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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.
Project Description:
Figure 1. Segmentation of the internal carotid artery (left). Vessel tree with the common, internal and external carotid arteries (right).
Accurate vessel segmentation is required in many clinical applications, such as identifying the degree of stenosis (narrowing) of a vessel to assess if blood flow to an organ is sufficient, quantification of plaque buildup (to determine the risk of stroke, for example), and in detecting aneurisms which pose severe risks if ruptured. Proximity to bone can pose segmentation challenges due to the similar appearance of bone and contrasted vessels in CT (Figure 1 – the internal carotid has to cross the skull base); other challenges are posed by low X-ray dose images, and pathology such as stenosis and calcifications.
Figure 2. Cross section of vessel segmentation from CT data, shown with straightened centerline. A typical segmentation consists of a centerline that tracks the length of the vessel, lumen surface and vessel wall surface. Since for performance reasons most clinical applications use only local vessel models for detection, tracking and segmentation, in the presence of noise the results can become physiologically unrealistic – for example in the figure above, the diameter of the lumen and wall cross-sections vary too rapidly.
Figure 3. Vessel represented as a centerline with periodically sampled cross-sections in the planes orthogonal to the centerline. Note that some planes intersect, which makes this representation problematic. The in-plane cross-sections of the vessel are shown on the right. The goal of this project is to design a method for refining a vessel segmentation based on the following general approach:
References:
Project Description:
Modern commercial airplanes are assembled out of millions of different parts. While many of these parts are rigid, many of them are not. For example, the hydraulic lines and flexible electrical conduits that supply an airplane's landing gear change their shape as the landing gear goes through its motion (you can see some of these lines in the accompanying photograph). These shapes can be modeled by minimizing the potential energy of the rest state of one of these flexible lines as the ends of the lines are moved by the landing gear. While this problem is amenable to solution through direct optimization of individual finite elements, the method often proves to be slow and unreliable. In this investigation, we will explore the use of variational methods (i.e. the calculus of variations) in an attempt to discover a more elegant and robust approach to modeling these flexible airplane parts.
Reference:
Any textbook on the calculus of variations. My favorite is The Variational Principles of Mechanics, by Cornelius Lanczos.
Keywords: Geometrical modeling, calculus of variations, boundary value problems
Prerequisites: Calculus of variations, optimization, numerical methods for ODEs and 2-point boundary value problems, Matlab
Project Description:
Driven by rapid advances in many fields including Biology, Finance and Web Services, applications involving millions or even billions of data items such as documents, user records, reviews, images or videos are not that uncommon. Given a query from a user, fast and accurate retrieval of relevant items from such massive data sets is of critical importance. Each item in a data set is typically represented by a feature vector, possibly in a very high dimensional space. Moreover, such a vector tends to be sparse for many applications. For instance, text documents are encoded as a word frequency vector. Similarly, images and videos are commonly represented as sparse histograms of a large number of visual features. Many techniques have been proposed in the past for fast nearest neighbor search. Most of these can be divided in two paradigms: Specialized data structures (e.g., trees), and hashing (representing each item as a compact code). Tree-based methods scale poorly with dimensionality, typically reducing to worst case linear search. Hashing based methods are popular for large-scale search but learning accurate and fast hashes for high-dimensional sparse data is still an open question.
In this project, we aim to focus on fast approximate nearest neighbor search in massive databases by converting each item as a binary code. Locality Sensitive Hashing (LSH) is one of the most prominent methods that uses randomized projections to generate simple hash functions. However, LSH usually requires long codes for good performance. The main challenge of this project is how to learn appropriate hash functions that take input data distribution into consideration. This will lead to more compact codes, thus reducing the storage and computational needs significantly. The project will focus on understanding and implementing a few state-of-the-art hashing methods, developing the formulation for learning data-dependent hash functions assuming a known data density, and experimenting with medium to large scale datasets.
Keywords: Approximate Nearest Neighbor (ANN) search, Hashing, LSH, Sparse data, High-dimensional hashing
References: For a quick overview of ANN search, review the following tutorials (more references are given at the end of the tutorials):
Prerequisites: - Good computing skills (Matlab or C/C++) - Strong background in optimization, linear algebra and calculus - Machine learning and computational geometry background preferred but not necessaryProject Description: A PC sold in 2010 had billions of transistors with 32 nm gate-length. In a year, that dimension will shrink to 22 nm. Light is essential to fabrication and quality control of such small semiconductor devices. Integrated circuits are manufactured by repeatedly depositing a film of material and etching a pattern in the deposited film. The pattern is formed by a process called photo lithography. An optical image of a photomask is projected on to a silicon wafer on which the integrated circuits will be formed. Presently, 193 nm wavelength deep UV light is used to project the image. Photomasks are inspected for defects by deep UV microscopy, which is subject to the same resolution limit as lithography. Manufacturing next generation integrated circuits and inspecting their photomasks require higher-resolution imaging. The leading candidate for next generation lithography, EUV (extreme ultraviolet) lithography uses 13.5 nm wavelength. At that wavelength, no material is highly reflective or transparent. The photomask and mirrors of the projector are coated with a multi-layer Bragg reflector to achieve sufficient reflectance. The photomask has a patterned absorber film over the Bragg reflector. The thickness of the absorber and the lateral dimensions of the mask pattern are on the order of four EUV wavelengths. Rapid yet accurate simulation of EUV image formation is essential to optimizing the photomask pattern and interpreting the images of an inspection microscope. This project concerns the key step of image simulation, which is calculating the diffracted near-field of the EUV photomask. References:
Prerequisites: Basic optics, electromagnetics, computational methods for wave equations
Project Description:
In the next generation electrical grid, or "smart grid", there will be many heterogeneous power generators, power storage devices and power consumers. This will include residential customers who traditionally are only part of the ecosystem as consumers, but will in the foreseeable future increasingly provide renewable energy generation through photovoltaics and wind energy and provide energy storage through plug-in hybrid vehicles. What makes this electrical grid "smart" is the capability to insert a vast number of sensors and actuators into the system. This allows a wide variety of information about all the constituents to be collected and various aspects of the electrical grid to be controlled via advanced electric meters, smart appliances, etc. Information gathered consists of e.g. amount of energy use, planned energy consumption, efficiency and status of equipment, energy generation costs, etc and this information is then used by all constituents to optimize certain objectives. This necessitates communication and information technology to transmit and process this information. The goal of this project is to focus on the optimization of local objectives in a smart grid. In particular, we study various centralized and decentralized optimization algorithms to determine the optimal matching and maintain stability between energy producers, energy storage, and energy consumers all connected in a complex and dynamic network.
Technical prerequisites: scripting languages (Matlab, python), optimization, linear and nonlinear programming.
Preferred but not necessary: graph theory, combinatorics, computer programming, experience with CPLEX, R.
Wednesday | Thursday | Friday | Saturday | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | | |||
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Wednesday August 03, 2011 | |||
Projects | |||
9:00am-9:30am | Coffee and Registration | Keller Hall 3-176 | |
9:30am-9:40am | Welcome to the IMA | Fadil Santosa (University of Minnesota, Twin Cities) | Keller Hall 3-180 |
9:40am-10:00am | Team 1: Geometric and appearance modeling of vascular structures in CT and MR | Stefan Atev (ViTAL Images, Inc.) | Keller Hall 3-180 |
10:00am-10:20am | Team 2: Modeling aircraft hoses and flexible conduits | Thomas Grandine (The Boeing Company) | Keller Hall 3-180 |
10:20am-10:40am | Team 3: Fast nearest neighbor search in massive high-dimensional sparse data sets | Sanjiv Kumar (Google Inc.) | Keller Hall 3-180 |
10:40am-11:00am | Break | Keller Hall 3-176 | |
11:00am-11:20am | Team 4: Diffraction by photomasks | Apo Sezginer (KLA - Tencor) | Keller Hall 3-180 |
11:20am-11:40am | Team 5: Optimizing power generation and delivery in smart electrical grids | Chai Wah Wu (IBM) | Keller Hall 3-180 |
11:45am-1:30pm | Lunch | Lind Hall 400 | |
1:30pm-4:30pm | Afternoon - start work on projects Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | ||
Thursday August 04, 2011 | |||
Students work on the projects. Mentors guide their groups through the modeling process, leading discussion sessions, suggesting references, and assigning work. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | |||
Friday August 05, 2011 | |||
Students work on the projects. Mentors available for consultation. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | |||
Saturday August 06, 2011 | |||
Students work on the projects. All teams may work on the 4th floor of Lind hall. | |||
Sunday August 07, 2011 | |||
Students work on the projects. All teams may work on the 4th floor of Lind hall. | |||
Monday August 08, 2011 | |||
9:00am-9:30am | Coffee | Keller Hall 3-176 | |
9:30am-9:50am | Team 4 progress report | Keller Hall 3-180 | |
9:50am-10:10am | Team 2 progress report | Keller Hall 3-180 | |
10:10am-10:30am | Team 5 progress report | Keller Hall 3-180 | |
10:30am-11:00am | Break | ||
11:00am-11:20am | Team 1 progress report | Keller Hall 3-180 | |
11:20am-11:40am | Team 3 progress report | Keller Hall 3-180 | |
12:00pm-1:30pm | Picnic at Lind Hall court yard area | ||
1:30pm-2:00pm | Group Photos | ||
2:00pm-5:00pm | Students work on the projects. Mentors available for consultation. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | ||
Tuesday August 09, 2011 | |||
Students work on the projects. Mentors available for consultation. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | |||
Wednesday August 10, 2011 | |||
Students work on the projects. Mentors available for consultation. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | |||
Thursday August 11, 2011 | |||
Students work on the projects. Mentors available for consultation. Team 1- 4-125A Mechanical Engineering Team 2- 401 Lind Hall Team 3- 305 Lind Hall Team 4- 340 Lind Hall Team 5- 321 Mechanical Engineering | |||
Friday August 12, 2011 | |||
8:30am-9:00am | Coffee | Keller Hall 3-176 | |
9:00am-9:30am | Team 3 final report | Keller Hall 3-180 | |
9:30am-10:00am | Team 1 final report | Keller Hall 3-180 | |
10:00am-10:30am | Team 4 final report | Keller Hall 3-180 | |
10:30am-11:00am | Break | ||
11:00am-11:30am | Team 5 final report | Keller Hall 3-180 | |
11:30am-12:00pm | Team 2 final report | Keller Hall 3-180 | |
12:00pm-1:30pm | Pizza party |
NAME | DEPARTMENT | AFFILIATION |
---|---|---|
Baha Alzalg | Department of Mathematics | Washington State University |
Catalina Anghel | Department of Mathematics | University of Toronto |
Stefan Atev | ViTAL Images, Inc. | |
Qichuan Bai | Department of Mathematics | The Pennsylvania State University |
Jorge Banuelos | Department of Mathematics, Statistics and Computer Science | Macalester College |
Richard Braun | Department of Mathematical Sciences | University of Delaware |
Weitao Chen | Department of Mathematics | The Ohio State University |
Miles Crosskey | Department of Mathematics | Duke University |
Brittan Farmer | Department of Mathematics | University of Michigan |
Eric Foxall | Department of Mathematics & Statistics | University of Victoria |
Xing (Margaret) Fu | Computational Math and Engineering | Stanford University |
Wenying Gan | Department of Mathematics | University of California, Los Angeles |
Thomas Grandine | Department of Applied Mathematics | The Boeing Company |
Ke Han | Department of Mathematics | The Pennsylvania State University |
Huiyi Hu | Department of Mathematics | University of California, Los Angeles |
Qing Huang | Department of Mathematics & Statistics | Arizona State University |
Jens Christian Jorgensen | Department of Mathematics | New York University |
Sunnie Joshi | Department of Mathematics | Texas A & M University |
Eunkyung Ko | Department of Mathematics & Statistics | Mississippi State University |
Sanjiv Kumar | Google Inc. | |
Rosemonde Lareau-Dussault | Department of Mathematics | University of Sherbrooke |
Gilad Lerman | School of Mathematics | University of Minnesota, Twin Cities |
Oumar Mbodji | Department of Mathematics | McMaster University |
Dennis Moore | Department of Mathematics | University of Kentucky |
Ahmet Ozkan Ozer | Department of Mathematics | Iowa State University |
Mustazee Rahman | Department of Mathematics | University of Toronto |
Anton Sakovich | Department of Mathematics & Statistics | McMaster University |
Shirin Sardar | Department Computational and Applied Mathematics | Rice University |
Apo Sezginer | KLA - Tencor | |
Alex Shum | Department of Applied Mathematics | University of Waterloo |
Cory Simon | Institute of Applied Mathematics | University of British Columbia |
Changhui Tan | Department of Mathematics | University of Maryland |
Hsiao-Chieh Tseng | Department of Mathematics | University of California, Davis |
Chai Wah Wu | Thomas J. Watson Research Center | IBM |
Haley Yaple | Department of Engineering Sciences and Applied Mathematics | Northwestern University |
Kidist Zeleke | Department of Mathematics | University of Houston |
Zhou Zhou | University of Michigan |
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