A $\operatorname{prox}$-Based Semi-Smooth Newton Method for TV-Minimization

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method for the non-differentiable TV-minimization problem. To this end, the primal-dual optimality conditions are reformulated as a nonlinear operator equation with Newton-(type-)differentiable structure. We investigate the question of well-posedness of the resulting semi-smooth Newton scheme in the infinite-dimensional setting and identify structural properties of the associated Newton-type derivatives. For a conforming finite element discretization, we prove that the resulting semi-smooth Newton method is globally well-posed and locally super-linearly convergent. The approach extends to a large class of convex minimization problems, coincides with established semi-smooth Newton methods for obstacle problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is well-posed in the infinite-dimensional setting. Numerical experiments indicate a robust practical performance of the proposed method, including reliable reduction of the discrete primal-dual gap estimator to machine precision, robustness with respect to the choice of proximity parameters, an improved convergence basin compared to a canonical primal semi-smooth Newton method, and effective performance even for quadratically graded meshes using only a mesh-independent initialization criterion.