Volume 51, pp. 274-314, 2019.

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

Christian Clason and Vu Huu Nhu

Abstract

In this paper, we consider a modified Levenberg-Marquardt method for solving an ill-posed inverse problem where the forward mapping is not Gâteaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates which are then used to prove convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fréchet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.

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Key words

inverse problem, iterative regularization, Levenberg-Marquardt method, non-smooth equation

AMS subject classifications

49K20, 49K40, 90C31

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