Mathematics of Information Technology and Complex Systems


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Milestones

Research Objectives

  • To develop a spectrum of computationally efficient state of the art conditional probability density and parameter  estimation algorithms that collectively solve the gamut of real time detection, tracking, path-space filtering,   prediction,and image processing problems based upon possibly incompletely determined predictive models.
  • To advance the art of predictive modeling through increased realism, mathematical complexity, and efficient computer tractable approximations.
  • To analyze, compare, and evaluate our algorithms and models on prototype problems suggested by our corporate sponsors or with real world applications.
  • To promote the use of sophisticated mathematics in industry.
  • To determine the additional skills necessary for graduates of the mathematical sciences to be sought by industry.     Subsequently, to train students accordingly.
  • To develop a set of course materials to help train graduate students in applications of stochastic analysis.



Sub-Topics

  • Develop and advance efficient, computer workable filtering algorithms;
  • Develop combined parameter and state estimation algorithms for tracking, prediction and image processing;
  • Create computer workable nonlinear filtering and estimation algorithms using our SERP, REST, IDEX and combined
  • parameter-state estimation methods as well as other particle filter, convolutional, chaos, or Markov chain techniques.  Compare methods empirically on benchmark problems;
  • Prove consistency and rates of convergence for the algorithms in 1) and 2);
  • Develop prediction systems to control lighting, microphones, etc. for live theatre performers. (Acoustic Positioning Research problem);
  • Improve model approximation and robustness;
  • Generate filters for signals in random environments;
  • Further basic filtering theory including uniqueness, particle representation, existence, and the innovations theorem.



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