A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations

Abstract

George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution xαδ of the method of Lavrentiev regularization, i.e., F(x)+α(x-x0)=yδ, approximating the solution xˆ of the ill-posed problem F(x)=y where F:D(F)⊆X⟶X is a nonlinear monotone operator defined on a real Hilbert space X. The convergence analysis of the method was based on a majorizing sequence. In this paper we are concerned with the problem of expanding the applicability of the method considered by George and Elmahdy (2012) by weakening the restrictive conditions imposed on the radius of the convergence ball and also by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as George and Elmahdy (2012), Mahale and Nair (2009), Mathe and Perverzev (2003), Nair and Ravishankar (2008), Semenova (2010) and Tautanhahn (2002). We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining order optimal error estimate. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem.