August 5 - 14, 2009
Project Description:
Birefringence refers to
a different index of refraction for orthogonal light polarizations in
a transparent material. In stress-free glasses (which are isotropic
and can be made homogeneous) the birefringence is zero by symmetry.
When such a glass is subjected to stress, even by squeezing with your
fingers, stress-induced birefringence is readily observed. In real
glasses a certain amount of stress is unavoidably frozen in during
glass forming. It is of interest in a number of applications needing
low or nearly zero birefringence to control and minimize the level of
frozen-in stress birefringence.
The goal of this
project is to develop computational tools in Matlab to read limited
sets of birefringence measurements and approximately reconstruct a
stress distribution within the glass part that would be consistent
with the measured birefringence scans. The general mathematical
jargon for this procedure is "tensor tomography," but we are not
trying to solve the problem at its most exact and sophisticated
level. Instead we seek to make the absolutely simplest model for
stresses within a sample that is approximately or adequately
representative of the real stresses in the sample. Such an
approximate reconstruction of stress would be useful to understand
what stresses have developed in the sample and also how the
birefringence would be altered if glass were removed, changing the
stress boundary conditions. The model stress would have to obey the
usual requirements of material continuity and force balance as well
as the force-free boundary condition on the surface. Part of our
goal is to achieve an adequate approximate description of stress
using the fewest birefringence measurements possible.
We have in mind a
real-life application where reconstruction of the stress field from
limited birefringence measurements would be useful. The application
is in the manufacture of lens blanks, or blocks of extremely pure and
highly homogeneous glass used to make the diffraction-limited optics
for computer chip manufacture. Here the problem is fully
three-dimensional, and at minimum several directions of birefringence
measurement will be required.
I am interested in
possibly using Green function methods to solve for a stress
distribution based on a set of initial strains. The strain field
would constitute the unknown degrees of freedom for which we solve.
This would automatically satisfy material continuity and force
balance within the interior, and can be arranged also to satisfy the
boundary conditions on faces. However, we may elect to pursue finite
element methods or other choices depending on student interests and
experience.
References:
Background on linear
elastic theory and stress-induced birefringence can be found in many
sources, including the web or textbooks in your university library.
Note that we will work only in the linear regime and only with
perfectly isotropic and homogeneous samples (when in their
stress-free condition), so much of the mathematics is simplified.
1. One useful set of notes on linear elastic theory can be found at
http://www.engin.brown.edu/courses/en224.
See the Lecture Notes and especially the "Kelvin solution" of
section 3.2 which is the basis of the Green function method.
2. Some basics of
birefringence are included in the IMA Mathematical Modeling in
Industry Workshop 2006 report found at
http://www.ima.umn.edu/2005-2006/MM8.9-18.06/abstracts.html.
See the link to the "Team 1 report" pdf .
Prerequisites:
Required:
computing skills, numerical analysis skills, familiarity with Fourier
analysis and convolution, ability to manipulate data arrays.
Desired: some optics,
some physics, familiarity with continuum elastic theory (stress and
strain); the needed optical and glass-forming background will be
supplied.
Keywords:stress-induced birefringence, optical
properties of glass, data analysis algorithms, tensor tomography,
linear elastic theory
Project Description:
Active portfolio management has developed substantially since the formulation of the Capital Asset Pricing Model (CAPM). While the original methodology of portfolio optimization has been lauded, it is essentially an academic exercise, with practitioners eschewing the suggested weightings. There are myriad reasons for this: nonstationarity of data, insufficiency of modeling parameters, sensitivity of optimization to small perturbations, and assumption of uniform investor utility all indicate potential failures in the model.
We will follow the work of Goldfarb and Iyengar and address some of the issues raised above. In particular, we will consider robust portfolio selection problems. These, still, suffer from the features of nonstationarity and potential misalignment of true investor risk aversion. However, they add flexibility and attempt to remove parameter specification sensitivity. Under this framework, we will also consider how a factor model may enhance our desired results. To be consistent with current conceptions and literature, we will attempt to assimilate the work of Fama and French into our model.
References:
Goldfarb, D. and Iyengar, G. 2003. Robust portfolio selection problems. Mathematics of Operations Research 28: 1-38
Goldfarb, D., Erdogan, E., and Iyengar, G. 2007. Robust portfolio management. Computational Finance 11: 71-98
Fama, E. and French, K. 1993. Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics 33: 3–56
Nocedal, J. and Wrigth, S. 1999. Numerical Optimization. Springer-Verlag, New York.
Prerequisites:
Familiarity with mean-variance optimization, constrained optimization methods, and regression. Desired: Coursework in mathematical finance, statistics and optimization; Matlab programming; and some work with second order cone programs.
Projection Description:
The earth’s atmosphere is a swirling ball of gas. The cause of
the swirling, especially near the surface, is due to different
temperatures of the air. These different air temperatures
change the index of refraction for the air in the atmosphere.
Thus when light travels through this turbulent/random medium
the light ends up getting speckled. It is these speckles,
caused by the turbulent atmosphere that limited the resolution
of earth-bound astronomical observations until the invention of
adaptive optics. You have observed this phenomenon any time
you’ve looked at a star. It is the motion of these speckles
over our eyes that causes the stars to twinkle. The graphic
below illustrates how light from a source ends up distorted by
the atmosphere resulting in a specular image.
Our problem focuses on a particular aspect of imaging through
turbulence. In the early 1970’s it was shown by Lawrence,
Clifford and Oochs and Lee and Harp that the primary source of
the variation of the intensity of light on a pair of photo
detector was from the wind. This observation can be used to
create a poorly posed inverse problem that if one can solve,
permits one to compute the cross wind profile along the path of
the light beam. The specific relationship relating time-lagged
cross covariance and wind speed is given by:
where:
τ – is the time lag between adjacent pixels
L – is the length of the flight path.
k – is wave number of the light used (the light is assumed to
be monochromatic.)
K – has units of 1 / length, is the reciprocal of the size of a
turbulent eddy ball.
ρ – spacing between detectors
v(z) – wind speed parallel to the line connecting the
detectors
C
_{n}^{2} (z) – scintillation
coefficient
Several different authors since then have advertised an ability
to measure the gross average wind over long periods of time.
(10 minute intervals is a common metric.) Here are several
questions that I currently have on this phenomenology. The
team will answer any questions that I don’t answer between now
and this summer.
- What is the impact of assuming C_{n}^{2} (z) is constant? Most
practitioners make this assumption. How is the inverse problem
affected if it is not constant? Is it possible to distinguish
between affects caused by variable C_{n}^{2} (z) and varying wind?
- The environment is constantly changing, thus measuring the
time lagged cross covariance is a very noisy measurement, and
difficult to do. In particular, given the underlying noise
assumptions of the inputs to this integral equation, what kind
of noise does one observe when measuring
C_{χN} (ρ,τ)?
- Question 2 above can be attacked in two ways. The first
is
an analytic approach the second is a simulation based approach.
Can we create a simulation to permit us to assess question 2?
A standard technique is based on phase screens. This would
require examining the literature, possibly grabbing code off
the net, and implementing a simulation in Matlab or similar
system.
- Related to question 2, how long a period of time can one
measure C_{χN} (ρ,τ)? Most authors use 10 minutes, can it be done in 10
seconds? 1 second? 0.1 seconds?
Key References: (a much longer list will be provided
this
summer):
Laser Beam Propagation through Random Media, Second Edition
Larry C. Andrews and Ronald L. Phillips, SPIE Press, 2005.
Imaging Through Turbulence, Michael C. Roggemann, Byron M.
Welsh, CRC Press, 1996.
Lawrence, Ochs, and Clifford, “Use of Scintillations to Measure
Average Wind Across a Light Beam”, Applied Optics, 1972, Volume
11, #2, Page 239-243.
Barakat, and Buder, “Remote Sensing of Crosswind profiles using
the correlation slope method”, Journal of the Optical Society
of America, 1979, volume 69, #11, Pages 1604-1608.
Lee, Harp, “Weak Scattering in Random Media, with Applications
to Remote Probing”, Proceedings of the IEEE, 1969, Volume 57,
#4. Pages 375+
Image copied and cropped from:
http://www.aanda.org/articles/aa/full/2003/47/aa3613/img193.gif
Project Description:
Reservoir simulations are used in the oil industry for field development and for production forecast. The heart of a simulator is a computer program that solves for the fluid flow within the reservoirs. The flow of fluid is modeled by a system of coupled, nonlinear partial differential equations (PDEs). These equations are then discretized in space and time. When using an implicit time discretization, a system of nonlinear algebraic equations needs to be solved at each time step. This is typically done using Newton’s method on a set of linearized state equations. At each Newton iteration, a linear system must be solved to update the set of state variables.
The challenge of performing accurate and realistic simulation is that the number of unknowns can be large, requiring the solution of a large system of nonlinear algebraic equations at each time step. The task in this project is to understand the bottleneck in the calculation and find ways to speed it up.
We will conduct our research using a MATLAB based, 2-phase flow simulator with fixed spatial discretization and adaptive time stepping. We consider two different time discretization schemes. The first scheme is fully implicit, while the second is based on an operator splitting method.
References:
Fundamentals of Numerical Reservoir Simulation
Donald W. Peaceman
Elsevier Science Inc. New York, NY, USA
Finite Volume Methods for Hyperbolic Problems
Randall J. LeVeque
Cambridge University Press
Prerequisites:
Background in numerical analysis, numerical linear algebra, scientific computation, and numerical methods for partial differential equations. Experience in MATLAB programming.
Project Description:
In recent years, the structure of complex networks has become
object of intense study by scientists from various disciplines;
see e.g. [1], [2] and [3], or the book-form paper collection
[4]. One often studied mechanism of growth and evolution in
such networks, e.g. social networks, is preferential attachment
[2]. In communications network engineering, network protocols
have been modeled mathematically using tools from optimization
[5] and game theory [6]. A picture has emerged of layered
networks (modeled as graphs) where each layer of the whole acts
non-cooperatively, implicitly optimizing its own objective,
treating other network layers largely as a black box. The
network layers interact dynamically, and implicit cooperation
towards a common overall objective is achieved by a suitable,
modular decomposition of tasks to the individual layers.
In this project, we will focus on the interaction between
social networks and communication networks. Given the
communication network, how do social networks grow and evolve?
Does preferential attachment account for the structure
observed? How do communication networks and their (often
protocol-induced) ‘preferences’ affect the structure of social
networks, and vice versa? We will use mathematics
(optimization, game theory, graph theory) and computer
simulation to investigate these questions.
Prerequisites:
Background: Optimization, Probability, Differential Equations.
Computer skills: Matlab, R, Python.
References:
[1] M.E. Newman, "The Structure and Function of Complex
Networks," SIAM Review, Vol. 45, No. 2, pp. 167-256, 2003.
[2] L.-A. Barabasi, R. Albert, "Emergence of Scaling in Random
Networks," Science, Vol. 286, No. 5439, pp. 509-512, 1999.
[3] D.J. Watts, "The ‘New’ Science of Networks," Ann. Rev.
Sociology Vol. 30, pp. 243-270, 2004.
[4] M.E. Newman, L.-A. Barabasi, D.J. Watts, "The Structure and
Dynamics of Networks," Princeton, 2006.
[5] M. Chiang, S.H. Low, A.R. Calderbank and J.C. Doyle,
"Layering as Optimization Decomposition: A Mathematical Theory
of Network Architectures," Proceedings of the IEEE, Vol. 95,
No. 1, pp. 255-312, January 2007
[6] E. Altman, T. Boulogne, R. El-Azouzi, T. Jimenez and L.
Wynter, "A Survey of Networking Games in Telecommunications,"
Computers and Operations Research, Vol. 33, No. 2, pp. 286-311,
2006.
llustration of SIFT
features computed using VLFeat library
Project Description:
Large collections of image and video data are becoming
increasingly common in a diverse range of applications,
including
consumer multimedia (e.g.
flickr and
YouTube), satellite
imaging,
video surveillance, and medical imaging. One of the most
significant
problems in exploiting such collections is in the retrieval of
useful
content, since the collections are often of sufficient size to
make a
manual search impossible. These problems are addressed in
computer
vision research areas such as
content-based image
retrieval, automatic image tagging, semantic video
indexing, and
object detection. A sample of the exciting work being done in
these
areas can be obtained by visiting the websites of leading
research
groups such as
Caltech
Computational Vision,
Carnegie
Mellon Advanced Multimedia Processing Lab,
LEAR,
MIT CSAIL Vision
Research,
Oxford
Visual Geometry Group, and
WILLOW.
One of the most promising ideas in this area is that of
visual
words, constructed by quantizing invariant image features
such as
those generated by
SIFT.
These visual word representations allow text document analysis
techniques (
Latent
Semantic Analysis, for example) to be applied to computer
vision problems, an interesting
example being the use of
Probabilistic
Latent Semantic Analysis or
Latent
Dirichlet allocation to learn to recognize categories of
objects
(e.g. car, person, tree) within an image, using a training set
which
is only labeled to indicate the object categories present in
each
image, with no indication of the location of the object in the
image. In this project we will explore the concept of visual
words,
understand their properties and relationship with text words,
and
consider interesting extensions and new applications.
References:
[1] Lowe, David G., Distinctive Image Features from
Scale-Invariant Keypoints,
International Journal of Computer Vision, vol. 60,
no. 2, pp. 91-110, 2004. doi:
10.1023/b:visi.0000029664.99615.94
[2] Leung, Thomas K. and Malik, Jitendra, Representing and
Recognizing the
Visual Appearance of Materials using Three-dimensional Textons,
International Journal of Computer Vision, vol. 43, no.
1, pp. 29-44, 2001. doi:
10.1023/a:1011126920638
[3] Liu, David and Chen, Tsuhan, DISCOV: A Framework for
Discovering
Objects in Video, IEEE Transactions on Multimedia,
vol. 10, no. 2, pp. 200-208, 2008. doi:
10.1109/tmm.2007.911781
[4] Fergus, Rob, Perona, Pietro and Zisserman, Andrew, Weakly
Supervised Scale-Invariant Learning of Models for Visual
Recognition,
International Journal of Computer Vision, vol. 71, no.
3, pp. 273-303, 2007. doi:
10.1007/s11263-006-8707-x
[5] Philbin, James, Chum, Ondřej, Isard, Michael, Sivic, Josef
and
Zisserman, Andrew, Object retrieval with large vocabularies and
fast
spatial matching, Proceedings of the IEEE Conference on
Computer Vision and Pattern Recognition, 2007. doi:
10.1109/CVPR.2007.383172
[6] Yang, Jun, Jiang, Yu-Gang, Hauptmann, Alexander G. and Ngo,
Chong-Wah,
Evaluating bag-of-visual-words representations in scene
classification, Proceedings of the international workshop
on multimedia information retrieval (MIR '07), pp.
197-206, 2007. doi:
10.1145/1290082.1290111
[7] Yuan, Junsong, Wu, Ying and Yang, Ming, Discovery of
Collocation
Patterns: from Visual Words to Visual Phrases, IEEE
Conference on Computer Vision and Pattern Recognition
(CVPR), pp. 1-8, 2007. doi:
10.1109/cvpr.2007.383222
[8] Fergus, Rob, Fei-Fei, Li, Perona, Pietro and Zisserman,
Andrew,
Learning object categories from Google's image search, IEEE
International Conference on Computer Vision (ICCV), vol.
2, pp. 1816-1823, 2005. doi:
10.1109/iccv.2005.142
[9] Quelhas, P., Monay, F., Odobez, J.-M., Gatica-Perez, D.,
Tuytelaars,
T. and Van Gool, L., Modeling scenes with local descriptors and
latent
aspects, IEEE International Conference on Computer Vision
(ICCV), pp. 883-890, 2005. doi:
10.1109/iccv.2005.152
[10] Sivic, Josef, Russell, Bryan C., Efros, Alexei A,
Zisserman, Andrew
and Freeman, William T., Discovering objects and their location
in
images, IEEE International Conference on Computer Vision
(ICCV), pp. 370-377, 2005. doi:
10.1109/iccv.2005.77
Prerequisites:
A strong computational background is essential, preferably with
significant experience in Matlab programming. (While experience
with
other programming languages such as C, C++, or Python may be
useful,
Matlab is likely to be the common language when individual team
member
contributions need to be integrated into a joint code.)
Some background in areas such as image/signal processing,
optimization theory, or statistical inference would be highly
beneficial.