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_n² (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.

1. What is the impact of assuming C_n² (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_n² (z) and varying wind?
2. 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)?
3. 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.
4. 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.

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.

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.