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Abstracts and Talk Materials
Mathematical Modeling in Industry XII - A Workshop for Graduate Students
August 6 - 15, 2008

Project Description: Due to the complexity of financial markets, financial derivative modeling requires both an ability to understand and implement theoretical mathematical objects as well as a reliance on simulation techniques. A well known economist, Eugene Fama, once said, “We know all models are false.” This notwithstanding, an approximate model allows the practitioner to understand her position in terms of widely used market parameters such as volatility or correlation. Additionally, insight may be gained into the approximate distribution of payoffs as a function of such parameters once a model has been designated.

This project will present and model a financial instrument dubbed a ‘dispersion option’. Such an option has a payoff structure contingent on how much individual stock returns within a basket diverge from the average return of the basket. As a first step, we will simulate such an option with a variety of real market data, and examine the distribution of payoffs, thereby gaining insight into the historical behavior of such instruments. We will then attempt to examine the distribution of payoffs of such an option based on multiple models of the underlying names. This may be done using simulation techniques or via a mathematical proof depending on the complexity of the model assumed. Of primary interest would be to understand the payoff structure of the option as a function of easily identifiable parameters.

Reference:

Options, Futures, and Other Derivatives, J. C. Hull, Prentice Hall. Especially chapters titled "Numerical Procedures", and "More on Models and Numerical Procedures" in the sixth edition.

Prerequisites: Knowledge of options pricing theory (especially Risk-Neutral Valuation), statistics, some numerical analysis, and ability to write simulation code. Desired: Coursework in mathematical finance and statistics, Matlab programming, and a familiarity with model selection techniques and evaluation.

Project Description: The goal of this research is to revisit the stability results of Yeow (1974) on extending flows with free surfaces. The eigenvalue problem will be formulated and solved for the flow of a Newtonian film such as the one encountered in film casting. The stable and unstable region boundaries will be obtained. The analysis will be extended to a non-isothermal case similar to the work by Shah and Pearson (1972). Stability boundaries for different draw velocity (at the inlet and the exit of the process), and viscosity ratios will be sought. For the solutions which are unstable (or marginally unstable), time-dependent solutions (oscillating or growing in time) describing the free surface motion will be obtained.

References:

Y. L., Yeow: On the stability of extending films: a model for the film casting process (J. Fluid Mech. 1974 v66 (3) 613-622.

Y. T. Shah & J. R. A. Pearson: On the stability of non-isothermal fiber spinning - general case. Industrial & Engineering Chemistry Fundamentals. 1972 v11 (2) 150-153.

Additional References: D. Silagy, Y. Demay, J-F. Agassant: Study of Stability of the Film Casting Process. Polymer Engineering and Science, 1996 V36 (21) 2614-2625.

Y. Shah & J.R. Pearson: Stability of Fiber Spinning of Power-law fluids. Ind. Eng. Chem. Fundam. 1972 v11 (2)

G. Lamberti, G. Titomanlio, V. Brucato: Measurement and modeling of the film casting process 1. Width distribution along draw direction. Chemical Engineering Science 2002, v56, 5749-5761

G. Lamberti, G. Titomanlio, V. Brucato: Measurement and modelling of the \$lm casting process 2. Temperature distribution along draw direction. Chemical Engineering Science 2002, v57, 1993-1996.

Prerequisites: Computing skills, asymptotic analysis, numerical methods, familiarity with conservation laws in particular Navier-Stokes system of Newtonian fluids. Matlab, Maple and Comsol knowledge could be beneficial but not required.

Welcome and Introduction
August 6, 2008

Project Description: The process of laying out the curves and surfaces needed to describe free form shapes in mechanical design is called lofting. Examples of lofting include shapes such as ship hulls, airplane wings and bodies, automobile exteriors, and so on. The best lofting procedures take a vector of inputs, which can contain items like wing span, wing sweep angle, aspect ratios, wing leading edge curvatures, etc., and produce a mathematical model of the geometric shape. Good lofting procedures necessarily have to process the input data nonlinearly in order to produce acceptable shapes.

Additionally, it is frequently important to solve the inverse problem. Specifically, one is given a mathematical model of a geometric shape and, with any luck, a lofting code and wants to know what vector of inputs to the lofting code will produce the given shape. This problem has been called the unlofting problem, and it can usually be solved with with standard techniques in non-linear least squares and non-linear parameter estimation. Just as frequently, though, the unlofting problem arises in contexts where no lofting code exists, requiring such a code to be produced as part of the solution. So far, the requirement to produce a lofting code as part of the solution to the unlofting problem has ruined all attempts to produce a fully automatic solution.

This project will attempt to construct a prototype unlofting code given only a final geometric shape with no accompanying lofting code. Some recent developments in multiresolution modeling have suggested a promising approach to this problem that we will explore during the workshop, focusing initially on 2D curves and then migrating to simple 3D shapes if time permits.

References:

"Multiresolution morphing for planar curves," by S. Hahmann, G.-P. Bonneau, M. Cornillac, and B. Caramiaux. Computing 79 (2-4), pp. 197-209 (2007)

Prerequisites: Required: 1 semester of numerical analysis and computing skills. Desired: Knowledge of non-linear least squares, splines, and Python programming.

Keywords: Lofting, geometric morphing, inverse problems, multiresolution modeling, nonlinear parameter estimation.

Project Description: Some electrical interconnections in medical devices are made by forming and welding piece of thin flat ribbon (or wire) between two electrical terminals. Current equipment for forming the ribbon allows for a virtually an infinite set of motions between the two terminals to be programmed. Currently the only method for determining what the resulting shape of the ribbon will be from a set of machine motions is to program the machine, form a ribbon, visually observe the resultant shape, and iterate until the "desired shape" is obtained. The problems proposed are 1) Given some data regarding the ribbon shapes that result from a very limited set of tool motions, can a more general model be developed that can predict the shape of the loop based on the machine motions, 2) Can this model be improved by incorporating the material response behavior of the ribbon or other physical relationships that govern ribbon formation, 3) Can this model be inverted so that if a particular ribbon shape is desired, a corresponding set of machine parameters can be identified, and 4) If only the spacing, positioning, and clearance around two terminals are known, can an optimal shape be identified that minimizes the stress induced in the ribbon from relative motion between the terminals while avoiding interference with any of the surrounding geometric constraints.

References:

1. "Wire-Bonding Loop Profiles" http://www.siliconfareast.com/wirebond-loop-profiles.htm

2. "Apparatus and method for laser welding of ribbons" US Patent 6,717,100

Prerequisites: Familiarity with mechanics of materials, plastic deformation of thin metal, curve fitting, data analysis, optimization, & machine control would all be helpful.

Keywords: wire bonding, interconnect ribbon, tool path control, plastic deformation of thin wire or ribbon

Images:

Project Description: Instruments for chemical spectroscopy are finding key application in fields of homeland security, healthcare and manufacturing of chemicals and machine parts [1]. The need to automatically analyze large amounts of data quickly and to calibrate these instruments in an unbiased way is thus becoming ever more important. In many applications, for example healthcare and law enforcement, both calibration [2] and data analysis ([3,4]) should be performed with as little operator input as possible.

One of the most important chemical spectroscopy instruments is the Matrix Assisted Laser Desorption Absorption Time of Flight (MALDI-TOF) mass spectrometer. A schematic of the instrument is shown above, and sample data output is shown below. The MALDI-TOF produces a collection of 2-tuples (usually between 50,000-100,000 pairs of data points), from which one should identify peaks and then integrate the area underneath each peak. A major challenge is the development of an automated peak peaking and peak integration algorithm requiring no operator input. A second and closely related challenge is the development of an operator independent calibration scheme.

I will outline an approach to the data analysis problem and present some very precursory work involving Standard Reference Materials (SRM). I will also present a first attempt at automatic instrument calibration. Data from larger molecules will be used as a litmus test. If time permits, I will will present at least one other spectroscopy instrument.

References:

[1] Introduction to Mass Spectrometry, J. T. Watson, Lippencott-Raven, 1997.

[2] Wallace, W. E., Guttman, C. M., Flynn, K. M., Kearsley, A. J., Numerical optimization of matrix-assisted laser desorption/ionization time-of-flight mass spectrometry: Application to synthetic polymer molecular mass distribution measurement’ ANALYTICA CHIMICA ACTA Volume: 604 Issue: 1 Special Issue: Pages: 62-68 NOV 26 2007

[3] Wallace, W. E., Kearsley, A. J., Guttman, C. M., An operator-independent approach to mass spectral peak identification and integration’ ANALYTICAL CHEMISTRY. Volume: 76 Issue: 9 Pages: 2446-2452. MAY 1 2004

[4] Wallace, W. E., Kearsley, A. J., Guttman, C. M., `MassSpectator: Fully automated peak picking and integration - A Web-based tool for locating mass spectral peaks and calculating their areas without user input. ‘ ANALYTICAL CHEMISTRY Volume: 76 Issue: 9 Pages: 183A-184A MAY 1 2004

Prerequisites: A programming language, (Fortran 90, C, C++, or Matlab); a course in optimization or signal processing is helpful but not necessary.

Project description: Peer-to-peer networks are decentralized computing architectures that promise to deliver scalability in data sharing and streaming applications under dynamic network conditions. In these architectures peers are connected to the network and contribute resources in return for some useful services delivered by the network. Some questions that determine the performance and robustness of the peer-to-peer network are: what is the capacity of the network? How robust is the network behavior with respect to flashcrowds and random peer failures and departures? In this project we study the performance and robustness of various peer-to-peer networks by studying various algorithms for constructing the overlay network and for determining the data packets that are transmitted . We study the properties of the complex network resulting from these algorithms in order to identify peer-to-peer networks which are both robust and efficient.

Prerequisites: computer programming (C, Matlab or Python), discrete mathematics. Desired: computer networks, graph theory, probability.

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