Ill-posed problems -- or when the inexact is more exact
This is a new and interesting paper aimed at introducing to the practice of inverse problems based on a rather elementary level of mathematical elements. In particular, the authors present the phenomenon of ill-posedness as the typical property of inverse problems, where physical quantities as causes are not observable directly and have to be calculated from effects, that means from indirect measurements. On the one hand, they consider ill-posed problem in form of examples including a simple variant of computerized tomography. Here, the text is also appropriate for undergraduate students similar to the book of \textit{C. W.~Groetsch} [Inverse problems. Activities for undergraduates (1999; Zbl 1011.00003)] (see also the former book [Inverse problems in the mathematical sciences (Vieweg, Braunschweig) (1993; Zbl 0779.45001)]). On the other hand, they present basic ideas of regularization, in particular the Tikhonov regularization method, and they show the chances and limitations of such regularization technique by means of numerical case studies. Under different viewpoints some first ideas of numerical analysis of inverse problems are presented (for much more details see, e.g., \textit{A.~Kirsch} [An introduction to the mathematical theory of inverse problems (Springer, Berlin) (1996; Zbl 0865.35004)]). The second half of the paper focuses on the mathematical background. The variational viewpoint, the spectral viewpoint and also convergence rates under source conditions are briefly discussed and illustrated by tables.
scientific article; zbMATH DE number 2138140