A convergence analysis of the Landweber iteration for nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems
zbMATH Open document ID: 0840.65049
author: Martin Hanke
author: Otmar Scherzer
author: Andreas Neubauer
published in: Numerische Mathematik
publication date: 1996-06-30T00:00:00Z
review text: Let \(X\) and \(Y\) be Hilbert spaces, \(F : {\mathcal D}(F) \subset X \to Y\) a nonlinear differentiable operator. To solve the equation \(F(x) = y\), the authors apply the Landweber method \[ x_{k + 1} = x_k + F'(x_k)(y - F(x_k)),\quad k = 0,1,2,\dots. \] The authors prove the convergence of the method (with the discrepancy stopping corresponding to inexact data \(y\)) under the conditions \[ |F'(x)|\leq 1,\quad |F(x) - F(\widetilde{x}) - F'(x) (x - \widetilde{x}) |\leq \eta |F(x) - F(\widetilde{x})|,\quad \eta < 1/2. \] They present examples of different problems ( Hammerstein integral equation, distributed parameter problem) where the last condition is fulfilled. Under further conditions, they also study the convergence rate of the method.
Mathematics Subject Classification ID: 65J20
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 839147
zbMATH Keywords: Landweber iteration
zbMATH Keywords: nonlinear ill-posed problems
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: convergence
zbMATH Keywords: Hammerstein integral equation
zbMATH Keywords: distributed parameter problem
reviewed by: G. Vainikko
MaRDI profile type: Publication
full work available at URL: s002110050158
OpenAlex ID: W2080888649
DOI: 10.1007/S002110050158
description: scientific article; zbMATH DE number 839147