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Continuous methods of stable approximation of solutions to non-linear equations in Hilbert space based on a regularized Gauss-Newton scheme. (Russian, English) Zbl 1083.47515
Zh. Vychisl. Mat. Mat. Fiz. 44, No. 1, 8-17 (2004); translation in Comput. Math. Math. Phys. 44, No. 1, 6-15 (2004).
The nonlinear equation of the form \[ F(x)=0, \qquad x \in H_1, \tag{1} \] where \(F(x)\) is a nonlinear operator acting from the Hilbert space \(H_1\) to the Hilbert space \(H_2\), is considered. It is supposed that instead of the exact operator \(F(x)\) in (1), an approximate operator \(\tilde F : H_1 \to H_2\) is available. Here, a class of approximation methods for the solution of nonlinear equations with a smooth non-exactly given operator in Hilbert space is constructed and investigated. It is supposed that a derivative of the operator has no regularity. The regularized Gauss-Newton scheme is used for the construction of continuous methods of stable approximation.

47J06 Nonlinear ill-posed problems
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization