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Nonlinear Inverse Problems: Theoretical
Aspects and Some Industrial Applications
Heinz W. Engl
1
,
2
and Philipp K¨ugler
2
1
Johann Radon Institute for Computational and Applied Mathematics, Austrian
Academy of Sciences, A–4040 Linz, Austria
[email protected]
2
Institut f¨ur Industriemathematik, Johannes Kepler Universit¨
at, A–4040 Linz,
Austria
[email protected]
1 Introduction
Driven by the needs from applications both in industry and other sciences,
the Feld of inverse problems has undergone a tremendous growth within the
last two decades, where recent emphasis has been laid more than before on
nonlinear problems. This is documented by the wide current literature on reg
ularization methods for the solution of nonlinear illposed problems. Advances
in this theory and the development of sophisticated numerical techniques for
treating the direct problems allow to address and solve industrial inverse prob
lems on a level of high complexity.
Inverse problems arise whenever one searches for causes of observed or de
sired eﬀects. Two problems are called inverse to each other if the formulation
of one problem involves the solution of the other one. These two problems
then are separated into a direct and an inverse problem. At Frst sight, it
might seem arbitrary which of these problems is called the direct and which
one the inverse problem. Usually, the direct problem is the more classical
one. E.g., when dealing with partial diﬀerential equations, the direct problem
could be to predict the evolution of the described system from knowledge of
its present state and the governing physical laws including information on all
physically relevant parameters while a possible inverse problem is to estimate
(some of) these parameters from observations of the evolution of the system;
this is called ”parameter identiFcation”. Sometimes, the distinction is not so
obvious: e.g., diﬀerentiation and integration are inverse to each other, it would
seem arbitrary which of these problems is considered the direct and the in
verse problem, respectively. But since integration is stable and diﬀerentiation
is unstable, a property common to most inverse problems, one usually consid
ers integration the direct and diﬀerentiation the inverse problem. Note also
that integration is a smoothing process, which is inherently connected with
the instability of diﬀerentiation.
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Heinz W. Engl and Philipp K¨ugler
Other important classes of inverse problems are
•
(Computerized) tomography
(cf. [Nat86]), which involves the reconstruc
tion of a function, usually a density distribution, from values of its line
integrals and is important both in medical applications and in nondestruc
tive testing [ELR96b]. Mathematically, this is connected with the inversion
of the Radon transform.
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