August 2  11, 2010
Project Description:
In mathematical finance, financial risk is nearly always a
statistic of the underlying assets in question. That is, some
salient property of the joint density function of a basket of
assets is chosen to proxy risk. One of the first such
formulations was the Capital Asset Pricing Model, wherein
assets were assumed to be jointly normal, and volatility was
therefore assumed as a risk metric. As time has informed us,
though, the normal distribution may not be the best choice to
describe equity returns on all time scales. Alternate risk
measures have been created, all with varying degrees of success
and at least a modicum of failure.
One promising risk metric is Conditional Value at Risk, or
CVaR. This measure has many appealing properties, one of which
seems paramount: it leaves the task of defining the joint
density to the practitioner. We will study CVaR as an
objective function in a set of optimization problems.
Rockafellar and Uryasev (1999) suggested a linear programming
formulation to solve such a problem. However, we note a curse
of dimensionality issue with their proposal. We will examine
alternative methods for solving the basic CVaR optimization
problem with linear constraints; viz., we will consider a
smooth approximation to the objective as well as a
reformulation of the problem. We will also, of necessity, be
concerned with modeling a joint density for the assets we will
concern ourselves with. In the end, we will test our
methodology expost on real market data.
References:
R.T. Rockafellar and S. Uryasev, Optimization of Conditional
ValueAtRisk. The Journal of Risk, Vol. 2, No. 3, pp. 2141,
2000.
S. Alexander, T. F. Coleman, and Y. Li, Minimizing VaR and
CVaR for
a Por tfolio of Derivatives, Journal of Banking and Finance,
Vol. 30,
no. 2, pp. 583605, 2006.
G. Iyengar and A.K.C. Ma, Fast gradient descent method for meanCVaR
optimization, 2009,
preprint.
Prerequisites:
Familiarity with meanvariance optimization, constrained
optimization methods, and some statistics.
Desired: Coursework
in mathematical finance, statistics and optimization; Matlab
programming.
Project Description:
Our project is a flow modelling project. It would be a project
using similar calculations to those used in Computational Fluid
Dynamics or Sedimentation Engineering.
We have a solution mining process, that has specific
application to potash mining. The process uses interconnected
horizontal wells drilled through a high grade potash zone to
develop a horizontal development that should eventually
dissolve out all the potash in the zone, while leaving the salt
in the caverns created.
Potash will be dissolved out of the ore using a brine saturated
in sodium chloride, but substantially undersaturated in
potassium chloride. Sodium Chloride will collect in the well
bore and the intersections similar to sand and rock in a
developing meandering stream. In fact the whole process is
similar to the mechanics of a meandering stream, except that
the flow is bounded on the vertical surface by ore (rather than
air).
The calculations involve consideration of the dissolution rate
(we can provide dissolution rate curves related to temperature,
flow velocity, and brine saturation). Actual flow patterns will
need to be modelled based on the tendency of a fluid to flow in
a sinusoidal pattern even in a clean pipe, but amplified as the
sine flow erodes the cavern wall at the extremities of the sine
wave. Sedimentation occurs inside the curve on the point bar.
One well will intersect the next well at angles of 30 to 45
degrees. This will cause a larger case of the meander in the
well bore. This will be the major extension of the cavern
action. Sedimentation occurs in the vortex created.
While the meander and intersections will add to the horizontal
extension, the low flow area over the point bars will continue
to add to the cavern height and will eventually be the major
production zone. Salt liberated will simply fall to the point
bar as the cavern height increases.
Note:
The project can be handled as a simple computational problem,
determining the amount of potash dissolved down the length of
the horizontal well bore and around the intersection cavern
based on the changing cross sectional area over time, the
change accompanying change in flow velocity in the meander, the
change in saturation of the brine and change in temperature of
the brine (due to heat of solution of the potash). I will
supply curves and data from literature to provide equations and
experimental curves for these factors. The more complicated
project would involve predicting the form of the flow from CFD
engineering models as well as the calculations described.
References:
National Centre For Computational Hydroscience And Engineering,
University of Mississippi
http://www.ncche.olemiss.edu/COMPUTATIONAL METHODS FOR FLUID DYNAMICS
Ferziger, Joel H., Peric, Milovan
3rd, rev. ed., 2002, XIV, 423 p. 128 illus., Softcover
ISBN: 9783540420743
Sedimentation Engineering Theory, Measurements, Modeling, and
Practice (ISBN: 0784408238 /0784408238) Vanoni Vito A.
Prerequisites:
Required: differential equations, computing skills.
Keywords:
Stream meander modelling, sedimentation engineering
Project Description:
The management of water resources is one of the big future
challenges of our world. Besides the "big picture" how to
provide the humankind with a sufficient amout of high quality
potable water, there are a lot of problems to be solved on an
operational level. For example, local water suppliers have to
make sure that water is available when needed in sufficient
quantity and quality. They have to deal both the requirement
for the lowest utility prices and the fiscal requirements of
their municipalities which might possibly be in a precarious
financial situation. Usually, the energy expended for pumps in
wells and pumping stations is their major cost factor. The
complex rates of electricity providers have highcost and
lowcost periods as well as limitations of total energy
consumption; violations result in considerable additional fees.
The goal of the project is to develop a software which is able
to find an tradeoff between mimimum energy costs and the
reduction of pump switches, where this last aspect minimizes
service and maintenance efforts. The degree of freedom for the
optimization is essentially an intelligent water storage
management.
The project requires basically three steps for the team to be
performed:

Modelling of water nets as well as the corresponding
production and delivery problem, based on resources and demand
forecasts

A concept how to deal conflicting goals, like cost
optimization and reduction of pump switches

Definition and implementation of solution algorithms. This
includes to decide which libraries of basic optimization
algorithms should be used.
Besides mathamatics, it is intended to introduce some basic
aspects of project management. For example, there will be
assigned "roles" of the team members, like project manager,
systems architect, or sales (I am the customer). A project
schedule has to be set up in the beginning.
References:
G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial
Optimization, John Wiley & Sons, 1988
R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows 
Theory, Algorithms, and Applications, Prentice Hall, 1993
http://zibopt.zib.de/ The optimization suite of the Zuse
Zentrum Berlin is one possibility to attack the optimization
problems of the project. The advantage is that it is free for
academic purposes. However, the team members are free to choose
any other tool for linear and integer optimization, provided
they will have solved the licensing requirements. (I strongly
recommend not to start from scratch, by implementing the
Simplex algorithm.)
Prerequisites:
Methods of Linear Optimization as well as programming skills
are a must (preferably in C/C++, especially if the team will
decide to choose the ZIB Optimization Suite), experience in
Discrete Optimization and Graph Theory a real advantage.
The correct and accurate modeling of two phaseflow inside a
ProtonExchange Membrane Fuel Cell (PEMFC) is one of the key
elements of a realistic simulation. A schematic drawing of the
cross section of a PEMFC can be seen in Figure 1.
Figure 1: Schematic crosssection of a
PEMFC.
The reactant
gases are supplied to the fuel cell through the gas channels
and
diffuse into the layered porous medium to the chemically active
catalyst layers on the anode and cathode side, where they are
consumed. The ProtonExchange Membrane is (PEM) as gas tight
as possible, and supposed to function purely as a proton
conductive medium. In the cathode catalyst layer, oxygen reacts
with electrons (supplied through the electrically conductive
porous
medium) and protons (transported through the membrane and the
catalyst layer) and water is produced. The water is produced in
either liquid or vapor form, depending on the local conditions,
and
is transported through the porous network structure and removed
by the flow in the gas channels.
The optimal performance of a fuel cell depends to a high degree
on maintaining optimal saturation (liquid water content) and
relative humidity (water vapor content). A certain humidity
level is
required to guarantee high proton conductivity of the Proton
Exchange Membrane (PEM), but if the humidity is too high, the
pores of the network begin to fill with water, and the
transport of
the gases becomes more and more difficult. One way to mitigate
the flooding problem is to assign part of the pore network to
the
liquid phase and the other part to the gas phase by the
introduction
of hydrophobic and hydrophilic compounds the the porous
structure, and by the introduction of multilayer diffusion
media
with varying material properties. Simple model simulation do
not
agree well with experimental values of the liquid water
saturation,
see Figure 2.
Figure 2: Experimental vs. theoretical
water content (Image taken from Adam Z. Webers conference
presentation).
It is therefore necessary to describe the porous
diffusion media in more detail and with greater understanding
of
their structural and chemical properties.
The function of the porous structure of a PEMFC can be split in
a
structural part, i.e. the distribution of the pore sizes, and a
chemical part, i.e. the wettability of the material depending
on the
contact angle. The chemical treatment leads to multiple contact
angles, idealized by a prescribed contact angle distribution. A
further complication arises from the fact that the contact
angle is
different for the advancing and the receding edge of an
interface,
and therefore different whether water is drained or produced.
Given
a pore size distribution and the contact angle distribution,
the
saturation profile depending on the capillary pressure can be
calculated and compared to experimental values. Due to the
complex interaction in this kind of porous network, a
hysteresis in
the capillary pressure vs. saturation curve can be observed, as
shown in Figure 3.
Figure 3: Contact angle hysteresis
(Image taken from Adam Z. Webers
conference presentation).
Figure 4: Lattice Boltzmann simulation
of liquid water flow through a porous
medium (Movie obtained from Palabos
image galery).
The objective of this project is to study the twophase flow
through a porous network with the previously stated structural
and
chemical properties, to obtain the characteristic saturation
curves
and to properly quantify the energy stored in the hysteresis
loop.
References:
 Fuel cell systems explained (2nd edition), Larminie, James;
Dicks, Andrew, 2003 John Wiley & Sons
 Wettability and capillary behavior of fibrous gas diffusion
media for polymer electrolyte membrane fuel cells, J.T.
Gostick et al., Journal of Power Sources 194/1(2009),
dx.doi.org/10.1016/j.jpowsour.2009.04.052
 Characterization of internal wetting in polymer electrolyte
membrane gas diffusion layers, P. Cheung et al., Journal of
Power Sources, 187/2, 2009, 487492,
dx.doi.org/10.1016/j.jpowsour.2008.11.036
Prerequisites:
Required: 1 semester of ordinary differential equations, 1
semester
of partial differential equations, 1 semester numerical
analysis,
computing skills (python/scipy/sage or Matlab or C/C++).
Desired: 1 semester of physics or fluid dynamics in general.
Keywords: Fuel cell, twophase flow, hysteresis, porous media
Keywords: Geodesy, special and general relativity, gravimetry, GPS.
Background:
The exact computation of the gravitational field of the Earth
is the point of departure to obtain a complete
threedimensional image of the surface of the Earth, which is
used in various applications. The latter include geology,
hydrology, civil engineering, global positioning systems, among
others. The ideal surface obtained through these techniques is
called the geoid, and it provides the best reference model for
the above mentioned applications.
Geoid determination depends on accurately measuring the values
of gravitational forces at different points on Earth. To
understand the precision of existing instruments let it be
known that phenomena as the following have to be taken into
account:
 The rotation of the Earth that generates a centrifugal
force
that reduces the gravity force,
 The tides which change the distribution of the planet's
mass and thus the values of gravitational forces.
Problem:
It is a fairly recent technique to perform gravimetric
measurements on moving platforms, for example with gravimeters
mounted on aircrafts and satellites. Considering the above
mentioned required accuracy, one has to consider the following
factors:

Even the smallest external forces on aircraft mounted
gravimeters,

The noninertial nature of the reference frames that
correspond to moving platforms which is due to the Earth's
curvature.
There already exist some models and formulas that take into
account such effects, but we do not know of a complete model
that fully considers them. Also, it does not seem to exist a
model that has taken care of the relativistic effects of a
noninertial platform. We recall that global positioning
systems (GPS) already incorporate relativistic effects to
achieve the required accuracy of about 5 meters for
nonmilitary users. In contrast, an important goal is to
determine the geoid with high precision (centimeters).
Project:
Our main goal is to study the models used for gravimetric
measuring and geoid computation on moving and noninertial
platforms. We will first estimate the effective accuracy of the
results obtained with the use of classical mechanics. Next, we
will consider the models that take into account special and
general relativistic effects. Our main tool will be the already
known theory and formulas as well as some that we can develop
as the project advances. Another relevant component will be the
use of software and graphing tools to understand and visualize
the phenomena involved. These should suggest corrections and
improvements to the current techniques.
Feasibility
As mentioned above, we already have at our disposal theoretical
models in the classic mechanics setup. Also, we have obtained
some preliminary theoretical computations that evidence the
influence of relativistic effects in the gravimetric
measurements on moving platforms. Finally, we can compare the
modeling results obtained with the measurements available at
some government instances.
References:
 Geodesy 3rd Edition, Torge, Wolfgang, 2001, de Gruyter, ISBN
3110170728
 Gravitation, Misner, C., Thorne, K., Wheeler, J., 1973,
Freeman, ISBN 0716703440
 Satellite Geodesy 2nd Edition, Seeber, Gunter, 2003, de
Gruyter, ISBN 3110175495
Prerequisites:
Required: 1 semester course of differential geometry, computing
skills
(e.g. C/C++).
Desired: (one or more of the following) 1 semester course of
introductory physics or classical mechanics, 1 semester course
of Riemannian geometry, 1 semester course of special
relativity.
Keywords: Image stabilization, Image segmentation, Classification,
Tracking, Particle filters.
Project Description:
The project consists in detecting moving objects in
suburban
and forest firing areas that could indicate people in danger.
Automatic monitoring systems can help in this situation,
either, making a fully automatic analysis or sending a
prealarm, to indicate situations that require operator's
attention. The data are video sequences acquired, supposedly,
with an unmanned aerial vehicle (UAV), or remotely piloted
vehicle.
There are several advantages of using UAV in fire monitoring
with respect to manned planes (or helicopters); they have a
relative the lower cost of operation; they can fly in dangerous
situations; they can fly for longer period of time and they can
be equipped with video cameras for performing autonomous scene
analysis.
Challenges of the project are:
 We have an unstable source video. The UVS motion introduces
shifts and shakes of the scene in the video. Hence, a
preprocessing for video
stabilization could be useful [2].

The fire's smoke is moving and must not be confused with
objects' motion. We will require classifying moving objects
between smoke and interesting blobs. Color, texture and motion
features can support such a classification [4,5].
 Static objects in the scene, as trees or houses, can occlude
the moving objects (MOs). Moreover smoke can partially or fully
occlude such MOs. For managing this situation, a motion
analysis could alert us for occlusions and allows tracking
partially occluded object [3].
 Noise may produce false positive. Thus, the motion analysis
could provide us from the information for classifying among
noise, smoke and MOs and then to eliminate false positives [5].
During the development of our algorithms, we need to take into
account that the monitoring system is pretended to be
implemented in real time and executed autonomously in the UAV,
i.e. we will prefer computationally efficient methods.
References:
 Handbook of Mathematical Models in Computer Vision, N.
Paragios, Y Chen, O. Faugeras Eds. Springer NY, 2006
 R. Szeliski, "Image Alignment and Stitching," Chap 17 in
[1], pp 273292.
 A. Blake, "Visual tracking: a short research roadmap," Chap
18 in [1], pp 293307.
 M. Rivera, O. Ocegueda, J.L.Marroquin, "Entropy controlled
quadratic Markov measure fields for efficient image
segmentation," EEE Trans. on Image Process., vol 16.(12),
30473057, 2007.
 C.M. Bishop Pattern Recognition and Machine Learning,
Springer NY, 2006. In particular chapters: 7,9,12
Prerequisites:
Required: 1 semester of image processing, 1 semester numerical
methods, 1 semester numerical optimization, computing skills
(Matlab or C/C++).
Desirable:
1 semester pattern recognition.