A Lucid Interval
Intervals at Work
The interval community can point to a number of success stories. In 1995 Joel Hass, Michael Hutchings and Roger Schlafly proved part of the "double-bubble conjecture" by a method that entailed extensive numerical calculations; they used interval methods to establish rigorous bounds on computational errors. The conjecture concerns soap films enclosing a pair of volumes, and states that the common configuration of two conjoined quasi-spherical bubbles has the smallest surface-to-volume ratio. Hass, Hutchings and Schlafly proved the conjecture for the case of two equal volumes, essentially by calculating the best possible ratio for all configurations. The calculations did not have to be exact, but any errors had to be smaller than the differences between the various ratios. Interval methods provided this guarantee. (The general case of the double-bubble conjecture was proved a few years later by Hutchings, Frank Morgan, Manuel Ritoré and Antonio Ros—without interval arithmetic and indeed without computers, using "only ideas, pencil, and paper.")
A quite different application of interval methods was reported in 1996 by Oliver Holzmann, Bruno Lang and Holger Schütt of the University of Wuppertal. Instead of trying to control the errors of a calculation, they were estimating the magnitude of errors in a physical experiment. The experiment was a measurement of Newton's gravitational constant G, done with two pendulums attracted to large brass weights. The interval analysis assessed various contributions to the uncertainty of the final result, and discovered a few surprises. An elaborate scheme had been devised for measuring the distance between the swinging pendulums, and as a result this source of error was quite small; but uncertainties in the height of the brass weights were found to be an important factor limiting the overall accuracy.
Would we be better off if intervals were used for all computations? Maybe, but imagine the plight of the soldier in the field: A missile is to be fired if and only if a target comes within a range of 5 kilometers, and the interval-equipped computer reports that the distance is [4,6] kilometers. This is rather like the weather forecast that promises a 50-percent chance of rain. Such statements may accurately reflect our true state of knowledge, but they're not much help when you have to decide whether to light the fuse or take the umbrella. But this is a psychological problem more than a mathematical one. Perhaps the solution is to compute with intervals, but at the end let the machine report a definite, pointlike answer, chosen at random from within the final interval.
© Brian Hayes
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