ISSN:
1572-932X
Keywords:
variational inequality
;
inverse ill-posed problem
;
monotone operator
;
degree of ill-posedness
;
nonlinear
;
convex set
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008643727926
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