Convergence rates of a penalized variational inequality method for nonlinear monotone ill-posed equations in Hilbert spaces

TL;DR

This paper analyzes the convergence rates of a penalized variational inequality method with Lavrentiev regularization for solving nonlinear monotone ill-posed equations in Hilbert spaces, providing new error estimates under specific conditions.

Contribution

It introduces new error estimates for a penalized variational inequality approach with Lavrentiev regularization, applicable when the operator is cocoercive and the solution has an adjoint source representation.

Findings

Derived new convergence rate estimates for the method.

Validated the approach with numerical experiments.

Abstract

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.