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 <A u − f, v − u> ≥ 0 for all v ∈ 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 uεδ, η ∈ Kη for which <A uεδ, η + ε uεδ, η − fδ, v − uεδ, η> ≥ 0 for all v ∈ 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.