Semismooth Newton and Quasi-Newton Methods in Weighted l−Regularization of Nonlinear Inverse Problems

In this paper, we investigate the semismooth Newton and quasi-Newton methods for the minimization problem in the weighted `−regularization of nonlinear inverse problems. We propose the conditions for obtaining the convergence of two methods. The semismooth Newton method is proven to locally converge with superlinear rate and the semismooth quasi-Newton method is proven to locally converge at least with linear rate. Two methods are presented as active set methods as well. For using the semismooth quasi-Newton method in practice, we propose two specific cases. The first one returns to a gradient-type method with Barzilai-Borwein rule for step-sizes. The second one based on Broyden’s method is proven to converge and its convergence rate is superlinear in finite dimensional spaces. Finally, the efficiency of the methods are illustrated in a parameter identification problem in elliptic equations.