Faculty Publications
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Item An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems(2014) Vasin, V.; George, S.In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F(x)=y where F:D(F)?X?X is a nonlinear monotone operator or F?(x0) is nonnegative selfadjoint operator defined on a real Hilbert space X. We assume that only a noisy data y??X with ?y- y???? are available. Further we assume that Fréchet derivative F? of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x0-x?, the error ?x?-xn,??? between the regularized approximation xn,??(x0,??;=x0) and the solution x? is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem. © 2013 Elsevier Inc. All rights reserved.Item Expanding the applicability of Tikhonov's regularization and iterative approximation for ill-posed problems(Walter de Gruyter GmbH, 2014) Vasin, V.; George, S.Recently, Vasin [J. Inverse Ill-Posed Probl. 21 (2013), 109-123] considered a new iterative method for approximately solving nonlinear ill-posed operator equation in Hilbert spaces. In this paper we introduce a modified form of the method considered by Vasin. This paper weakens the conditions needed in the existing results. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in [J. Inverse Ill-Posed Probl. 21 (2013), 109-123]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Order optimal error bounds are given in case the regularization parameter is chosen a priori and by the adaptive method of Pereverzev and Schock [SIAM J. Numer. Anal. 43 (2005), 2060-2076]. A numerical example of a nonlinear integral equation proves the efficiency of the proposed method. © 2014 by De Gruyter.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 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.Item On the Implementation of Iterative Methods Without Inverse Updating for Solving Equations in Banach Spaces(World Scientific, 2025) Argyros, I.K.; George, S.The implementation of iterative methods using inverses to solve equations is a computationally expensive or impossible task in general. This is the case, since the analytical form of the inverse is difficult to find in practice. That is why, we replace the inverse by a sum of linear operators which is well defined. The convergence of the resulting hybrid methods is studied based on majorizing sequences under generalized continuity assumptions on the operators involved and in the setting of a Banach space. It is demonstrated by numerical experimentations that the convergence order as well as the number of iterations required to obtain a predetermined error tolerance when comparing the original to the hybrid method is essentially the same. © 2025 World Scientific Publishing Company.