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University of Minnesota, Twin Cities |
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 7 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.
Application Procedure
Graduate students and advanced undergraduates are invited to apply. An application form must be submitted to the IMA. In addition, two letters of recommendation are required; one must be from the student's advisor, director of graduate studies, or department chair. Prerequisites vary and depend on the project, but computational skills are important.
The IMA will cover local living expenses but not travel (IMA Participating Institution may use their PI funds for this purpose). Selection criteria will be based on background and statement of interest, as well as geographic and institutional diversity. Women and minorities are especially encouraged to apply. Applications must be completed by April 15, 2005 for full consideration. Early submissions are encouraged. Successful applicants will be notified by April 30, 2005.
Completed application forms and Letters of Recommendation should be addressed to "Math Modeling Committee" and emailed to mm-applications@ima.umn.edu. Text, pdf or postscript files are prefered.
Standard 2-D SAR image formation techniques require data samples that have been collected on a regular rectangular grid (with respect to the viewpoint). A target that happened to move through the imaged scene will be displaced and smeared across the final image. Imaging a moving target is equivalent to collecting data that is not on a regular grid (although more generally this generates data in a cube, rather than data in a square). Surprisingly little is understood about how to form an image in this more general case. This topic encompasses a myriad of research areas. One starting point would be to use standard scattering center models to predict the scattering at new locations. This approach requires the ability to extract viewpoint stable features from the measured data. Filtering, tracking, and dynamic programming are key elements to this approach. Alternately, knowing the final product is a 3D image, it may be possible produce the end result without requiring exp! licit feature extraction. Data will be available to test newly developed approaches.
Solving an inverse problem means determining the parameters of a model given a set of measurements. In solving many practical inverse problems, accounting for the uncertainty of the solution is very important to aid in decision-making. A standard approach to do this begins by choosing a model parametrization and then using a Bayesian approach to make inferences on the model parameters from measurement data. However, this quantified uncertainty is a function of the model parametrization and for many inverse problems, there are many model parametrizations that account for the data equally well. A well known approach to accounting for model uncertainty is Bayesian Model Averaging where many model parametrizations are considered. Wavelet model updates provide an efficient means of sifting through a family of model parametrizations given by decimated wavelet bases. By decimated wavelet basis we mean a subset of the model's coordinates in a wavelet basis.
When working with measurement data sets which are particularly noisy or large in terms of their data storage size, it is natural to consider denoising or compressing the data also using a decimated wavelet representation. Kalman filters can be used to update the solution (including its uncertainty) when denoising or locally changing the resolution of the measurement data by changing the decimation.
We shall explore how to compute likely representations of both model and measurements in the context of solving a few model geophysical inverse problems.
Consider the dynamics of interacting elastic disks in the plane. This is an experimentally realizable 2-d model of dry granular flow where the stresses can be visualized using the photoelastic effect. As the elastic disks move in a vacuum they interact through collisions with each other and with the surrounding geometry. Because of the finite propagation speed of deformations inside each grain it can be difficult to capture computationally even simple experiments involving just a few interacting grains.
The goal of this project is to improve our ability to simulate dense granular flow in complex geometry.
Some project ideas: (A) Pursue full numerical simulations of of the interacting elastic disks driven by contact mechanics. (B) Develop simplified discrete element models for tracking the center of mass of each disk using classical mechanics including heuristics for the delay arising from the finite propagation speed of elastic deformations in each disk. (C) Compare with experimental data.
Problem Statement: Integrated Circuits are designed using a polygon representations of the wiring and transistors layers known as "layout". Layout must conform to strict design rules. Typically the design rules describes the minimum width of any polygon, minimum spacing between adjacent polygons, directionality (normally only vertical or horizontal runs are allowed) and corner angles (normally only 90 degrees). Multiple wiring layers are common, with modern ICs having as many as ten wiring levels. Different wiring layers are connected through a contact or "via" level. The via levels must also obey their own set of design rules on size and spacing.
The process of analyzing the design of an integrated circuit involves imaging the various layers and then using image processing to reconstruct the layout of the different layers. Built-in biases, both in the manufacturing process and the analysis process prevent the reconstructed layout from being a faithful representation of the original layout. For example, sharp corners become rounded, line widths may either expand or contract and straight lines become roughened with many false vertices. Typical extracted layout is shown in figure 1.
Figure 1: Unsimplified Layout
Our problem is, given the raw polygon data from image processing, how can we as faithfully as possible recreate the original layout?
There are a number of potential errors that must be avoided in any reconstruction scheme including: creation of shorts between adjacent polygons, creation of a break or "open" in an existing polygon, creation of self intersecting polygons, creation of "via opens" (a break in the connection to an upper or lower layer).
The ideal algorithm should also significantly compaction the layout data by eliminating spurious vertices.
The layout reconstruction problem has significant overlap with the general problem of line simplification which has been extensively studied for Geographic Information Systems (GIS). One of the first and most effective methods for line simplification is the algorithm proposed by Douglas and Peucker[1] in 1973. A straightforward implementation of the algorithm runs in time O(k2) for a chain of size k; Hershberger and Snoeyink[2] show how to improve this to O(k log k). This algorithm is very efficient, however it is a general line simplification solution and not optimized to the highly specific characteristics of IC layout.
Unique features of our problem are the strict design rules governing permissible polygon shapes as well as the constraints imposed by the via points.
References:
[1] DOUGLAS, D. H., AND PEUCKER, T. K. Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. The Canadian Cartographer 10 (1973), 112-122.
[2] HERSHBERGER, J., AND SNOEYINK, J. Speeding up the Douglas-Peucker line simplification algorithm. Proc. 5th Internat. Sympos. Spatial Data Handling (1992), pp. 134-143.
Randomized clinical trials are the central feature of modern medical methodology. In an RCT, two or more populations of patients who are statistically identical on the relevant dimensions are given different medical treatments. The results are then recorded, and are used to establish medical practice. The Archimedes Project of Kaiser Permanente has undertaken the development of a mathematical model of human physiology that will complement RCTs. The purpose of the model is to integrate data from various RCTs, then to predict health outcomes for populations not specifically considered in any particular RCT, perhaps for treatments not specifically considered in any particular RCT. What sort of model(s) of human physiology are appropriate for this purpose?
References:
Eddy, D., and Schlessinger, L.,Diabetes Care 26:3102-3110, 2003
Eddy, D., and Schlessinger, L.,Diabetes Care 26:3093-3101, 2003
If you have ever looked closely at an image in a magazine or a newspaper, you will see that the image, which looks like it has many shades and colors, is actually composed of dots of ink of only a handful of colors. Digital halftoning is the art of representing a full color picture with only a few colors and is necessary in all digital printers. Halftoning is possible due to the low-pass frequency characteristic of the human visual system. Several types of algorithms exist in modern digital printers with usually a tradeoff between speed and quality. This tradeoff is important since digital printers can operate at speeds of a few pages a minute (an inkjet printer for home use) to thousands of pages a minute (a production class digital printer). This project studies some questions regarding the stability and efficiency of algorithms used to perform digital halftoning. One of the goals of this project is to design an algorithm which has superior output quality, yet operates at speeds close to speedier algorithms. The type of mathematics used include linear algebra, dynamical systems theory, convex analysis, and mathematical programming.
Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday Monday | Tuesday | Wednesday | | |||
---|---|---|---|
Monday August 01, 2005 | |||
Workshop Outline: Posing of problems by the 6 industry mentors. Half-hour introductory talks in the morning followed by a welcoming lunch. In the afternoon, the teams work with the mentors. The goal at the end of the day is to get the students to start working on the 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) Fadil Santosa (University of Minnesota, Twin Cities) | EE/CS 3-180 |
9:40am-10:00am | Team 1: Sparse Aperture Imaging | D. Gregory Arnold (US Air Force Research Laboratory) | EE/CS 3-180 |
10:00am-10:20am | Team 2: Uncertainty Quantification in Geophysical Inverse Problems | Nicholas Bennett (Schlumberger-Doll) | EE/CS 3-180 |
10:20am-10:40am | Team 3: Contact Algorithms for Dry Granular Flow with Elastic Grains | Petri Fast (Lawrence Livermore National Laboratory) | EE/CS 3-180 |
10:40am-11:00am | Break | EE/CS 3-176 | |
11:00am-11:20am | Team 4: Integrated Circuit Layout Reconstruction | Edward Keyes (Semiconductor Insights) | EE/CS 3-180 |
11:20am-11:40am | Team 5: Models of Human Physiology, and Randomized Clinical Trials | David Ross (Kaiser Permanente) | EE/CS 3-180 |
11:40am-12:00pm | Team 6: Algorithms for Digital Halftoning | Chai Wah Wu (IBM) | EE/CS 3-180 |
12:00pm-1:00pm | Lunch at Orchid Cafe, Thai Cuisine, 304 Oak St. Minneapolis | ||
1:30pm-4:30pm | afternoon - start work on projects | EE/CS 3-180 | |
Tuesday August 02, 2005 | |||
Students work on the projects. Mentors guide their groups through the modeling process, leading discussion sessions, suggesting references, and assigning work. | |||
Wednesday August 03, 2005 | |||
Students work on the projects. Mentors guide their groups through the modeling process, leading discussion sessions, suggesting references, and assigning work. | |||
Thursday August 04, 2005 | |||
Students and mentors work on the projects. | |||
Friday August 05, 2005 | |||
9:30am-9:50am | Team 2 Progress Report | Lind 305 | |
9:50am-10:10am | Team 4 Progress Report | Lind 305 | |
10:10am-10:30am | Team 1 Progress Report | Lind 305 | |
10:30am-11:00am | Break | Lind Hall 400 | |
11:00am-11:20am | Team 6 Progress Report | Lind 305 | |
11:20am-11:40am | Team 3 Progress Report | Lind 305 | |
11:40am-12:00pm | Team 5 Progress Report | Lind 305 | |
12:15pm-2:00pm | Picnic | TBA | |
Saturday August 06, 2005 | |||
Students and mentors work on the projects. | |||
Sunday August 07, 2005 | |||
Students and mentors work on the projects. | |||
Monday August 08, 2005 | |||
Students and mentors work on the projects. | |||
Tuesday August 09, 2005 | |||
Students and mentors work on the projects. | |||
Wednesday August 10, 2005 | |||
9:00am-9:30am | Team 6 Final Report | EE/CS 3-180 | |
9:30am-10:00am | Team 3 Final Report | EE/CS 3-180 | |
10:00am-10:30am | Team 4 Final Report | ||
10:30am-11:00am | Break | EE/CS 3-180 | |
11:00am-11:30am | Team 1 Final Report | EE/CS 3-180 | |
11:30am-12:00pm | Team 5 Final Report | EE/CS 3-180 | |
12:00pm-12:30pm | Team 2 Final Report | EE/CS 3-180 | |
12:30pm-2:00pm | Pizza party | Lind Hall 400 |
NAME | DEPARTMENT | AFFILIATION |
---|---|---|
D. Gregory Arnold | Air Force Research Laboratory | US Air Force Research Laboratory |
Douglas Arnold | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Katherine Bartley | Department of Mathematics | University of Nebraska |
Nicholas Bennett | Schlumberger-Doll | |
Ian Besse | Department of Applied Mathematical and Computational Sciences | The University of Iowa |
Sagar Bhatt | Department of Computational and Applied Mathematics | Rice University |
Henry Boateng | Department of Mathematics | University of Michigan |
Richard Braun | Department of Mathematical Sciences | University of Delaware |
Patrick Campbell | Department of Mathematical Modeling and Analysis | Los Alamos National Laboratory |
Pengwen Chen | Department of Mathematics | University of Florida |
Julianne Chung | Department of Mathematics and Computer Science | Emory University |
James Ci | University of Minnesota, Twin Cities | |
Shilpa Das Gupta | Department of Mathematics | Mississippi State University |
Valjean Elander | Department of Mathematics | University of Nevada |
Malena Espanol | Department of Mathematics | Tufts University |
Qirong Fang | Department of Mathematics | Purdue University |
Petri Fast | BAD ADDRESS | Lawrence Livermore National Laboratory |
Harshini Fernando | Department of Mathematics & Statistics | Texas Tech University |
John Harlim | Department of Mathematics | University of Maryland |
Alfa Heryudono | Department of Mathematical Sciences | University of Delaware |
Mark Iwen | Department of Mathematics | University of Michigan |
Lijian Jiang | Department of Mathematics | Texas A & M University |
Chao Jin | Department of Applied Mathematics | University of Colorado |
Edward Keyes | Semiconductor Insights | |
Joel Lepak | Department of Mathematics | University of Michigan |
Yan Li | Department of Mathematics | Texas A & M University |
Michael Minardi | US Air Force Research Laboratory | |
Kiran Naidu | US Air Force Research Laboratory | |
Jayoung Nam | Department of Mathematics | Indiana University |
Jorge Ramirez | Department of Mathematics | Oregon State University |
Fernando Reitich | School of Mathematics | University of Minnesota, Twin Cities |
Robert Ronkese | University of Delaware | |
David Ross | The Archimedes Project | Kaiser Permanente |
John Sabino | Department of Computational and Applied Mathematics | Rice University |
Rina Santos | Department of Mathematical Sciences | University of Nevada |
Fadil Santosa | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Arnd Scheel | Institute for Mathematics and its Applications | University of Minnesota, Twin Cities |
Qingshuo Song | Department of Mathematics | Wayne State University |
Skyler Speakman | Department of Mathematics | University of Kentucky |
David St John | University of Illinois, Chicago | |
David Tello | Department of Mathematics & Statistics | Arizona State University |
Paula Vasquez | Department of Mathematical Sciences | University of Delaware |
Lei Wang | Department of Mathematics | University of Michigan |
Thomas Williams | Department of Mathematics | Mississippi State University |
Chai Wah Wu | Thomas J. Watson Research Center | IBM |
Chuan Xue | Department of Mathematics | University of Minnesota, Twin Cities |
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