Faculty Publications
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Item A derivative free iterative method for the implementation of Lavrentiev regularization method for ill-posed equations(Kluwer Academic Publishers, 2015) Shubha, V.S.; George, S.; Padikkal, P.In this work, we develop a derivative free iterative method for the implementation of Lavrentiev regularization for approximately solving the nonlinear ill-posed operator equation F(x) = y. Convergence analysis shows that the method converges quadratically. Apart from being totally free of derivatives, the method, under a general source condition provide an optimal order error estimate. We use the adaptive method introduced in Pereverzyev and Schock (SIAM J. Numer. Anal. 43, 2060–2076, 2005) for choosing the regularization parameter. In the concluding section the method is applied to numerical solution of the inverse gravimetry problem. © 2014, Springer Science+Business Media New York.Item A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations(Elsevier Inc. usjcs@elsevier.com, 2015) Padikkal, P.; Shubha, V.S.; George, S.George and Elmahdy (2012), considered an iterative method which converges quadratically to the unique solution x?? of the method of Lavrentiev regularization, i.e., F(x) + ?(x - x0) = y?, approximating the solution x 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. © 2014 Elsevier Inc. All rights reserved.Item Finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill-posed equations with monotone operators(Elsevier Inc. usjcs@elsevier.com, 2016) Shubha, V.S.; George, S.; Padikkal, P.; Erappa, M.E.Recently Jidesh et al. (2015), considered a quadratic convergence yielding iterative method for obtaining approximate solution to nonlinear ill-posed operator equation F(x)=y, where F: D(F) ? X ? X is a monotone operator and X is a real Hilbert space. In this paper we consider the finite dimensional realization of the method considered in Jidesh et al. (2015). Numerical example justifies our theoretical results. © 2015 Elsevier Inc. All rights reserved.Item 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 Iterative regularization methods for ill-posed operator equations in Hilbert scales(Cambridge Scientific Publishers jonathan.mckenna@touchbriefings.com, 2017) Argyros, I.K.; George, S.; Padikkal, P.In this paper we report on a method for regularizing a nonlinear ill-posed operator equation in Hilbert scales. The proposed method is a combination of Lavrentiev regularization method and a Modified Newton's method in Hilbert scales . Under the assumptions that the operator F is continu- ously differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulfils a general source condition, we give an optimal order convergence rate result with respect to the general source function. © CSP - Cambridge, UK; I & S - Florida, USA, 2017.Item Third-order derivative-free methods in Banach spaces for nonlinear ill-posed equations(Springer Verlag service@springer.de, 2019) Shubha, V.S.; George, S.; Padikkal, P.We develop three third order derivative-free iterative methods to solve the nonlinear ill-posed oprerator equation F(x) = f approximately. The methods involve two steps and are free of derivatives. Convergence analysis shows that these methods converge cubically. The adaptive scheme introduced in Pereverzyev and Schock (SIAM J Numer Anal 43(5):2060–2076, 2005) has been employed to choose regularization parameter. These methods are applied to the inverse gravimetry problem to validate our developed results. © 2019, Korean Society for Computational and Applied Mathematics.