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In this paper, we consider a modified Levenberg--Marquardt method for solvingan ill-posed inverse problem where the forward mapping is not G\^ateauxdifferentiable. By relaxing the standard assumptions for the classical smoothsetting, we derive asymptotic stability estimates that are then used to provethe convergence of the proposed method. This method can be applied to aninverse source problem for a non-smooth semilinear elliptic PDE where aBouligand subdifferential can be used in place of the non-existing Fr\'echetderivative, and we show that the corresponding Bouligand-Levenberg-Marquardtiteration is an iterative regularization scheme. Numerical examples illustratethe advantage over the corresponding Bouligand-Landweber iteration.

Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem