Regularization Theory for Ill-posed Problems: Selected Topics: 58

The theory of inverse problems has a wide variety of applications because any mathematical model needs to be calibrated before it can be used, and such a calibration is a typical inverse problem. The regularization theory, in its turn, is the algorithmic part of the theory of inverse problems. It provides and analyzes the methods for dealing with ill-posedness that is one of the main issues for inverse problems.

In spite of a growing number of monographs on the regularization theory there are quite a few topics that have been developed only recently and are not reflected yet in this literature. The present book is motivated by some of them. The first novelty of this book is that it analyzes simultaneously the ill-posed problems with deterministic and stochastic data noises. The chapter on regularization algorithms in the learning theory and a chapter on multi-parameter regularization are other features, which also distinguish this book from existing monographic literature on inverse problems. Furthermore Chapter 5 of the present book is the first attempt in the monographic literature to analyze the adaptive choice of the regularization space. It also demonstrates a meta-learning based approach to regularization on a problem from diabetes technology, but it will be also seen how its main ingredients can be exploited in other applications. At the same time the material of the chapter describes one of the first applications of the regularization theory in the diabetes therapy management.

Such a context makes the book of interest for a wide audience within the inverse problems community and beyond. The first part of the book can be also recommended for the use in lecture courses while the second part can be seen as a presentation of some further developments of the basic theory. This material is new in monographic literature on regularization theory and can be used in student’s seminars.