First we explain the concept of an inverse problem. It begins with the sup-
position that there is a direct problem, i.e., a well-posed problem of mathe-
matical physics. In other words, we have a good mathematical description of
a given "device". If this device is in part unknown and supplementary infor-
mation about a solution of the direct problem is given we obtain an inverse
For example, for any mass distribution one can find its Newtonian poten-
tial. Moreover, this problem is well-posed although not very elementary. If
a mass distribution is not known but its potential outside a certain ball is
given and the aim is to determine this mass distribution, we find ourselves
studying the inverse problem of potential theory. This inverse problem was
formulated by Laplace 200 years ago. More generally, given a linear dif-
ferential operator and boundary or initial data one can find a solution to a
related well-posed boundary value problem. An inverse coefficients (identifi-
cation) problem is to find coefficients or the right-hand side of this equation
if additional boundary data are given.
Many inverse problems arise naturally and have important applications.
As a rule, these problems are rather difficult to solve for two reasons: they
are nonlinear and they are improperly posed. Probably, the second reason is
more serious. Only in recent decades have we made certain progress in both
the analytical and computational aspects.
Since our main goal is determining existing but unknown objects it is very
important to be sure we have sufficient data. So uniqueness questions are of
exceptional importance here.
Most direct problems can be reduced to finding values y — F(x) of an
operator F acting from a topological space X into a topological space Y .
Usually F is continuous and X, Y are Banach spaces with norms || ||^
and || ||
. The inverse problem is then connected with the inverse operator
or with solving the equation
Many direct problems themselves are equivalent to such an equation. A
related problem is said to be well-posed (with respect to the pair X, Y) if