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 Discretized Newton-Tikhonov method for ill-posed hammerstein type equations(International Publications internationalpubls@yahoo.com, 2016) Argyros, I.K.; George, S.; Erappa, M.E.George and Shobha (2012) considered the finite dimensional realization of an iterative method for non-linear ill-posed Hammerstein type operator equation KF(x) = f, when the Fréchet derivative F' of the non-linear operator F is not invertible. In this pa- per we consider the special case i.e., F'-1 exists and is bounded. We analyze the convergence using Lipschitz-type conditions used in [10], [13], [22] and also analyze the convergence using a center type Lipschitz condition. The center type Lipschitz con- dition provides a tighter error estimate and expands the applicability of the method. Using a logarithmic-type source condition on F(x0)-F(?) (here ? is the actual solution of KF(x) = f) we obtain an optimal order convergence rate. Regularization param- eter is chosen according to the balancing principle of Pereverzev and Schock (2005). Numerical illustrations are given to prove the reliability of our approach.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.