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University of Minnesota, Twin Cities | |
University of Minnesota, Twin Cities |
Description
The IMA is holding a 10-day workshop on Mathematical Modeling in Industry. The workshop is designed to provide graduate students and qualified advanced undergraduates with first hand experience in industrial research.
Format
Students will work in teams of up to 6 students under the guidance of a mentor from industry. The mentor will help guide the students in the modeling process, analysis and computational work associated with a real-world industrial problem. A progress report from each team will be scheduled during the period. In addition, each team will be expected to make an oral final presentation and submit a written report at the end of the 10-day period.
Designing affordable, efficient, quiet supersonic passenger aircraft has been under investigation for many years. Obstacles to designing such aircraft are also many, both in fundamental physics and in computational science and engineering. The problem of design is multidisciplinary in its nature and the goals of the constituent disciplines that govern the behavior of an aircraft are often at odds. In particular, aircraft that yields low sonic boom may not be attractive aerodynamically, while aerodynamically optimized aircraft may produce unacceptable sonic boom. One of the essential difficulties in using direct optimization methods to design for low boom and low drag is in modeling the design problem. For instance, it is not clear what objective functions to use.
Some Early Boom Shaping Developments (Ferri, 1969)
This project will use simple aerodynamic and sonic boom models to examine modeling of the design problem itself. We will attempt to establish a meaningful direct functional dependence between the shape of the aircraft and aerodynamic and noise quantities of interest by studying the sensitivity of these quantities to changes in shape. We will experiment with several direct multiobjective optimization problem formulations.
References:
Seebass, R., Argrow, B.; "Sonic Boom Minimization Revisited", AIAA Paper 98-2956
Shepherd, K.P., Sullivan, B.M.; "A Loudness Calculation Procedure Applied to Shaped Sonic Booms", NASA Technical Paper 3134, 1991
Carlson, H.W., Maglieri, D.J.; "Review of Sonic Boom Generation Theory and Prediction Methods", J. Acoust. Soc. Amer., 51, pp. 675-685 (1972)
Alonso, J.J., Kroo, I.M., Jameson, A. "Advanced Algorithms for Design and Optimization of Quiet Supersonic Platform", 40th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper 2002-0144, Reno, NV, January 2002
Raymer, D.P.; "Aircraft Design: a Conceptual Approach", Third Edition, AIAA, 1999
Prerequisites:
-Required: Scientific computing skills (Matlab or Fortran 90/95 or C), 1 semester in nonlinear optimization
-Desired: Some background in statistical modeling, numerical analysis, multiobjective optimization
802.11 Wireless Local Area Networks (WLANs) have become as ubiquitous as Internet access for personal computers. The basic unit of a WLAN is composed of one Access Point (AP), and several mobile stations (STAs), all forming a Basic Service Set (BSS). A typical WLAN setup is depicted in Figure 1.
Figure 1: A typical Basic Service Set (BSS), with one AP, and several mobile stations.
The IEEE Standard governing WLANs describes two modes of operation: Distributed Coordination Function (DCF), and Point Coordination Function (PCF). By and large, chipset manufacturers implement only the DCF mode, and compatibility testing is done for this mode exclusively. The DCF is a contention-based mechanism where each wireless device (AP, or STA) competes for air time. More specifically, the 802.11 standard is implemented as follows:
1. At regular intervals (typically hundreds of ms) the AP broadcasts a beacon signal, which resets all devices internal clocks;
2. Assume a transmission opportunity ended at time t0. Depending on the status of the internal Backoff counter (BCK) of the device, the following actions can take place:
a. If BCKr=0, and there is no activity on air for DIFS (Distributed Inter Frame Spacing = 50us in 802.11b) time, then station starts transmitting its data packet;
b. If BCK=0 and during the DIFS period there is activity on air then device generates a random BCK between 0 and CW-1 (initially CW=CWmin = 16, in 802.11b); Then the following rules apply:
-For each slot time (Ts) of medium inactivity, the BCK decrements;
-The countdown is stopped whenever medium is busy, and the the countdown is resumed only after a supplemental AIFS (Arbitration Inter Frame Spacing, =DIFS in 802.11b) wait;
-When BCK reaches 0, the device transmits its packet data;
-If receiver (AP, or STA) receives successfully the packet, then it sends back on the air an Acknowledgement (ACK) frame, after a SIFT (Short Inter Frame Spacing = 10us) period after transmitter finishes its transmission;
-If transmitter receives the ACK correctly, then it assumes data was received correctly, and transmission ends; On the other hand, if ACK is not broadcast, or the transmitter does not receive correctly the ACK, then it assumes the transmission was not successful, and the following rules apply:
i. If current number of retransmissions has not reached a max threshold, then increment the Number of Retransmissions counter
ii. If CW
Project description:
Astronomical telescopes detect the passage of an earth-orbiting object as a streak in an image. Over a period of months, it is possible that many objects will pass through the field of view, some appearing more than once. There are estimates of 100,000 objects in orbit that might be detected by high resolution telescopes. A large field of view telescope may see 100 streaks a night. Most of these objects are space debris that pose a hazard to operational satellites. There is keen interest within the space community to discover and track all these objects.
If the telescope sensor is properly instrumented, it is possible to obtain time-tagged pairs of angles that relate the space object position to the sensor. With enough angle pairs, it is possible to estimate the position and velocity (the state) of the object, along with estimates of the uncertainties of these parameters. The workshop problem is to develop techniques to identify all the streaks made by each object. Streaks created by an object must somehow be associated with one another and disassociated from those made by other objects. One solution approach treats the state data as vectors in R6 and uses statistical clustering techniques for the association. A variation on this approach addresses physical properties of the orbits, sorting according to those least likely to change with small state variations.
Regardless of the approach, there are several interesting aspects to the problem. Automatic streak detection is required, with transform techniques of interest. Orbit mechanics are essential to effective state estimation as well as clustering techniques. In addition, traditional clustering techniques are computationally taxing. A related problem is identification of asteroids that might pose a hazard to planet earth.
References:
Vallado, David A., Fundamentals of Astrodynamics and Applications, Edition 2, Microsoft Press, 2004; Milani, Andrea, "Three Short Lectures on Identifications and Orbit Determination," http://copernico.dm.unipi.it/~milani/preprints/preprint.html, 2006; Kaufman, L. and Rousseeuw, P., Finding Groups in Data - An Introduction to Cluster Analysis. Wiley Interscience 2005
Prerequisites:
-Required: computing proficiency demonstrated by knowledge of at least one compiler, one semester differential equations, one semester statistics
-Desired: one semester numerical analysis, familiarity with orbit mechanics and estimation theory.
The Pan-starrs telescope on Mount Haleakela in Hawaii will be used, among other tasks, to search for asteroids. However, using its 1.4 billion pixel sensor, it will also detect earth-orbiting objects.
Presently, when a physics motivated vehicle designer explores vehicle designs for a new concept, he is often faced with an enormous range of choices and constraints. For an example, an aircraft designer has Aircraft shape, fuel type, and engine as his main free variables. While his main constraints are dictated by the laws of physics (weight, size, power, lift, and stall). Additionally, he has his objective which is typically some combination/subset of acceleration, maneuverability, range, endurance, payload capacity (size, weight and power), max and min speeds, manufacturing cost, maintainability, reliability, development cost, takeoff length, landing length, noise footprint and other items.
I am interested in examining the following problem: Given a set of performance objectives, how does one determine the space of designs available to the designer and find the optimal designs? How does the designer best visualize this space of options? Because he doesn't want just "the" answer, he wants to understand many aspects of the answer. While I'm interested in the general vehicle design problem, we will focus on aircraft design using a baseline tool that is to be determined as a concrete example with which we can test our ideas.
Traditional non-invasive sensing technologies have generated information about only one or two dimensional projections of objects of interest. But the use of arrays of sensor components, and opportunities to rapidly move such arrays around objects of interest are enabling the practical generation of many forms of three-dimensional data. For example, in acoustics there has been steady progression from one-dimensional echo trains, to two-dimensional acoustic images, to modern three-dimensional reconstructions, on scales from ultrasound wavelengths to global seismic surveys. Similarly, three-dimensional tomographic reconstructions from x-rays are now commonly used to resolve ambiguities in traditional two-dimensional x-ray images.
As more three-dimensional data becomes available, the value of automatic tools for utilizing such data increases. Several desired applications need methods by which to automate the finding of correspondences between three-dimensional data sets. These three-dimensional data sets frequently share many geometric characteristics, but also have significant differences, due to differences in data collection geometries, changes in sensor capabilities, temporal changes in the object of interest, and noise in the data.
One approach to finding unknown coordinate transformations, which are needed to align multi-dimensional data sets, is to require an expert to examine each set and label certain common landmarks. If sufficient landmarks, having the same unique labels can be found in both sets, the three-dimensional coordinates of the landmarks enable the coordinate transformation to be estimated. This is like aligning images of faces, by first extracting the coordinates the tips of the noses, the left corners of the mouths, the bases of the right earlobes, etc.
But when no prior expertise is available, we need methods of estimating the transformation from set of automatically generated coordinates of 'interesting' locations (unlabeled landmarks). We expect that a significant subset of corresponding unlabeled landmarks may exist somewhere in the data set to which we need to compare. To solve our alignment problems, we need to devise automated methods to robustly find a pair of large subsets from a pair of sets of unlabeled landmarks, such that the subsets have similar geometric characteristics.
Does there exist a rigid motion mapping the configuration of red points onto a subset of the blue points? If so, what is the blue subset, and what is the rigid motion? If not, how much deformation of the red configuration is needed to make it so?
In principle, these problems can be solved by exhaustively comparing every possibility, but the level of effort grows exponentially fast with the number of landmarks. Our goal will be to find and test new approaches to this problem, seeking to devise algorithms which are robust and far more efficient.
References:
Oliver Faugeraus, Three-Dimensional Computer Vision, MIT Press, 2001
Ian L. Dyrden, Kanti V. Mardia, {Statistical Shape Analysis}, Wiley, 1998
D. G. Kendall, D. Barden, T. K. Carne, H. Le, Shape and Shape Theory, Wiley Series in Probability and Statistics, 1999
Gene H. Golub, Charles Van Loan, Matrix Computations, Johns Hopkins University Press, 1996
Prerequisites:
-Basics of linear algebra and matrix theory, basic computer programming skills, elementary Euclidean geometry
-Desired: Ability to bring relevant ideas from one or more of geometry, invariant theory, optimization theory, graph theory, combinatorics, or something else.
Today's optical telecommunication networks carry audio, video and data traffic over fiber optics at extremely high bit rates. The design of such networks encompasses a range of challenging combinatorial optimization problems. Typically, these problems are computationally hard even for restricted special cases. In this project we study how to assign wavelengths and place equipment so as to carry a set of traffic demands in large scale optical networks.
Our design problems are motivated by a popular optical technology called Wavelength Division Multiplexing (WDM). In this setting each fiber is partitioned into a fixed number of wavelengths and demands sharing a common fiber must be transported on distinct wavelengths. A demand stays on the same wavelength along its routing path as much as possible. When this is infeasible, we can either deploy an extra fiber for the demand to continue on the same wavelength; or place a wavelength converter for the demand to continue on a different wavelength. Both options incur cost. One objective is to assign wavelengths and place converters in an advantageous way so as to minimize the total cost.
In this project we explore algorithms and heuristics for assigning wavelengths and placing converters. The goals include studying the tradeoff between optimality and complexity and understanding the gap between theoretical bounds and practical performance.
References:
[1] Matthew Andrews and Lisa Zhang, Complexity of Wavelength Assignment in Optical Network Optimization. (Please see Section VI.) Proceedings of IEEE INFOCOM 2006. Barcelona, Spain, April 2006. http://cm.bell-labs.com/~ylz/2006.coloring4.pdf
[2] C. Chekuri, et al. Design Tools for Transparent Optical Networks. Bell Labs Technical Journal. Vol. 11, No. 2, pp. 129-143, 2006.
Prerequisites:
-Required: One semester of algorithms; One semester of theory of computing; One semester of programming.
-Desired: Knowledge of Python and CPLEX.
Wednesday | Thursday | Friday | Saturday | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | | |||||
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Wednesday August 08, 2007 | |||||
Projects | |||||
9:00am-9:30am | Coffee and Registration | EE/CS 3-176 | |||
9:30am-9:40am | Welcome and Introduction | Douglas Arnold (University of Minnesota, Twin Cities) Richard Braun (University of Delaware) Fernando Reitich (University of Minnesota, Twin Cities) Arnd Scheel (University of Minnesota, Twin Cities) | EE/CS 3-180 | ||
9:40am-10:00am | Team 1: Supersonic design | Natalia Alexandrov (NASA Langley Research Center) | EE/CS 3-180 | ||
10:00am-10:20am | Team 2: 802.11 WLAN MAC layer modeling | Radu Balan (Siemens Corporate Research, Inc.) | EE/CS 3-180 | ||
10:20am-10:40am | Team 3: Associating earth-orbiting objects detected by astronomical telescopes | Gary Green (The Aerospace Corporation) | EE/CS 3-180 | ||
10:40am-11:00am | Break | EE/CS 3-176 | |||
11:00am-11:20am | Team 4: High dimensional, nonlinear, non-convex optimization problems in the area of aircraft and vehicle design | John Hoffman (Lockheed Martin) | EE/CS 3-180 | ||
11:20am-11:40am | Team 5: Size and shape comparisons from noisy, unlabeled, incomplete configurations of landmarks in three-dimensional space | Mark Stuff (Michigan Technological University) | EE/CS 3-180 | ||
11:40am-12:00pm | Team 6: Wavelength assignment and conversion in optical networking | Lisa Zhang (Alcatel-Lucent Technologies Bell Laboratories) | EE/CS 3-180 | ||
12:00pm-1:30pm | Lunch | Lind Hall 400 | |||
1:30pm-4:30pm | Afternoon - start work on projectsBreak-out Rooms
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Thursday August 09, 2007 | |||||
Students work on the projects. Mentors guide their groups through the modeling process, leading discussion sessions, suggesting references, and assigning work. | |||||
Friday August 10, 2007 | |||||
Students work on the projects. | |||||
Saturday August 11, 2007 | |||||
Students and mentors work on the projects.
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Sunday August 12, 2007 | |||||
Students and mentors work on the projects.
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Monday August 13, 2007 | |||||
9:30am-9:50am | Team 6 Progress Report | EE/CS 3-180 | |||
9:50am-10:00am | Team 3 Progress Report | EE/CS 3-180 | |||
10:10am-10:30am | Team 2 Progress Report | EE/CS 3-180 | |||
10:30am-11:00am | Break | EE/CS 3-176 | |||
11:00am-11:20am | Team 4 Progress Report | EE/CS 3-180 | |||
11:20am-11:40am | Team 5 Progress Report | EE/CS 3-180 | |||
11:40am-12:00pm | Team 1 Progress Report | EE/CS 3-180 | |||
12:00pm-2:00pm | Picnic | UofM East River Flats Park | |||
Tuesday August 14, 2007 | |||||
Students and mentors work on the projects.
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Wednesday August 15, 2007 | |||||
Students and mentors work on the projects.
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Thursday August 16, 2007 | |||||
Students and mentors work on the projects.
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Friday August 17, 2007 | |||||
9:00am-9:30am | Team 5 Final Report | EE/CS 3-180 | |||
9:30am-10:00am | Team 1 Final Report | EE/CS 3-180 | |||
10:00am-10:30am | Team 4 Final Report | EE/CS 3-180 | |||
10:30am-11:00am | Break | EE/CS 3-176 | |||
11:00am-11:30am | Team 2 Final Report | EE/CS 3-180 | |||
11:30am-12:00pm | Team 6 Final Report | EE/CS 3-180 | |||
12:00pm-12:30pm | Team 3 Final Report | EE/CS 3-180 | |||
12:30pm-2:00pm | Pizza party | Lind Hall 400 |
NAME | DEPARTMENT | AFFILIATION |
---|---|---|
Haseena Ahmed | Department of Mathematics | Iowa State University |
Natalia Alexandrov | Langley Research Center | NASA Langley Research Center |
Douglas Arnold | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Radu Balan | Siemens Corporate Research, Inc. | |
Suman Balasubramanian | Department of Mathematics & Statistics | Mississippi State University |
John Baxter | School of Mathematics | University of Minnesota, Twin Cities |
Richard Braun | Department of Mathematical Sciences | University of Delaware |
Michael Case | Department of Mathematical Sciences | Clemson University |
Qiang Chen | Department of Mathematical Sciences | University of Delaware |
Prince Chidyagwai | Department of Mathematics | University of Pittsburgh |
Derek Dalle | Department of Mathematics | University of Minnesota, Twin Cities |
Lisa Driskell | Department of Mathematics | Purdue University |
Ying Wai Fan | Department of Mathematics and Computer Science | Emory University |
Brendan Farrell | Department of Applied Mathematics | University of California, Davis |
Yejun Gong | Department of Mathematical Sciences | Michigan Technological University |
Kun Gou | Department of Mathematics | Texas A & M University |
Gary Green | The Aerospace Corporation | |
Chad Griep | Department of Mathematics | University of Rhode Island |
Jeffrey Haack | Department of Mathematics | University of Wisconsin, Madison |
John Hoffman | Tactical Systems | Lockheed Martin |
Jingwei Hu | Department of Mathematics | University of Wisconsin, Madison |
Xueying Hu | Department of Mathematics | University of Michigan |
Yi Huang | Liquid Crystal Institute | Kent State University |
Mark Iwen | Department of Mathematics | University of Michigan |
Rashi Jain | Department of Mathematical Sciences | New Jersey Institute of Technology |
MoonChang Kim | Department of Mathematical Sciences | Seoul National University |
Mandar Kulkarni | Department of Mathematics | University of Alabama at Birmingham |
Yuen Yick Kwan | Department of Mathematics | Purdue University |
Yun Liu | Department of Mathematics | University of Minnesota, Twin Cities |
Timur Milgrom | Department of Mathematical Sciences | Clemson University |
Darin Mohr | Department of Mathematics | The University of Iowa |
Mechie Nkengla | Department of Mathematics, Statistics and Computer Science | University of Illinois, Chicago |
Vincent Quenneville-Bélair | Department of Mathematics & Statistics | McGill University |
Fernando Reitich | School of Mathematics | University of Minnesota, Twin Cities |
Arnd Scheel | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Chehrzad Shakiban | Institute of Mathematics and its Applications | University of Minnesota, Twin Cities |
Josef Sifuentes | Department of Computational and Applied Mathematics | Rice University |
Mark Stuff | Michigan Technological University | |
Helmi Temimi | Department of Mathematics | Virginia Polytechnic Institute and State University |
Ting Wang | Department of Mathematics | University of Michigan |
Jahmario Williams | Department of Mathematics & Statistics | Mississippi State University |
Zhenqiu Xie | Department of Mathematics | Purdue University |
Jinglong Ye | Department of Mathematics & Statistics | Mississippi State University |
Lisa Zhang | Bell Laboratories | Alcatel-Lucent Technologies Bell Laboratories |
Jintong Zheng | Department of Mathematics | University of Delaware |
Weifeng Zhi | Department of Mathematics | University of Kentucky |
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