Xukai Shen, a geophysicist working at BP Plc, had a hunch he could solve a riddle that had vexed the company: whether there was a lot of oil hidden beneath a salt dome 7,000 feet underwater in the Gulf of Mexico. So he asked to use the company’s supercomputer exclusively for two weeks to check it out.

Using an algorithm, the 33-year-old with a Stanford PhD harnessed the computer’s massive power last year to produce a clearer seismic image of what lay beneath. The result: a potentially massive oil find. With a clearer picture...