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• #### Probability and Statistical Physics / Connecting Microscopic and Macroscopic 53-58

to fly the plane past a radar detector thousands of times, from every possible angle and with every possible configuration of bombs or fuel tanks or other attachments. This would be prohibitively expensive and might be impossible for many enemy aircraft.

… 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.

Alternatively, you could try to compute mathematically what the plane’s radar signature should look like. If you could do that reliably, then it would be an easy matter to tweak the configuration to take into account bombs, fuel tanks, etc.

The problem of directly computing a radar reflection boils down to solving a system of differential equations called Maxwell’s equations, which describe the way that electric and magnetic fields propagate through space. (A radar pulse is nothing more than an electromagnetic wave.) There was nothing new about the physics; the Maxwell equations have been known since the 19th century. The difficulty was all in the mathematics. Before the Fast Multipole Method, it took a prohibitively large number of calculations to compute the radar signature of something as complicated as an airplane. In fact, scientists could not compute the radar reflection of anything other than simple shapes.

It was known in principle that Maxwell’s equations could be formulated as an integral equation, which is more tolerant of facets, corners, and discontinuities. To solve these, one must be able to calculate something called a Green’s function, which treats the skin of the plane as if it were made up of many point emitters of radar waves, adding up the contribution of each source. This approach reduces Maxwell’s equations from a three-dimensional problem to a two-dimensional one (the two dimensions of the plane’s surface). However, that reduction is still not enough to make the computation feasible. Using Green’s function to evaluate the signal produced by N pulses from N points on the target would seem to require N2 computations. Such a strategy does not work well for large planes because the larger the plane is, the more data one needs to include and this approach requires an impractically large computational effort.

The Fast Multipole Method builds on the insight that the problem becomes more manageable if the source points and target points are widely separated from one another. In that case, the radar waves produced by the sources can be approximated by a single “multipole” field. Although it still takes N computations to compute the multipole field the first time, after that you can reuse the same multipole function over and over. Thus, instead of doing 1 million computations 1 million times (a trillion computations), you do 1 million computations once, and then you do one computation 1 million times (2 million computations in total). Thus, Fast Multipole Method makes the more efficient Green’s function approach computationally feasible.

A second ingenious idea behind the Fast Multipole Method is that it can be applied even when the source points and target points are not widely separated. You simply divide up space into a hierarchical arrangement of cubes. When sources and targets lie in adjacent cubes, you compute their interaction directly (not with a multipole expansion). That part of the Fast Multipole Method is slow. But the great majority of source-target pairs are not in adjacent cubes. Thus their contributions to the Green’s function can be

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