Regularization and stabilization of inverse problems (Q2702473)

The paper is a review of the main topics of regularization theory for a certain class of inverse problems. First, the author gives a short overview of inverse problems for partial differential equations. The ill-posedness of implicit as well as explicit problems is illustrated by the example of the one-dimensional heat equation. NEWLINENEWLINENEWLINEThe ill-posed problems are considered in the framework of a general Hilbert space. The author surveys certain basic theories of the Tikhonov regularization for the implicit problem \(Kf=g\) including such questions as convergence, parameter choice rules, discretization, and iterated regularization. Moreover, for the evaluation of \(Lg\) where \(L\) is an unbounded linear operator and \(g\) is not directly available, related stabilization techniques are described. NEWLINENEWLINENEWLINEAt the end, a brief sketch of the convergence theory of Tikhonov regularization for nonlinear ill-posed problems is given. A rich list of references comprises 74 items.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].