A simplified iterated Lavrentiev regularization method for nonlinear ill-posed monotone operator equations under a heuristic rule

In this paper, we propose a simplified iterative Lavrentiev regularization approach for computing stable approximate solutions of the nonlinear ill-posed operator equation F(x)=y, where F : D(F)⊂ X → X is a nonlinear monotone operator defined on a Hilbert space X. For iterative regularization methods, the choice of a suitable stopping rule is a key issue since it strongly influences the stability and accuracy of the computed solution. In many practical situations, the exact level of noise present in the data is either unknown or unreliable, which makes classical a priori and a posteriori stopping rules difficult to apply. To address this difficulty, Q. Jin and W. Wang introduced in 2018 a heuristic stopping rule for the iteratively regularized Gauss–Newton method. Motivated by their work, we propose a heuristic selection rule for a simplified version of the Lavrentiev regularization method. The main advantage of the proposed scheme is that it requires the computation of the Fréchet derivative of the operator F only once, namely at an initial approximation x₀ of the exact solution x^. Under suitable assumptions that control the nonlinearity of the operator, we derive error estimates for the proposed method. Finally, we illustrate the practical behavior of the method by applying it to a nonlinear integral operator problem.