Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates

Abstract

Let H be a Hilbert space and K be a nonempty closed convex subset of H. For f ∈ H, we consider the (ill-posed) problem of finding u ∈ K for which 〈Au - f, υ - u〉 ≥ 0 for all υ ∈ K, where A: H → H is a monotone (not necessarily linear) operator. We study the approximation of the solutions of the variational inequality by using the following perturbed variational inequality: for fδ ∈ H, ∥fδ - f∥ ≤ δ, find uδ,ηε ∈Kη for which 〈Auδ,ηε + εuδ,ηε - fδ, υ - uδ,ηε〉 ≥ 0 for all υ ∈ Kη, where ε, δ, and η are positive parameters, and Kη, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(ε1/3) of convergence of the solutions of the regularized variational inequalities to a solution of the original variational inequality using the Mosco approximation of closed convex sets, where A is a weakly differentiable inverse-strongly-monotone operator.