“The Book... and (some of the) things we didn't know at that time”
In the fall of 1992, shortly after my PhD, Heinz Engl offered me the opportunity to come as a visiting scholar for a semester to the university of Linz. This trip, which provided a major boost to my academic career, ended with the invitation to join in as an author of a (now well-known) monograph on ill-posed problems, that appeared about three years later, eventually.
In this talk I will recapitulate some personal memories of that time, before turning to a more technical scientific matter that happens to be largely missing in this book, as its relevance had not been foreseen in those days – the issue of monotonicity.
Although monotonicity arguments were used in the ill-posed problems community already by Engl and Gfrerer in their seminal paper on the choice of the regularization parameter in Tikhonov's method, and by Defrise and de Mol in a note on the Landweber iteration, the full power of this argument only developed shortly after the publication of our book: For linear problems this culminated into the so-called monotonicity rule by Hämarik and Tautenhahn, whereas for nonlinear ill-posed problems this technique is now a standard argument in the analysis of various iterative regularization methods. We provide a brief account of this technique and review some of its major achievements.