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Zbl 0895.35109
Hohage, Thorsten
Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem. (English)
[J] Inverse Probl. 13, No.5, 1279-1299 (1997). ISSN 0266-5611

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.
[A.Kirsch (Karlsruhe)]

MSC 2000:: *35R30 Inverse problems for PDE
65J15 Equations with nonlinear operators (numerical methods)
35P25 Scattering theory (PDE)
35R25 Improperly posed problems for PDE

Keywords: regularized Gauss-Newton method; numerical experiments; convergence rates

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Zentralblatt MATH Berlin [Germany]