Iterative Regularization Methods for Nonlinear Ill-Posed Problems
This book deals with regularization techniques for stabilizing ill-posed non-linear inverse problems and their numerical realisation via iterative procedures. After a short introduction, chapter 2 is devoted to the nonlinear Landweber iteration and analysis of its convergence and convergence rates. Chapter 3 deals with modified Landweber methods showing how to get better convergence rates or to get results under weaker assumptions (regularization in Hilbert scales, iteratively regularized Landweber iteration and the Landweber-Kaczmarz method). To make numerical realization more efficient, the authors propose in chapter 4 several variants of Newton type methods (the Levenberg-Marquardt method, the iteratively regularized Gauss-Newton method and its generalisations and Broyden´s method for ill-posed problems). Chapter 5 is devoted to another approach increasing efficiency of numerical realisation of ill-posed problems, namely to multigrid methods. Since many practical problems are related to shape recovery, chapter 6 describes applications of the level set methods and their adaptation for solving inverse problems. Finally, chapter 7 presents two applications: reconstruction of transducer pressure fields from Schlieren tomography and a parameter estimation problem from nonlinear magnetics. The book is intended as a text book for graduate students and for specialists working in the field of inverse problems.