title:
The Mathematical Challenge of Multiscale Modeling in Biology: From signal transduction to spatial pattern formation
Abstract:
In the last two decades enormous progress has been made on
understanding molecular details in a number of cellular
processes such as signal transduction and gene control, but
frequently the objective in modeling is to understand the
population-level behavior of cells. This gives rise to the
problem of how to incorporate sufficient microscopic-level
information into macroscopic-level descriptions. In this talk we
will discuss two systems that involve chemotaxis, one for which
this has problem has been more-or-less solved, and one for which
a great deal remains to be done.
Chemotaxis in the bacterium E. coli is widely-studied
because of its accessibility and because it incorporates
processes that are important in the response of numerous sensory
systems to stimuli: signal detection and transduction,
excitation, adaptation, and a change in behavior. Quantitative
data on the change in behavior is available for this system, and
the major biochemical steps in the signal
transduction/processing pathway have been identified. We will
discuss a mathematical model of single cells that can reproduce
many of the major features of signal transduction, adaptation
and aggregation, and which incorporates the interaction of the
chemotactic protein CheY_p with the flagellar motor. We shall
then address the problem of how to obtain macroscopic equations
for population-level behavior that incorporate certain features
of the microscopic model.
Many cells such leukocytes (cells of the immune system) also respond
chemotactically to external signals, but the process by which
they determine directional information and alter their pattern
of movement is much more complex than in bacteria, and the
micro-to-macro step is much more difficult. In the remainder
of the talk we will discuss recent progress and open questions
in this area.
Title:
Target characterization using time reversal symmetry of wave propagation
Abstract: The fact that wave propagation looks the same whether time is going
forward or backward has been know theoretically since the formulation of
wave theory. Only recently, however, has array technology and computers
been developed to the point that time reversal of waves can actually be
performed in real systems. Experiments using ultrasonic and underwater
acoustical arrays have shown enhanced focusing, communications, and
imaging through complicated media. Better theoretical understanding of the
time reversal symmetry for acoustic and electromagnetic waves has
motivated new techniques for imaging and characterization of targets
applicable to more conventional array technology. In this talk, these new
techniques for target characterization and imaging will be discussed along
with examples using both experimental and computational data. It is shown
that the time reversal properties of an array system can be predicted by
performing a singular value decomposition of the multistatic data matrix.
The spectrum of singular values and the form of the singular vectors are
related to the physical properties of the target in the field of view of
the array. This relationship is described for a number of simple cases and
imaging techniques that exploit their properties are shown.
Title:
On the Regularity Conditions for the Navier-Stokes and the Related Equations.
Abstract: In this talk I present my
recent results on the regularity conditions for a solution to the 3D
Navier-Stokes equations with powers of the Laplacian, which
incorporates the vorticity direction and its magnitude
simultaneously. For the proof of the we exploit geometric properties
of the vortex stretching term as well as the estimate using the
Triebel-Lizorkin type of norms.
Title:
Some Mathematical Problems in Computational Systems Biology
Abstract: Biochemical reaction networks provide a paradigm for many dynamical
systems in biology. The paradigm can be generalized to describe
"variable-structure systems" in which objects larger than molecules (such
as cells) also change in number and in their relationships over time. In
the course of building mathematical and software tools for understanding
such networks and dynamical systems, we have identified some interesting
applied mathematical problems whose reformulation and solution would be
very useful in current computational biology. For example, we can
identify partial differential equations whose solution would be especially
instructive for enzyme kinetics. These problems arise at the level of
small reaction networks, multimolecular complexes, and the development of
multicellular tissues. Developmental examples include modeling the shoot
meristem of a plant.
A common mathematical framework for models at these different spatial
scales can be given in terms of "dynamical grammars". In a dynamical
grammar, an input/output syntax for an elementary chemical or biological
processes is mapped to an operator algebra expression for the generator of
the temporal dynamics associated with that process. Many processes act
simultaneously (in parallel) if their generators are summed. Contingent
spatial relationships are expressed in terms of dynamical graph grammars,
whose formulation could perhaps be improved by use of ideas from topology
and differential geometry. By solving such problems, we may hope to
construct a useful modeling language of sufficient generality to describe
multiscale, variable-structure dynamical systems that arise naturally in
biology.
Title:
Quenching of reaction by fluid flow
Abstract: We consider the problem of
quenching the flame in a framework of passive reaction-diffusion
model. We ask which flows are more efficient in supressing reaction,
and prove bounds on the relationship between flow strength and the
initial flame size for different classes of flows. The estimates we
prove agree very well with numerical experiments carried out in
collaboration with astrophysics ASC group at the University of
Chicago. The problem is closely related to proving norm bounds for the
evloution semigroup corresponding to the passive scalar model. The
techniques involve PDE and probability tools, and further natural
questions indicate interesting links with spectral theory of elliptic
operators and dynamical systems.
Title:
A mathematical model for the regulation of tumor dormancy based on
enzyme kinetics
Abstract: We
present a two compartment model for tumor dormancy based on an idea of
Zetter to wit: The vascularization of a secondary (daughter) tumor can
be suppressed by inhibitor originating from a larger primary (mother)
tumor. We apply this idea at the avascular level to develop a model
for the remote suppression of secondary avascular tumors via the
secretion of primary avascular tumor inhibitors. The model gives good
agreement with experimental observation (Derm. Surg. 29(2003)
664-667). The authors reported on the emergence of a polypoid
melanoma at a site remote from a primary polypoid melanoma after
excision of the latter . The authors observed no recurrence of the
melanoma at the primary site, but did observe secondary tumors at
secondary sites five to seven centimeters from the primary site within
a period of one month after the excision of the primary site. We
attempt to provide a reasonable biochemical/cell biological model for
this phenomenon. We show that when the tumors are sufficiently remote,
the primary tumor will not influence the secondary tumor while, if
they are too close together, the primary tumor can effectively prevent
the growth of the secondary tumor, even after it is removed. It should
be possible to use the model as the basis for a testable hypothesis
which could be checked in a controlled in vitro experiment.
Title:
Least squares minimization to
estimate the transport of alcohol in the human body
Abstract:
Experimental measurements of transdermal vapor alcohol concentration
are used to estimate alcohol concentration in the body using an
inverse problem approach. First we propose a model for the transport
of alcohol from blood compartments to the skin surface and use the
transdermal measurements to estimate the signal obtained by a
breathalyzer which is the standard for blood alcohol
concentration. Later we couple our skin model to a body model of the
human body. The human body is divided in several compartments to
facilitate the description of the transport of alcohol in the human
body from ingestion to elimination. The adjoint method is used for the
computation of the least squares functional gradient. Parameters of
the model are estimated using real breathalyzer and a transdermal
alcohol skin device data applied to individuals in a hospital. The
parameter values obtained are used to predict the evolution of alcohol
concentration for patients in the field. Kalman filtering techniques
can be used to correct predictions in real time.
Title:
Modeling insulin secretion ultradian oscillations with two time delays
Abstract:
In the glucose-insulin regulatory system, insulin secretion
oscillates with a period of 50-150 minutes. Over the
past decade, several mathematical models have been proposed to
model these ultradian oscillations as well as the metabolic
system producing them. However these existing models yield
profiles deviant from a normal physiological range. We introduce
a DDE (delay differential equation) model with two discrete
delays for better understanding and more accurately modeling
the glucose-insulin dynamics and the insulin secretory oscillations.
With the same set of experimental data used to test other existing
models, the simulation profiles obtained from this two time delay
model fall within a normal physiological range.
Title:
Divorcing pressure from viscosity in incompressible Navier-Stokes
dynamics
Abstract:
The pressure term has always created difficulties in treating the
Navier-Stokes equations of incompressible flow, reflected in the
lack of a useful evolution equation or boundary conditions to
determine it. In joint work with Bob Pego and Jie Liu, we
show that in bounded domains with no-slip boundary conditions,
the Navier-Stokes pressure can be determined in a such way that
it is strictly dominated by viscosity. As a consequence, in a
general domain with no-slip boundary conditions, we can treat the
Navier-Stokes equations as a perturbed vector diffusion equation
instead of as a perturbed Stokes system. We illustrate the
advantages of this view by providing simple proofs of (i) the
stability of a difference scheme that is implicit only in
viscosity and explicit in both pressure and convection terms,
requiring no solutions of stationary Stokes systems or inf-sup
conditions, and (ii) existence and uniqueness of strong solutions
based on the difference scheme.
A preprint is available at http://arxiv.org/abs/math.AP/0502549
Title:
Modeling biochemical systems with
differential equations, stochastic processes, and constraint-based
optimizations
Abstract: With
the demand from modeling systems level cellular biochemistry as a
reaction network, different applied mathematical approaches are now
being pursued. I will discuss three approaches based on (1) systems
of ODE with nonlinearity, (1) stochastic processes with
irreversibility, and (3) constraint-based optimization suggesting an
oriented matroid. A unifying theme of these approaches is the
nonequilibrium thermodynamics of living (open) systems.
Title:
Abstract:
Title:
Reactive fronts in Boussinesq flows
Abstract: I will describe some recent
results on front propagation in a fluid flow in the Boussinesq
approximation. A reaction-diffusion-advection equation is coupled to
the fluid flow equation by a temperature dependent buoyancy force. We
show that the problem admits non-planar travelling front solutions and
that the fluid coupling speeds up the fronts.
Title:
A Moving Mesh Method Based on Harmonic Mapping and Its Application
Abstract: In this talk, I will introduce briefly a moving mesh method based
on harmonic mapping. As a rare character, the unique existence of the harmonic
mapping is the basic motivation for us to develop this method. The method
is implemented in finite element and an iterative procedure is adopted to
avoid mesh tangling caused by numerical factors. Our method can move the mesh
interior the domain and mesh on the boundary in coupling for both 2D and 3D
problems. The moving mesh module can be a black box added on the whole solver
to the PDE under consideration that it is very convenient for coding - no
modifications to the solver of the PDE are required. The inter-mesh mesh
updation is implemented by a linear convection equation instead of generally
adopted interpolation methods, thus the method can be easily to applied to
problem as incompressible Navier-Stokes equation where the divergence free
interpolation can be a big problem, and problem as conservation laws where
the conservative interpolation is not trival to be implemented. Numerical
results including viscos Burgers equation, reaction-diffusion equation,
incompressible Navier-Stokes equation and its coupling with level set method
will be shown.
Title:
Distributed Linear System Solvers:
Mathematical Algorithms and Biological Applications
Abstract:
Partial differential equations were discretized using the mortar
finite element method, where the mortar space contains piecewise
quadratic and cubic functions. We first proved the wellposedness of
the saddle point system. To solve this saddle point system, the
existing domain decomposition (DD) algorithm of the linear system
solver requires the communication of both the interface solution and
the interface residual of every local problem. We have developed a new
algorithm where only the interface solution is communicated, to
accommodate globally nonconforming meshes. The resulting communication
complexity is reduced. The scalability and parallel efficiency of this
new algorithm were tested with a highly adaptive mesh. As the number
of processors increases, linear and logarithmic speed-up of the
solving time were observed with this new linear system solver, for the
convection-diffusion equation and the Poisson equation respectively.
Three-dimensional dynamic simulation in computational biology provides
an emerging field for the application of efficient distributed linear
system solvers. We developed a 3D continuum model to investigate the
role of structural and functional cellular components in regulating
synchronized calcium signaling (SCS), characterized by high gradient
near the t-tubule membrane and low gradient in the cytoplasm along the
transverse direction, which enables ventricular myocyte to respond
rapidly and forcefully to electrical andchemical stimuli. The
distributed linear system solver improved the simulation speed by ~10
folds. Simulation results suggest that both t-tubule structure and the
spatially heterogeneous distribution of calcium-handling-proteins are
important for SCS. The model also predicts that two aspects of
heterogeneous distribution are required: the concentration of
calcium-handling-proteins in the t-tubule membrane to be ~6 times of
that in the surface membrane; and the concentration of L-type calcium
channels, in the cytoplasmic end of the t-tubule, to be ~2.3 times of
that in the surface membrane end. These results have provided a
foundation for further studies on the effects of three- dimensional
t-tubule geometry and ion channel distribution on calcium dynamics.
Title:
Assessment of Ancestry
Probabilities in the Presence of Genotyping Errors
Abstract: This
talk discusses an extention of a Bayesian approach for estimating the
ancestry probability, the probability that an inbred line is an
ancestor of a given hybrid, to account for genotyping errors. The
effect of such errors on ancestry probability estimates is evaluated
through simulation. The simulation study shows that if
misclassification is ignored, then ancestry probabilities may be
slightly overestimated. The sensitivity of ancestry probability
calculations to the assumed genotyping error rate is also assessed.
Finally we briefly discuss approaches for estimating the error rate
from limited data.
Title:
Rigorous Shadowing of Numerical
Solutions of Ordinary Differential Equations by Containment
Abstract: An
exact trajectory of a dynamical system lying close to a numerical
trajectory is called a shadow. We present a general-purpose method
for proving the existence of finite-time shadows of numerical ODE
integrations of arbitrary dimension in which some measure of
hyperbolicity is present. Much of the rigor is provided automatically
by interval arithmetic and validated ODE integration software that is
freely available. The method is a generalization of a previously
published containment process that was applicable only to
two-dimensional maps. We extend it to handle maps of arbitrary
dimension, and finally to ODEs. The method involves building
$n$-cubes around each point of the discrete numerical trajectory
through which the shadow is guaranteed to pass at appropriate times.
The proof consists of two steps: first, the rigorous computational
verification of a simple geometric property we call the inductive
containment property; and second, a simple geometric argument showing
that this property implies the existence of a shadow. The
computational step is almost entirely automated and easily adaptable
to any ODE problem. The method allows for the rescaling of time,
which is a necessary ingredient for successfully shadowing ODEs.
Finally, the method is local, in the sense that it builds the shadow
inductively, requiring information only from the most recent
integration step, rather than more global information typical of
several other methods. The method produces shadows of comparable
length and distance to all currently published results. We will also
briefly mention how a cheaper, non-rigorous algorithm can be used to
bolster confidence in large numerical simulations of physical systems.
Title:
Modeling Cancer, the Immune System, and Metabolic Function
Abstract: We will
discuss two approaches for modeling cancer growth. One approach is to
employ a deterministic space-independent model that includes separate
components to represent specific and non-specific immune function. The
other approach employs a spatially dependent hybrid cellular-automata
(HCA) model whose rules are driven by cellular metabolic function.
For the determinstic model, numerical simulations of mixed
chemo-immuno therapy and vaccine therapy using both mouse and human
parameters are presented. We illustrate situations for which neither
chemotherapy nor immunotherapy alone are sufficient to control tumor
growth, but in combination the therapies are able to eliminate the
entire tumor burden. The HCA model is in its initial stages of
development, and preliminary results will be presented.
This is joint work with Ami Radunskaya (Pomona College) and Weiqing Gu
(Harvey Mudd College).
Sept 21 (Tuesday): SPECIAL SEMINAR!!! 4pm, Room 254. Speaker: Dieter Bothe (University of Paderborn, Germany)
Sept 27 : Speaker:
Zoltan Toroczkai (Los Alamos)
Oct 4 : Speaker:
Robyn Araujo (NIH)
Oct 11 : Speaker: David Wolpert (NASA)
Oct 18 : Speaker: Arnold Kim (UC Merced)
Oct 25 : Speaker:
Shaoqiang Tang (Peking University)
Nov 1 : Speaker: Don Saari (UCI)
Nov 8 : Speaker: Laura Biven (Max-Planck-Institut fur Physik Komplexer Systeme and Bard High School Early College New York)
Nov 15 : Speaker: Ziyad Muslimani (University of Central Florida)
Nov 22 : Speaker:
Darryl Holm (Los Alamos and Imperial
College London)
Nov 29 : Speaker: Simon Tavare (University of Southern California)
Dec 6 : Speaker: Natalia Komarova (UCI)
Winter 2005
Jan 10 : Speaker: Royce Zia (Virginia Tech)
Jan 18 (Tuesday): SPECIAL SEMINAR!!! 4pm, Room 254. Speaker: Donald Nelson (Worcester Polytechnic Institute)
Jan 24 : Speaker: Peter Dodds (Columbia University)
Jan 31 : Speaker: Alan Newell (University of Arizona)
Feb 7 : Speaker: Emmanuel Candes (Caltech)
Feb 14 : Speaker: Carlos Castillo-Chavez (Cornell University and Arizona State University)
Feb 21 : NO SEMINAR
Feb 28 : Speaker: Hans Othmer (University of Minnesota)
Mar 7 : Speaker: David Chambers (Lawrence Livemore National Lab)
Mar 11 (Friday), MSTB 254: SPECIAL SEMINAR!!! 4pm, Speaker: Dongho Chae (South Korea)
Mar 14 : Speaker: Eric Mjolsness (UCI)
Mar 21 : Speaker: Alexander Kiselev (Wisconsin)
Spring 2005
Apr 4 : Speaker: Howard Levine (Iowa State University)
Apr 11 : Speaker: Miguel Dumett (USC)
Apr 18 : Speaker: Yang Kuang (Arizona State University)
Apr 22 : Speaker: Jian-Guo Liu (University of Maryland)
Apr 25 : Speaker: Hong Qian (University of Washington)
May 2 : Speaker: No seminar
May 9 : Speaker: Lenya Ryzhik (Chicago)
May 16 : Speaker: Ruo Li (Caltech)
May 18 (Wednesday): SPECIAL SEMINAR!!! 4pm, Room 254. Speaker:
Kathy Lu (UCSD)
May 23 : Speaker: Hal Stern (UCI)
Date to be determined : Speaker: Wayne Hayes (UCI)
Jun 6 : Speaker: Lisette de Pillis (Harvey Mudd College)