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Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem. (English) Zbl 0895.35109
The author studies nonlinear equations of the form F(x)=y when F is Fréchet-differentiable between Hilbert spaces X and Y but F ' (x ) fails to be boundedly invertible at the solution x . This situation characterizes nonlinear illposed problems and arises in the study of inverse potential and inverse scattering problems. In this paper, logarithmic convergence rates of the iteratively regularized Gauß-Newton method are proven under a relatively weak source condition for the solution. For the inverse potential and the inverse scattering problem and the case of the obstacle being a circle, the author interpretes this condition as a smoothness assumption. A few numerical experiments show the applicability of the method with the expected convergence rates.
MSC:
35R30Inverse problems for PDE
65J15Equations with nonlinear operators (numerical methods)
35P25Scattering theory (PDE)
35R25Improperly posed problems for PDE