Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations

Abstract

Abstract Mahale and Nair [12] considered an iterated form of Lavrentiev regularization for obtaining stable approximate solutions for ill-posed nonlinear equations of the form F ⁢ ( x ) = y ${F(x)=y}$ , where F : D ⁢ ( F ) ⊆ X → X ${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate under a general source condition. However, the iterations defined in [12] require calculation of Fréchet derivatives at each iteration. In this paper, we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fréchet derivative only at the point x 0 ${x_{0}}$ , i.e., at the initial approximation of the exact solution x † ${x^{\dagger}}$ . Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fréchet derivative at the point x 0 ${x_{0}}$ , instead at the unknown exact solution x † ${x^{\dagger}}$ as in [12].