Many issues in science, technology, economics and ecology such as image
processing, computerized tomography, seismic tomography in engineering
geophysics, acoustic sounding in wave approximation, problems of linear
programming lead to solve problems having the following operator equation
type (A. Bakushinsky and A. Goncharsky, 1994; F. Natterer, 2001; F.
Natterer and F. W¨ubbeling, 2001):
A(x) = f; (0.1)
where A is an operator (mapping) from metric space E into metric space Ee
and f 2 Ee. However, there exists a class of problems among these problems
that their solutions are unstable according to the initial data, i.e., a small
change in the data can lead to a very large difference of the solution. It is
said that these problems are ill-posed. Therefore, the requirement is that
there must be methods to solve ill-posed problems such that the smaller
the error of the data is, the closer the approximate solution is to the correct
solution of the derived problem. If Ee is Banach space with the norm k:k
then in some cases of the mapping A, the problem (0.1) can be regularized
by minimizing Tikhonov’s functional:
Fδ
α(x) = kA(x) − fδk2 + αkx − x+k2; (0.2)
