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Regularization of Nonlinear Ill-Posed Variational Inequalities and Convergence Rates
Authors:
Fengshan Liu and M Zuhair Nashed
Institution:
(1) Department of Mathematics and Computer Science, Delaware State University, Dover, DE, 19901, U.S.A.;(2) Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, U.S.A.
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
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 f
H,
f
– f
, find u
,
K
for which
,
+
u
,
– f
, v – u
,
>
0 for all v
K
, where
,
, and
are positive parameters, and K
, a perturbation of the set K, is a nonempty closed convex set in H. We establish convergence and a rate O(
1 / 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.
Keywords:
variational inequality
inverse ill-posed problem
monotone operator
degree of ill-posedness
nonlinear
convex set
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