Faculty Publications
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Item Expanding the applicability of a two step Newton-type projection method for ILL-posed problems(Adam Mickiewicz University Press Nowowiejskiego 55 Poznan 61-734, 2014) Argyros, I.K.; Erappa, M.E.; George, S.There are many classes of ill-posed problems that cannot be solved with existing iterative methods, since the usual Lipschitz-type assumptions are not satisfied. In this study, we expand the applicability of a two step Newton-type projection method considered in [10], [11], using weaker assumptions. Numerical examples for the method and examples where the old assumptions are not satisfied but the new assumptions are satisfied are provided at the end of this study.Item Cubic convergence order yielding iterative regularization methods for ill-posed Hammerstein type operator equations(Springer-Verlag Italia s.r.l. springer@springer.it, 2017) Argyros, I.K.; George, S.; Erappa, S.M.For the solution of nonlinear ill-posed problems, a Two Step Newton-Tikhonov methodology is proposed. Two implementations are discussed and applied to nonlinear ill-posed Hammerstein type operator equations KF(x) = y, where K defines the integral operator and F the function of the solution x on which K operates. In the first case, the Fre´ chet derivative of F is invertible in a neighbourhood which includes the initial guess x0 and the solution x^. In the second case, F is monotone. For both cases, local cubic convergence is established and order optimal error bounds are obtained by choosing the regularization parameter according to the the balancing principle of Pereverzev and Schock (2005).We also present the results of computational experiments giving the evidence of the reliability of our approach. © 2016, Springer-Verlag Italia.