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Zbl 0842.65036
Scherzer, Otmar
Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. (English)
[J] J. Math. Anal. Appl. 194, No.3, 911-933 (1995). ISSN 0022-247X

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.\par 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.
[Yu Wenhuan (Tianjin)]

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
34A55 Inverse problems of ODE
47J25 Methods for solving nonlinear operator equations (general)
65J20 Improperly posed problems (numerical methods in abstract spaces)

Keywords: nonlinear ill-posed problem; convergence; iterative methods; nonlinear operator equation; Landweber iteration; inverse problem; diffusion coefficient

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Zentralblatt MATH Berlin [Germany]