Global convergence of damped semismooth Newton methods for ℓ1 Tikhonov regularization

Abstract

We are concerned with Tikhonov regularization of linear ill-posed problems with ℓ1 coefficient penalties. Griesse and Lorenz (2008 Inverse Problems 24 035007) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton methods are particularly competitive due to their superlinear convergence properties and their ability to solve piecewise affine equations exactly within finitely many iterations. However, the convergence of semismooth Newton schemes is only local in general. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for ℓ1 Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.