STOCHASTIC SYSTEMS SEMINAR, 2005-06
In Fall 2005 and Winter 2006, this seminar will meet at
3pm on Wednesdays commencing October 5, 2005 in AP&M 6218.
Commencing Spring 2006, the seminar will meet at 4pm
on Wednesdays in AP&M 7218.
DESCRIPTION:
This will be an interdisciplinary reading seminar commencing in Fall 2005.
The seminar will meet once per week on a day and time to be arranged.
Faculty, postdocs, visiting scholars and graduate
students will take turns presenting research papers
on stochastic systems arising in applications.
Areas of application include
biology, operations management, neuroscience and finance.
Students wishing to enroll in this
seminar for credit should enroll for Math 288, Lecture B -- Stochastic Systems.
One unit of credit involves attendance and two units involves giving
a presentation.
Please address all enquiries concerning this seminar to
Professor Williams at williams@math.ucsd.edu
SOME SAMPLE PAPERS
Biology
Epidemics and Immunology
Finance
Neuroscience
Operations Management
Communications
SCHEDULE FOR FALL 2005
October 5:
Introduction to modeling of chemical reactions, Karen Ball.
Click here for a related background paper:
Stochastic vs. deterministic modeling of intracellular virus kinetics, R. Srivastava, L. You, J. Summers and J. Yin, J. Theoretical Biology, 218(3), 2002, 309--321.
October 12: Stochastic models for multiscale chemical reactions, Karen Ball.
Click
here for the related paper by Ball, Kurtz, Popovic, Rempala.
See especially Sections 1, 2 and 3.
October 19:
A Simple Model of Spiking Neurons.
Eugene Izhikevich (Neurosciences Institute).
Abstract:
I present an introduction into the mechanisms of generation of
spikes and bursts in neurons and in Hodgkin-Huxley-type neuronal
models. Then I present a deterministic model that reproduces
spiking and bursting behavior of known types of cortical
neurons. The model combines the biologically plausibility of
Hodgkin-Huxley-type dynamics and the computational efficiency of
integrate-and-fire neurons. Using this model, one can simulate
tens of thousands of spiking cortical neurons in real time (1 ms
resolution) using a desktop PC.
MATLAB and C++ code, as well as pdf file of the paper are
available by clicking
here.
For a paper related to this talk, click here.
Another related work can be found by clicking
here.
October 26:
Stochastic Stability of Difference Equations arising in Adaptive Filtering,
Robert Bitmead, Mechanical and Aerospace Engineering, UCSD.
Abstract:
Parameter error equations in adaptive filtering naturally occur as vector difference equations
with random coefficients depending on the realization of the regressors. Our aim in this
seminar will be to understand the various concepts of and conditions for the exponential
stability of the homogeneous parts. Specifically, we shall analyze the scalar case in some
detail and then proceed to untangle the significantly weaker results available in the vector
case. The role of correlation and dependence will be explored to understand the unusual
distinction between the cases. Please bring your mobile phones, in their off mode, since they
will be used as very common example of adaptive filtering and I would like to point to them.
For slides from Professor Bitmead's talk click
here for quicktime,
here
for pdf, and
here for powerpoint.
November 2: Wireless channel parameters maximizing TCP throughput, R. Cruz (ECE, UCSD).
For a related paper, click
here.
November 9: Fluid and diffusion approximations for
service networks, Part I, A. Puha (Mathematics, UCSD and CSUSM).
For a related paper by Mandelbaum, Massey and Reiman, click here.
For another related paper by T. Kurtz,
in Stochastic Processes and their Applications, 6 (1978), 223-240,
click here.
November 16:
Fluid and diffusion approximations for service networks, Part II, A. Puha.
November 30:
The Zero-range process and its applications in transport,
networks and biology.
Erel Levine, Center for Theoretical Biological
Physics, Postdoc, UCSD.
Relevant papers:
M. R. Evans and T. Hanney,
Nonequilibrium statistical mechanics of the zero-range process and related models,
J. Phys. A: Math. Gen. 38 R195-R240 (2005).
E. Levine, D. Mukamel, G. M. Schutz,
Zero-Range Process with Open Boundaries,
Journal of Statistical Physics 120, 759 - 778 (2005)
A. G. Angel, M. R. Evans, E. Levine, and D. Mukamel,
Critical phase in nonconserving zero-range processes and rewiring networks,
Phys. Rev. E 72, 046132 (2005).
SCHEDULE FOR WINTER 2006
Wednesday, January 18, 2006.
Amber Puha, California State University, visiting UCSD.
Queues with many servers.
For the paper on which this talk is based, click here.
Wednesday, January 25, 2006.
Peter Rowat, Institute for Neural Computation, UCSD.
The stochastic Hodgkin-Huxley equations.
Abstract:
The deterministic Hodgkin-Huxley equations describe the
generation of the action potential, or voltage "spike",
in squid giant axon. These equations are fundamental to
theoretical neuroscience. The inter-spike interval (ISI),
which carries information in the nervous system, has
intrinsic jitter due to channel noise, when the applied
current is constant. For some ranges of the current this
jitter is very large (CV>1).
I have investigated a previously unnoticed anomaly in
the distribution of ISIs. When ISIs are converted to
instantaneous frequencies, the associated histogram
is bimodal. The underlying mechanism arises from the
interaction of channel noise with a sub-critical Hopf
bifurcation in the noise-free dynamics. This mechanism
and associated statistics of ISIs will be described.
A difficulty with defining the biological notion of
"threshold" will also be mentioned.
Wednesday, February 1, 2006.
Dr. Shirin Handjani, Center for Communications Research.
A Particle System on the Tree with Unusual Asymptotic
Behavior. Click here for an abstract.
Wednesday, February 8, 2006.
No seminar this week due to Information theory and applications
conference at UCSD, February 6-10, 2006.
Wednesday, February 15, 2006.
R. J. Williams, Mathematics Department,
Fluid approximation for an Internet congestion control model
with fair bandwidth sharing.
Wednesday, February 22, 2006.
Sumit Bhardwaj, graduate student, ECE.
Maxweight Scheduling in a Generalized Switch: State Space Collapse and
Workload Minimization in Heavy Traffic.
The paper on which this talk is based is by
Alexander L. Stolyar, published in the
Annals of Applied Probability 2004, Vol. 14, pp. 1-53.
For a copy of this paper, click here.
Wednesday, March 8, 2006
Professor Jason Schweinsberg, Mathematics Department, UCSD.
Title:
Mutation patterns in populations with large family sizes.
Abstract:
Suppose we take a sample of size $n$ from a population
and follow the ancestral lines backwards in time until
the most recent common ancestor. Under standard
assumptions, this process can be approximated by Kingman's
coalescent, in which two lineages merge at rate one.
Mutations that have occurred since the time of the most
recent common ancestor will lead to segregating sites,
which are positions in the DNA at which not all
individuals in the sample are the same. If we denote
by $M_k$ the number of mutations that affect $k$
individuals in the sample, then the sequence
$M_1, \dots, M_{n-1}$ is called the site frequency
spectrum. We will explain how the site frequency spectrum
would be affected if some individuals have large numbers
of offspring, so that the coalescent process that
describes the genealogy of the population can have
multiple mergers of ancestral lines. This model may
be realistic for certain marine species. This is joint
work with Julien Berestycki and Nathanael Berestycki.
Wednesday, March 15, 2006
Professor Tara Javidi, Department of Electrical and Computer
Engineering, UCSD
Cooperative and Non-cooperative Resource Sharing: Delay
Perspective
Abstract:
From multi-description/multi-path routing to content
distribution in P2P networks to community networking, many
forms of resource sharing have, recently, been proposed to
improve the network performance. From the perspective of any
one user and when ignoring the interaction among users, all
such schemes reduce to various forms of providing parallelism.
In this talk, we argue that focusing on parallelism is by no
means sufficient. When considering more users in the system,
these strategies provide forms of statistical multiplexing
advantage, while possibly increasing the network load via
increased redundancy, overhead increase, and even contention
inefficiency.
In this talk, we illustrate the issue of resource sharing in
the above context via a multi-queue multi-server problem. Even
though such model might not be perfectly realistic, it does
capture some of the above trade-offs. We use this model to
provide analytical results in a special case of homogeneous
users and servers. Furthermore, we prove the robustness of a
certain locally optimal strategy to non-cooperation in a Nash
equilibrium/strategy context.
SCHEDULE FOR SPRING 2006
Wednesday, April 19, 2006
Professor Emanuel Todorov, Cognitive Science, UCSD
Stochastic optimal control models of biological
movement.
Wednesday, April 26, 2006
Professor Yoav Freund, Computer Science and Electrical Engineering, UCSD
Title: Boosting and Brownian motion
Abstract:
The computational task that lies in the core of many machine learning
problems is the minimization of a cost function called the training
error. This problem is frequently solved by local search algorithms
such as gradient descent. The training error can usually be expressed
as a sum over many terms, each corresponding to the loss of the model
on a single training example. We show that the iteratively minimizing
a cost function of this form by local search is closely related to the
following game:
Imagine you are a shepherd in charge of a large herd of sheep and your
goal is to concentrate the sheep in a particular small area by
nightfall. Your influence on the sheep movements is represented by
vectors which define the direction in which you want each sheep
to move. The lengths of the vectors correspond to the fraction of your
"energy" that you spend on moving the particular sheep, and these
lengths sum to one. The sheep then have to respond by moving in a way
that has a slight correlation with the influence direction on average.
We characterize the min/max solution to this game and show that by
taking the appropriate small-step/continuous-time limit, this solution
can be characterized by a stochastic differential equation.
By solving this differential equation we re-derive some known boosting
as well as design some new ones with desirable properties.
Wednesday, May 3, 2006.
Professor J. F. Delmas, visiting UCSD.
Title: Limit theorems for bifurcating Markov chains and application to the
detection of cellular aging (following
Julien Guyon (CERMICS-ENPC)).
For a copy of the paper on which the talk is based,
click here.
Abstract:
A general method is given to study dependent data in a binary tree,
where an individual in one generation gives rise to two different
offspring, one ot type 0 and one of type 1 in the next generation. For
any specific characteristic of these individuals, we assume that the
characteristic is stochastic and depends on its ancestor only through
its mother's characteristic. The dependency structure may be described
by a transition probability P(x, dydz) which gives the probability that
the pair of daughters' characteristic is around (y,z) given that the
mother's is x, y (resp. z) corresponding to the characteristic of the
offspring of type 0 (resp. 1). This defines a bifurcating Markov chain.
A strong law a large numbers and central limit theorem are derived for
such a model. The results are then used to detect cellular aging of
E. Coli.
Wednesday, May 10, 2006.
Professor Massimo Franceschetti, ECE, UCSD
Title: Random Networks for Communications.
Abstract:
The theory of random graphs is a useful mathematical tool to describe
many
real world systems. Recently, the mathematical and engineering
communities
have shown a renewed interest in the geometric version of these
models. The
nodes are geometrically distributed at random, and pairs of nodes are
connected by edges, whose presence depends on the random positions of
the
nodes in the plane. One emerging application is in the field of
wireless
communications, where radio transmitting stations communicate by
radiating
electromagnetic waves.
In this talk, first I review several models of random networks for
communication that are directly related to continuum percolation.
Then, I
show some recent results on connectivity of dependent percolation
models of
interference limited networks. Finally, I introduce the Gupta-Kumar
concept
of throughput scaling, and argue how this can be obtained as a
natural
consequence of the RSW theorem in percolation theory.
Wednesday May 17, 2006.
Kristin Jehring, Graduate student, Mathematics Department, UCSD.
On genetic networks and queueing theory.
Paper on which this talk is based:
Arazi, Ben-Jacob, Yechiali, Bridging genetic networks and queueing theory, Physica A, 332 (2004), 585-616.
Wednesday, May 24, 2006.
Dr. Ben Raphael, Postdoc, Computer Science and Engineering, UCSD.
Title:
Models of Tumorigenesis.
Wednesday, May 31, 2006.
Professor Jun Liu (Rady School of Management)
Title: Long-Lived Private Information in a Continuous Time Economy:
Portfolio Choice, Optimal Consumption, and Utility Gain
Abstract: We study the consumption-investment problem of an agent with a
constant relative risk
aversion (CRRA) preference function, who possesses private information about the
future
prospects of a stock. We examine the value of the information to the agent by
comparing
the utility equivalent with and without the information of the agent. The value
of
private information to the agent depends linearly on the wealth of agents and
decreases
with both the propensity to intermediate consumption and risk aversion. Agents
with
low coefficients of relative risk aversion value private information more highly.
Consistent
with the empirical literature, the optimal portfolio holdings of informed agents
are
correlated with expected returns on the risky asset. Highly risk averse informed
agents
consume a greater fraction of their wealth when they are informed than when they
are
uninformed, but the opposite is true of agents with low degrees of risk aversion.
Last updated September 28, 2005.