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The normal business of physicists may be schematically thought of
as predic ting the motions of particles on the basis of known
forces, or the propagation of radiation on the basis of a known
constitution of matter. The inverse problem is to conclude what the
forces or constitutions are on the basis of the observed motion. A
large part of our sensory contact with the world around us depends
on an intuitive solution of such an inverse problem: We infer the
shape, size, and surface texture of external objects from their
scattering and absorption of light as detected by our eyes. When we
use scattering experiments to learn the size or shape of particles,
or the forces they exert upon each other, the nature of the problem
is similar, if more refined. The kinematics, the equations of
motion, are usually assumed to be known. It is the forces that are
sought, and how they vary from point to point. As with so many
other physical ideas, the first one we know of to have touched upon
the kind of inverse problem discussed in this book was Lord
Rayleigh (1877). In the course of describing the vibrations of
strings of variable density he briefly discusses the possibility of
inferring the density distribution from the frequencies of
vibration. This passage may be regarded as a precursor of the
mathematical study of the inverse spectral problem some seventy
years later."
Here is a clearly written introduction to three central areas of
inverse problems: inverse problems in electromagnetic scattering
theory, inverse spectral theory, and inverse problems in quantum
scattering theory. Inverse problems, one of the most attractive
parts of applied mathematics, attempt to obtain information about
structures by nondestructive measurements. Based on a series of
lectures presented by three of the authors, all experts in the
field, the book provides a quick and easy way for readers to become
familiar with the area through a survey of recent developments in
inverse spectral and inverse scattering problems. In the opening
chapter, Paivarinta collects the mathematical tools needed in the
subsequent chapters and gives references for further study.
Colton's chapter focuses on electromagnetic scattering problems. As
an application he considers the problem of detecting and monitoring
leukemia. Rundell's chapter deals with inverse spectral problems.
He describes several exact and algorithmic methods for
reconstructing an unknown function from the spectral data. Chadan
provides an introduction to quantum mechanical inverse scattering
problems. As an application he explains the celebrated method of
Gardner, Greene, Kruskal, and Miura for solving nonlinear evolution
equations such as the Korteweg_de Vries equation. Each chapter
provides full references for further study.
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