Convergence rates of a penalized variational inequality method for nonlinear monotone ill-posed equations in Hilbert spaces
Provides theoretical convergence guarantees for a regularization method applicable to monotone problems where standard Lavrentiev fails, benefiting inverse problems and optimization communities.
The paper establishes new error estimates for Lavrentiev regularization of variational inequalities for nonlinear monotone ill-posed equations in Hilbert spaces, under cocoercivity and source conditions. Numerical experiments confirm the theoretical rates.
We consider perturbed nonlinear ill-posed equations in Hilbert spaces, with operators that are monotone on a given closed convex subset. A simple stable approach is Lavrentiev regularization, but existence of solutions of the regularized equation on the given subset can be guaranteed only under additional assumptions that are not satisfied in some applications. Lavrentiev regularization of the related variational inequality seems to be a reasonable alternative then. For the latter approach, in this paper we present new error estimates for suitable a priori parameter choices, if the considered operator is cocoercive and if in addition the solution admits an adjoint source representation. Some numerical experiments are included.