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Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions. (English) Zbl 1129.65043

The authors consider the nonlinear ill-posed operator equation F(x)=y, where F:D(F)Y is injective and continuously FrĂ©chet differentiable on its domain D(F)X; X, Y are Hilbert spaces. They assume that there exists a solution x of the equation, and that only noisy data y δ satisfying y δ -yδ are available. To iteratively compute an approximation to x they replace the n-th Newton step by the linearized equation

F ' [x n δ ]h n =y δ -F(x n δ ),n=0,1,2,

where h n =x n+1 δ -x n δ . Since the linearized equation inherits the ill-posedness of the initial equation the authors apply a Tikhonov regularization with the initial guess x 0 -x n δ , called the regularized Gauss-Newton method (IRGNM). The following problems are studied: Convergence of the IRGNM for exact data; IRGNM with discrepancy principle for nonlinear problems; Solving the linearized equation.

MSC:
65J15Equations with nonlinear operators (numerical methods)
65J20Improperly posed problems; regularization (numerical methods in abstract spaces)
65J22Inverse problems (numerical methods in abstract spaces)
47J06Nonlinear ill-posed problems