A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. (English) Zbl 0840.65049

Hanke, Martin; Neubauer, Andreas; Scherzer, Otmar

Numer. Math. 72, No. 1, 21-37 (1995).

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.

Reviewer: G.Vainikko (Espoo)

MSC:

65J20	Numerical solutions of ill-posed problems in abstract spaces; regularization
65J15	Numerical solutions to equations with nonlinear operators
45G10	Other nonlinear integral equations
65R30	Numerical methods for ill-posed problems for integral equations
47J25	Iterative procedures involving nonlinear operators

Keywords:

Landweber iteration; nonlinear ill-posed problems; Hilbert spaces; convergence; Hammerstein integral equation; distributed parameter problem