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Inverse und schlecht gestellte Probleme. (Inverse and ill-posed problems). (German) Zbl 0667.65045
Teubner Studienb├╝cher: Mathematik. Stuttgart: B. G. Teubner. 205 S. DM 26.80 (1989).
In the first chapter the author describes linear inverse problems, at hand of some examples including computer tomography and an inverse scattering problem. He outlines typical difficulties arising in solving these problems. Inverse problems can be effectively solved only by use of regularization techniques. The mathematical tools for deriving such methods are summarized in chapter two.
The next chapter is devoted to the classification and stabilization of linear ill-posed problems. In chapter four the most important regularization methods are discussed such as filtering techniques, Tikhonov-Philipps regularization, Landweber iteration, conjugate gradient method, stochastic regularization techniques and projection methods.
The numerical realization of these methods is treated in chapter five. Finally, the last chapter is devoted to the study of computer tomography, especially to the ill-posedness of the Radon transform and the filtered backprojection method as a reconstruction technique.
Reviewer: A.Neubauer

65J10 Numerical solutions to equations with linear operators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Rxx Numerical methods for integral equations, integral transforms
44A15 Special integral transforms (Legendre, Hilbert, etc.)
45B05 Fredholm integral equations
65R20 Numerical methods for integral equations
47A50 Equations and inequalities involving linear operators, with vector unknowns
93Exx Stochastic systems and control