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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]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.

65J15Equations with nonlinear operators (numerical methods)
34A55Inverse problems of ODE
47J25Iterative procedures (nonlinear operator equations)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)