What is widely considered in engineering circles as an elegant but otherwise useless mathematical strategy from the nineteenth century gets a makeover from a professor and graduate student pair from the Johns Hopkins University.
The Jacobi iterative method is a method introduced by German mathematician Carl Gustav Jacob Jacobi in 1845 to solve diagonally dominant systems of linear equations by using an approximate value and iterating a series of operations to arrive at a solution. By the early 1900s, well before the rise of supercomputers, "human computers," or people who were each assigned a small portion of a more complex equation were using Jacobi's strategy to solve large math problems.
Unfortunately, the Jacobi method took too long to become a practical problem solver. One mathematician at that time incorporated tweaks to make the method five times faster, but it was still too slow to become a favored strategy. Eventually, the method fell out of favor with the introduction of faster strategies.
But the latest improvements could easily turn things around for the Jacobi iterative method, which is now seen more as an artifact to be filed away at a museum than a practical technique for solving math problems. Rajat Mittal, a professor of mechanical engineering at JHU, says he and his student Xiang Yang have found a way to make the method work 200 times faster.
"For people who want to use the Jacobi method in computational mechanics, a problem that used to take 200 days to solve may now take only one day," says Mittal, who penned with Yang a paper [pdf] on the Jacobi update published in the Journal of Computational Physics. "Our paper provides the recipe for how to speed up this method significantly by just changing four or five lines in the computer code."
The updated method, which the researchers have dubbed a scheduled relaxation Jacobi method, will have numerous applications in the industry, says Mittal, especially in creating modern simulations in large-scale parallel computers. As an example, he speaks of an aerospace engineer who wants to test wing designs for an airplane can use the updated Jacobi method to make the process faster.
Mittal first introduced the Jacobi method to his class, where Yang was a first-year graduate student, before quickly shifting the discussion to faster modern methods. Yang, however, was interested to see how tweaking with Jacobi's strategy could speed up the process of iterating numerical estimates, eventually coming up with the updated method which he showed to his professor.
"Instead of saying that this method has been around for 169 years, and that everyone has already tried to improve it without much success, Professor Mittal told me that he felt my idea was very promising and he encouraged me to work on it," says Yang, lead author of the paper.
