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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 $F'(x^\dag)$ fails to be boundedly invertible at the solution $x^\dag$. 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.

35R30Inverse problems for PDE
65J15Equations with nonlinear operators (numerical methods)
35P25Scattering theory (PDE)
35R25Improperly posed problems for PDE
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