# Problem Solver

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Thirty minutes west of Illinois Institute of Technology, at Argonne National Laboratory, lurks a gargantuan machine called "Mira." The recently inaugurated supercomputer—one of the most powerful on Earth—boasts a peak performance of 10 petaflops, meaning it can execute 10 quadrillion (1016) calculations per second.

*Fred Hickernell*

Photo: Michael Goss

Photo: Michael Goss

Yet, according to IIT Department of Applied Mathematics Chair and Professor Fred Hickernell, lightning speed alone is not always sufficient for today's most demanding applications.

Hickernell works in the area of computational mathematics, often focusing on improving the underlying means by which calculations are performed. These stepwise recipes for solving problems are known as numerical algorithms. Fine-tuning them to improve their efficiency and reliability is critical for complex calculations in many branches of science and engineering.

One critical issue Hickernell has explored is that complex algorithms may fail to provide answers within specified ranges of error. Worse yet, it has often been impossible to predict when a program will perform a given calculation accurately and when it will fail, yielding spurious results.

"The sine function on your calculator can be trusted, because it's very well understood," Hickernell notes. "But the next simplest thing is finding the integral of a function, and up until our recent work in the last year, none of those algorithms had real guarantees."

As Hickernell explains, problems involving many variables can be particularly stubborn to solve with speed and accuracy. In addition to myriad scientific applications, such calculations are critical for the world of financial risk management.

-"Although one cannot simultaneously consider the infinite number of possible scenarios, one can get a good approximation of portfolio performance or trading strategy success by generating thousands or millions of possible future scenarios and looking at the distribution of the outcomes," he says.

Hickernell has devoted much of his mathematical career to this technique, known as Monte Carlo simulation, publishing numerous technical papers on various aspects.

Monte Carlo allows mathematicians to evaluate problems consisting of hundreds or even thousands of variables. The results of Monte Carlo scenarios are pooled and averaged. The process is suggestive of repeated random bets and outcomes in a casino; hence, the technique's name.

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