New Life for the Jacobi Method

Winter 2015

Jacobi-imgThe 169-year-old math strategy called the Jacobi iterative method is widely dismissed today as too slow to be useful. But thanks to graduate student Xiang Yang and mechanical engineering professor Rajat Mittal, it may soon get a new lease on life.

With a few tweaks, the duo says they’ve made the rarely used Jacobi method work up to 200 times faster. The result, they say, could speed up the performance of computer simulations used in aerospace design, shipbuilding, biomechanics, and other engineering tasks.

“I expect this to be adopted very quickly,” says Mittal, senior author of the paper published last June in the online Journal of Computational Physics. “Everyone is competing for access to powerful computer systems, and the new Jacobi method will save time. In fact, the beauty of this method is that it is particularly well suited for the large-scale parallel computers that are being used in most modern simulations.”

The dramatic makeover emerged quietly in the fall of 2012, after Mittal told students in his Numerical Methods class about the mathematically elegant but practically useless method. Yang, then a first-year grad student, was listening intently.

Carl Gustav Jacob Jacobi, a prominent German mathematician, had 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 until a useful solution appeared. A noted 20th-century mathematician managed to make the method proceed five times faster, but that was still considered rather slow. With the advent of speedier strategies and electronic computers, the Jacobi method fell out of favor.

In class, Mittal moved on to more modern topics. But Yang’s interest was piqued, and he began tinkering with the method. He returned to Mittal and proposed a way to make
the process of repeating numerical estimates move more efficiently—speeding up the arrival of a solution.

Yang spent a couple of weeks honing the updated math strategy, then the two began working together on a paper, with Yang as lead author.