Bouligand-Landweber iteration for a non-smooth ill-posed problem
This work provides a theoretical foundation for iterative regularization of non-smooth ill-posed problems, which is a challenging class of inverse problems.
The paper develops a convergence analysis for a modified Landweber method using Bouligand subderivatives to regularize non-smooth nonlinear ill-posed problems where the forward mapping is not Gâteaux differentiable. The method is shown to converge strongly for an inverse source problem with Lipschitz continuous nonlinearity under a generalized tangential cone condition.
This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand--Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated