Painful as it might be, think back to your junior high school math class when they taught you to solve problems like 2 x Y = 6.

Easy, right? That Y, called an "unknown," obviously stands for 3. Now think about solving a problem with 20 million unknowns – that is, 20 million of those Ys.

Forget about using a calculator. Even getting the time to run the problem on a supercomputer is going to be hard to do.

But some University of Illinois researchers have developed a way to solve a problem with 20 million unknowns on a small "cluster" of 10 relatively inexpensive computer workstations networked to calculate in concert.

"What we predict is, in the future, a 20 million unknown can be solved on a PC desktop," UI electrical and computer engineering Professor Weng Cho Chew said recently, by using the UI technique and taking advantage of the constantly growing power of personal computers.

"That's really where I see some of this work going, being able to solve these large problems on a low budget," said Larkin Hastriter, an Air Force captain who has been working with Chew while completing his doctorate at the UI.

Previously, solving a problem with 10 million unknowns was considered a monumental task.

"Most universities can't touch a problem (of the kind solved by the UI researchers) above 1 million or 2 million unknowns today," Hastriter said.

The UI achievement is about more than bragging rights in math and computer circles, however.

For example, Chew has been working with local firm Science Applications International Corp. on a Defense Department project to build a library of radar signatures for airplanes, missiles and surface craft, such as tanks and ships, that could allow military personnel to identify them almost instantly. The idea is to better target the enemy and to minimize incidents of friendly fire.

Physically measuring the signals given off by a target to create a radar signature is expensive, and sometimes impossible in the case of enemy craft. Nor is it always accurate, since measurements taken by different people can vary.

The researchers can get around that by extrapolating the signature of, say, a fighter plane from its picture and double checking it with less extensive physical measurements.

But doing such an extrapolation involves solving problems with unknowns. The more unknowns you can solve, the more accurate the representation is likely to be.

Hastriter, who will become a professor and researcher at the Air Force Institute of Technology in the fall, said the enhanced problem-solving capability also might allow radar systems to operate at higher frequencies with better resolution, making it easier to identify targets.

Likewise, Chew said, operating at higher frequencies could allow the use of smaller components, shrinking the size and improving the portability of the detection devices.

While the technique may be applicable to his radar work, Chew said it has a variety of other potential uses as well, from remote sensing to computer chip design.

"We hope in the future this technique will be used more and more," he said.

Chew encouraged Hastriter, an electrical engineer, to take on the challenge of solving a 20-million-unknown problem when he came to the UI three years ago. Previous work by Chew and colleagues on solving such massive problems, especially as they relate to electromagnetics, the basis of radar, is one thing that attracted Hastriter to the university.

So far, the researchers have solved problems involving "canonical shapes," like spheres, which have well-known characteristics, in order to validate their method.

"The next step is to take these algorithms and apply them to aircraft, tanks, other complex structures," Hastriter said. They've already solved a 10-million-unknown problem involving an aircraft target.

You can reach Greg Kline at (217) 351-5215 or via e-mail at gkline@news-gazette.com.