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Abstracts and Talk Materials
Mathematical Modeling in Industry XI - A Workshop for Graduate Students
August 8-17, 2007


Natalia Alexandrov (NASA Langley Research Center)

Team 1: Supersonic design
August 8, 2007

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:

  1. Seebass, R., Argrow, B.; "Sonic Boom Minimization Revisited", AIAA Paper 98-2956


  2. Shepherd, K.P., Sullivan, B.M.; "A Loudness Calculation Procedure Applied to Shaped Sonic Booms", NASA Technical Paper 3134, 1991


  3. Carlson, H.W., Maglieri, D.J.; "Review of Sonic Boom Generation Theory and Prediction Methods", J. Acoust. Soc. Amer., 51, pp. 675-685 (1972)


  4. 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


  5. 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

Keywords: multidisciplinary optimization, supersonic design, low boom, aerodynamic optimization

Radu V. Balan (Siemens Corporate Research, Inc.)
http://www.math.umd.edu/~rvbalan/

Team 2: 802.11 WLAN MAC layer modeling
August 8, 2007

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:
    • 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;
    • 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:
        1. If current number of retransmissions has not reached a max threshold, then increment the Number of Retransmissions counter
        2. If CW

Gary B. Green (The Aerospace Corporation)

Team 3: Associating earth-orbiting objects detected by astronomical telescopes
August 8, 2007

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.

Keywords: orbit mechanics, astronomical telescopes, statistical clustering



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.

John R. Hoffman (Lockheed Martin)

Team 4: High dimensional, nonlinear, non-convex optimization problems in the area of aircraft and vehicle design
August 8, 2007

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.

Mark A. Stuff (Michigan Technological University)

Team 5: Size and shape comparisons from noisy, unlabeled, incomplete configurations of landmarks in three-dimensional space
August 8, 2007

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:
  1. Oliver Faugeraus, Three-Dimensional Computer Vision, MIT Press, 2001
  2. Ian L. Dyrden, Kanti V. Mardia, {Statistical Shape Analysis}, Wiley, 1998
  3. D. G. Kendall, D. Barden, T. K. Carne, H. Le, Shape and Shape Theory, Wiley Series in Probability and Statistics, 1999
  4. 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.

Lisa Zhang (Alcatel-Lucent Technologies Bell Laboratories)
http://netlib.bell-labs.com/who/ylz/

Team 6: Wavelength assignment and conversion in optical networking
August 8, 2007

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.



Prerequisite:

Required: One semester of algorithms; One semester of theory of computing; One semester of programming.

Desired: Knowledge of Python and CPLEX.

Keywords: Analysis of algorithms, combinatorial optimization, implementation of heuristics

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