Summer Institute 2007
Carnegie Mellon University
May 31 - July 17



Projects:

• Green Oxidation
  Participants: Tarek Elgindi (University of Wisconsin)
  Project description:
  
  • Understand how a system of ODEs can arise in green chemistry oxidation processes
  • Investigate existence of exact solution by looking at some special cases
  • If no exact solution can be found, analyse its numerical properties
  • Try to predict correct values for rate constants from the equations
  • Analyze phase space behaviour, identify any possible limit cycles and try to see what implications that can have for the chemical problem
  Current plan: pdf
  Progress report: pdf
  
  
• Voronoi
  Participants: Morgan Shaffer (Mount Holoyoke College), Michelle Baker (Shippensburg University)
  Project description:
  
  • Understand Newton's method and its use in accelerating numerical methods.
  • Study the concept of centroidal Voronoi tessellations (CVTs) and its use in physical and engineering applications, such as data compression and segmentation.
  • Develop the set of routines for accelerating data analysis by coupling Lloyd algorithm with Newton, explore possible shortcomings and alternatives.
  Current plan:pdf
  Progress report: pdf
  
  
• Random walks: finance applications
  Participants: Jian Wang (UT Knoxville), Keith Rogers (Alabama State)
  Project description:
  
  • Review binomial options pricing
  • Study the principles of discrete time finance focusing on European calls and random trees
  • Develop understanding of the Brownian motion representation for a random walk and the use Black-Scholes equation in modeling option pricing
  • Taking a model problem, formulate and analyze corresponding Black-Scholes equation
  • Develop numerical techniques for solving the derived PDEs
  Current plan:pdf
  Progress report: pdf
  
  
• PageRank
  Participants: Alexander Chun (Northwestern University)
  Project description:
  • Understand the math behind Google's million-dollar worth search engine.
  • Study the relationship between the structure of the network and convergence of the numerical algorithms as well as their well-posedness.
  • Investigate possible ways to compute dominant eigenvalues and their advantages and drawbacks.
  • Taking as an example some model large matrix representation of the internet, develop a set of routines for computing page rank using existing methods such as power method and Gaussian elimination, as well as their possible modifications.
  • Compare the results of the test and try to answer some of the open questions.
  Current plan: pdf
  Progress report: pdf