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11 / Simulation of two-dimensional radar scattering from a stealthy airplane, similar in shape to the B-2 bomber. The front of the plane is at the top; two simulated radar signals are plotted in red and purple. On the left (red) is a computation using a low-order discretization. It incorrectly shows a considerable radar signal to the front and side of the airplane. On the right (purple) is a more accurate reconstruction of the radar signal, which would in practice be computed with the Fast Multipole Method. Note the near absence of a radar signal to the front and side of the airplane. (Actual three-dimensional data for the B-2 bomber are classified.) Reprinted with permission from Mark Stalzer, California Institute of Technology. /

computed quickly, using the multipole approximation. Because most of the calculation is accelerated and only a tiny part is slow, the overall effect is a great speedup.

In practice, the Fast Multipole Method has meant the difference between computing the radar signature of a coarse approximation to an airplane and computing the radar signature of a particular model of aircraft. While it would be tempting to say that it has saved the Air Force millions of dollars, it would be more accurate to say that it has enabled them to do something they could not previously do at any price (see Figure 11).

The applications of the Fast Multipole Method have not been limited to the military. In fact, its most important application from a business perspective is for the fabrication of computer chips and electronic components. Integrated circuits now pack 10 billion transistors into a few square centimeters, and this makes their electromagnetic behavior hard to predict. The electrons don’t just go through the wires they are supposed to, as they would in a normal-sized circuit. A charge in one wire can induce a parasitic charge in other wires that are only a few microns away.

Predicting the actual behavior of a chip means solving Maxwell’s equations, and the Fast Multipole Method has proved to be the perfect tool. For example, most cell phones now contain components that were tested with the Fast Multipole Method before they were ever manufactured.

At present the semiconductor companies use a slightly simpler version of the Fast Multipole Method than the original algorithm developed for Defense Advanced Research Projects Agency. The simpler version is applicable to static electric fields or



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