Mathematics of Information Technology and Complex Systems - MITACS

Mathematical researchers on the front lines of non-linearity

The plume of a pollution spill in water may be visible at the surface, yet all but impossible to measure properly, especially in underground or underwater regions, a limitation that can hamper monitoring and cleanup efforts. In much the same way, aircraft circling an airport can be tracked on a radar screen yet their individual movements remain hard to anticipate, placing stress on traffic controllers responsible for avoiding collisions.

One of the main challenges posed by these problems is imperfect corrupted measurements, which contain only partial information about a system, whether it be a polluted lake or a busy airspace. Such simple problems are rooted in some of today's most daunting abstract mathematics, where a network of Canadian researchers is making new inroads toward practical solutions.

Their solutions could lay the foundation for computer-driven models capable of accurately tracking and predicting the flow of a spill based upon limited data or avoiding a mid-air collision. The key to solving these problems is filtering, a mathematical process for estimating the current state of a random dynamic system by fusing corrupted information collected from a limited number of sources over a limited time period. Prompted by potential military applications, filtering was first studied during World War II, but the mathematical results did not lend themselves to real-time implementation, especially on the primitive computational hardware of that day.

The latest generation of powerful computers are much more capable of coping with this computational obstacle. And in the last decade the mathematics of implementable nonlinear filtering algorithms has proceeded to tackle the most complex features of non-linear equations, which can describe subtle physical phenomena such as the movement of gases and liquids.

"It's state-of-the-art mathematics, using new ideas of the 90s and today," says University of Alberta mathematician Dr. Michael Kouritzin. "It's both theoretical and applied, so it's where I like to be."

He is a project leader within MITACS, the Mathematics of Information Technology and Complex Systems Network, which was announced in October 1998. One of 22 federally funded Networks of Centres of Excellence, MITACS includes 21 projects developed around five main themes - biomedical, industrial, information technology, finance and manufacturing - which should be vital to Canada's success in the 21st century. The network received $14 million for its first three years of operation, linking 182 leading mathematical scientists and more than 150 graduate students at 21 Canadian universities, as well as 34 private firms.

Dr. Kouritzin's interest in filtering was prompted by his post-doctoral work with aerospace giant Lockheed Martin, which was looking for mathematical techniques to manage air traffic in civilian and military settings. Since then he has devoted much of his attention to creating and analysing nonlinear filtering algorithms. One such algorithm employs "smart Monte Carlo methods", a simulation of many particles mimicking the system's motion. The resulting observations are used to select "good" particles to be duplicated and "bad" ones to be removed. These algorithms provide the best estimates of the system's current and future state, which is the essence of reliable tracking and prediction.

"Nobody is certain about what might be the best method for any given filtering problem," he says. "It turns out that our branching particle method is one of the best around."

His earlier work on filters enhanced the effectiveness of radar, sonar, infrared and electrical optical sensors. In particular, these accomplishments helped Lockheed Martin refine its tracking systems to detect highly manoeuvrable targets.

Through MITACS, the techniques being introduced by Dr. Kouritzin and his colleagues should have an even wider range of uses, in computer networks, communications systems and financial analysis.

MITACS is one of 20 federally funded Networks of Centres of Excellence, the objectives of which are to enhance the Canadian economy and our quality of life. The program is funded by the Natural Sciences and Engineering Research Council (NSERC), the Canadian Institutes of Health Research (CIHR) and the Social Sciences and Humanities Research Council (SSHRC), in partnership with Industry Canada.

For more information on MITACS, visit www.mitacs.math.ca.