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Non-linear Inverse

Non-linear Inverse - Nonlinear Inverse Problems Theoretical...

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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 field 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 ill-posed 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 first 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 identification”. 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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2 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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