Levenberg–Marquardt method in Banach spaces with general convex regularization terms

We propose a Levenberg–Marquardt method with general uniformly convex regularization terms to solve nonlinear inverse problems in Banach spaces, which is an extension of the scheme proposed by Hanke in (Inverse Probl 13:79–95, 1997) in Hilbert space setting. The method is so designed that it can be used to detect the features of the sought solutions such as sparsity or piecewise constancy. It can also be used to deal with the situation that the data is contaminated by noise containing outliers. By using tools from convex analysis in Banach spaces, we establish the convergence of the method. Numerical simulations are reported to test the performance of the method.