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Short Summaries of Articles about Mathematics in the Popular Press

"Inverse Boundary-Value Problems," by Margaret Cheney. American Scientist, September-October 1997, pages 448-455.

This article discusses a class of mathematical problems known as "inverse problems." An inverse problem is one in which the desired quantity is not obtained by direct measurement but deduced from related quantities. For example, seismologists hunt for oil fields under the ocean floor not by drilling everywhere, but by bouncing acoustic waves off rocks at the bottom of the ocean. Examining the behavior of the reflected waves allows them to deduce whether there is oil underneath the rocks. This article discusses inverse problems in oil recovery and in an emerging medical imaging technique called electrical-impedance tomography. In addition, it describes methods for solving these problems, in particular methods that have been made possible by today's powerful computers. While computers have provided the brute calculating strength necessary to solve these problems, the author notes, "we should not forget that the key technology---both for telling the computer what to do and interpreting what it tells us---is still mathematics."