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Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. (English) Zbl 0842.65036
The author considers the convergence of iterative methods for solving a nonlinear operator equation \(F(x)= y\) by using the method of Landweber iteration, which is defined by the iterative scheme \[ x_{k+ 1}= x_k- f'(x_k)^* [f(x_k)- y]\equiv U(x_k). \] 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.

MSC:
65J15 Numerical solutions to equations with nonlinear operators
34A55 Inverse problems involving ordinary differential equations
47J25 Iterative procedures involving nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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