Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems.

*(English)*Zbl 0842.65036The author considers the convergence of iterative methods for solving a nonlinear operator equation $F\left(x\right)=y$ by using the method of Landweber iteration, which is defined by the iterative scheme

$${x}_{k+1}={x}_{k}-{f}^{\text{'}}{\left({x}_{k}\right)}^{*}[f\left({x}_{k}\right)-y]\equiv U\left({x}_{k}\right)\xb7$$

The author shows that if the functions $U$ and $f$ satisfy some conditions then the iterative schemes (weakly or strongly) converge to a solution of the original equation. Moreover, the author gives conditions guaranteeing that the iterative scheme is convergent in the case of inexact data $y$ and gives the convergence rates for ill-posed problems.

Finally, the author applies the results to an inverse problem for identifying the diffusion coefficient in a boundary-valued problem of an ordinary differential equation of order two.

Reviewer: Yu Wenhuan (Tianjin)

##### MSC:

65J15 | Equations with nonlinear operators (numerical methods) |

34A55 | Inverse problems of ODE |

47J25 | Iterative procedures (nonlinear operator equations) |

65J20 | Improperly posed problems; regularization (numerical methods in abstract spaces) |