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

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, where F : D(F)subset of X -> 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 Frechet derivatives at each iteration. In this paper, we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Frechet derivative only at the point x(0), i. e., at the initial approximation of the exact solution x(+). Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Frechet derivative at the point x 0, instead at the unknown exact solution x(+) as in [12].