Jacobi Iterative Method Gets A Makeover

The Jacobi iterative method, a 169-year-old math strategy, may soon get a new lease on life, thanks to a curious, numbers-savvy Johns Hopkins engineering student and his professor.

Carl Gustav Jacob Jacobi, a prominent German mathematician, unveiled it in 1845 as a way to solve systems of linear equations by starting with a guess and then repeating a series of math operations over and over until a useful solution appeared. By the early 20th century, the Jacobi iterative method was being used by human computers, groups of men and women who were each assigned to perform small pieces of larger math problems.

The problem was, as you can imagine, that it took a long time. A noted mathematician during that era managed to make the method proceed five times faster but that was still considered rather slow. In the age of computers, it went by the wayside.

But with just a few modern-day tweaks, the researchers say they’ve made it work up to 200 times faster and that could speed up the performance of computer simulations used in aerospace design, shipbuilding, weather and climate modeling, biomechanics and other engineering tasks.

Johns Hopkins graduate student Xiang Yang teamed up with Rajat Mittal, a professor of mechanical engineering to revamp the 169-year-old math strategy, making it work up to 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,” said Rajat Mittal, a mechanical engineering professor in the university’s Whiting School of Engineering and senior author of the journal article. “Our paper provides the recipe for how to speed up this method significantly by just changing four or five lines in the computer code.”

This dramatic makeover emerged quietly in the fall of 2012, after Mittal told students in his Numerical Methods class about the Jacobi method. Mittal cited Jacobi’s strategy as a mathematically elegant but practically useless method and then moved on to faster methods and more modern topics. Xiang Yang, then a first-year grad student in the class was listening intently.

“It just took so much time and so many computations to get to the answer you wanted,” said mechanical engineering grad student Yang. “And there were better methods. That’s why this Jacobi method isn’t being used much today.”

But after learning about the method in Mittal’s class, Yang began tinkering with it. He returned to Mittal and proposed a way to make the process of repeating numerical estimates move more efficiently, speeding up arrival at a solution. “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,” Yang said, “and he encouraged me to work on it.”

Yang spent a couple of weeks honing the updated math strategy which he and his professor called a ‘scheduled relaxation Jacobi method’. Then the grad student and Mittal began working together on a paper about the work that could be submitted to a peer-reviewed journal with Yang as lead author.

Now that it has been published and is being shared freely, Mittal expects the modified method to be embraced in many industry applications particularly those involving fluid mechanics.

Their paper describing this updated math tool was published June 27 in the online edition of the Journal of Computational Physics.

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Anne-Marie Jackson

Anna grew up on a steady diet of 20th century literature, science-fiction movies and Viking lore. Unsurprisingly she turned into a technology geek with an inclination for history & fantasy.