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A simplified generalized Gauss-Newton method for nonlinear ill-posed problems. (English) Zbl 1198.65101
Summary: Iterative regularization methods for nonlinear ill-posed equations of the form F(x)=y, where F:D(F)XY is an operator between Hilbert spaces X and Y, usually involve calculation of the Fréchet derivatives of F at each iterate and at the unknown solution x . In this paper, we suggest a modified form of the generalized Gauss-Newton method which requires the Fréchet derivative of F only at an initial approximation x 0 of the solution x . The error analysis for this method is done under a general source condition which also involves the Fréchet derivative only at x 0 . The conditions under which the results of this paper hold are weaker than those considered by B. Kaltenbacher [Numer. Math. 79, No. 4, 501–528 (1998; Zbl 0908.65042)] for an analogous situation for a special case of the source condition.
MSC:
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
35R30Inverse problems for PDE