Regularization of nonlinear ill-posed variational inequalities and convergence rates. (English) Zbl 0924.49009

Summary: Let \(H\) be a Hilbert space and \(K\) be a nonempty closed convex subset of \(H\). For \(f\in H\), we consider the (ill-posed) problem of finding \(u\in K\) for which \[ \langle Au- f,v-u\rangle\geq 0\quad\text{for all }v\in K, \] where \(A:H\to H\) 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 \(f_\delta\in H\), \(\| f_\delta- f\|\leq\delta\), find \(u^{\delta,\eta}_\varepsilon\in K_\eta\) for which \[ \langle Au^{\delta,\eta}_\varepsilon+\varepsilon u^{\delta,\eta}_\varepsilon- f_\delta, v- u^{\delta,\eta}_\varepsilon\rangle\geq 0\quad\text{for all }K_\eta, \] where \(\varepsilon\), \(\delta\), and \(\eta\) are positive parameters, and \(K_\eta\), a perturbation of the set \(K\), is a nonempty closed convex set in \(H\). We establish convergence and a rate \(O(\varepsilon^{1/3})\) 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.


49J40 Variational inequalities
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65J15 Numerical solutions to equations with nonlinear operators
47H05 Monotone operators and generalizations
Full Text: DOI