Newton-type regularization methods for nonlinear inverse problems

摘要: Inverse problems arise whenever one searches for unknown causes based on observation of their effects. Such are usually ill-posed in the sense that solutions do not depend contin- uously data. In practical applications, never has exact data; instead only noisy data available due to errors measurements. Thus, development stable methods solving inverse is an important topic. last two decades, many have been developed nonlinear problems. Due straightforward implementation and fast convergence property, more attention paid Newton-type regularization including general iteratively regularized Gaus-newton inexact Newton methods. The Gauss-Newton method was proposed by Bakushinski Hilbert spaces, quickly generalized its form. These produce all iterates some trust regions centered around initial guess. property explored under either a priori or posteriori stopping rules. We will present our recent results when discrepancy principle used terminate iteration. initiated Hanke then Rieder solve spaces. contrast Gauss- methods, such next iterate region current regularizing local linearized equations. An approximate solution output principle. Although numerical simulation indicates they quite efficient, long time it open problem whether order optimal. report work confirm indeed situations, formulated space setting may good since tend smooth thus destroy special feature solution. On other hand, can be naturally Banach spaces than Therefore, necessary develop framework By making use duality mappings Bregman distance we indicate how formulate corresponding results.