Faculty Publications
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Item Reconstruction of signals by standard Tikhonov method(2011) George, S.; Padikkal, P.In this work we propose a standard Tikhonov regularization approach for obtaining the signal f from the observed signal ye. The observed signal is distorted by an additive noise or error e. Deviating from the usual assumption on the bound onItem Finite dimensional realization of a Tikhonov gradient type-method under weak conditions(Springer-Verlag Italia s.r.l. springer@springer.it, 2016) Shubha, V.S.; George, S.; Padikkal, P.In this paper we consider projection techniques to obtain the finite dimensional realization of a Tikhonov gradient type-method considered in George et al. (Local convergence of a Tikhonov gradient type-method under weak conditions, communicated, 2016) for approximating a solution x^ of the nonlinear ill-posed operator equation F(x) = y. The main advantage of the proposed method is that the inverse of the operator F is not involved in the method. The regularization parameter is chosen according to the adaptive method considered by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005). We also derive optimal stopping conditions on the number of iterations necessary for obtaining the optimal order of convergence. Using two numerical examples we compare our results with an existing method to justify the theoretical results. © 2016, Springer-Verlag Italia.Item Convergence of a Tikhonov Gradient Type-Method for Nonlinear Ill-Posed Equations(Springer, 2017) George, S.; Shubha, V.S.; Padikkal, P.In this study Tikhonov Gradient type-method is considered for nonlinear ill-posed operator equations. In our convergence analysis, we use hypotheses only on the first Frec?het derivative of F in contrast to the higher order Frec?het derivatives used in the earlier studies. We obtained ‘optimal’ order error estimate by choosing the regularization parameter according to the adaptive method proposed by Pereverzev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005). © 2017, Springer (India) Private Ltd.