Faculty Publications
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Item Expanding the applicability of a modified Gauss-Newton method for solving nonlinear ill-posed problems(2013) Argyros, I.K.; George, S.We expand the applicability of a modified Gauss-Newton method recently presented in George (2013) [19] for approximate solution of a nonlinear ill-posed operator equation between two Hilbert spaces. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in earlier studies such as George (2013, 2010) [19,18]. 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. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 2013 Elsevier Inc. All rights reserved.Item Unified convergence domains of Newton-like methods for solving operator equations(Elsevier Inc. usjcs@elsevier.com, 2016) Argyros, I.K.; George, S.We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study. © 2016 Elsevier Inc. All rights reserved.Item Expanding the applicability of an iterative regularization method for ill-posed problems(Biemdas Academic Publishers, 2019) Argyros, I.K.; George, S.An iteratively regularized projection method, which converges quadratically, is considered for stable approximate solutions to a nonlinear ill-posed operator equation F(x) = y, where F : D(F) ? X ? X is a nonlinear monotone operator defined on the real Hilbert space X. We assume that only a noisy data y? with ky? y? k ? ? are available. Under the assumption that the Fréchet derivative F0 of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that, under a general source condition on x0 ? x, the error kxn h ? ? ? xk between the regularized approximation xn h ? ? , (x0 h ? ? := Phx0, where Ph is an orthogonal projection on to a finite dimensional subspace Xh of X) and the solution x is of optimal order. © 2019 Journal of Nonlinear and Variational AnalysisItem An apriori parameter choice strategy and a fifth order iterative scheme for Lavrentiev regularization method(Institute for Ionics, 2023) George, S.; Saeed, M.; Argyros, I.K.; Padikkal, J.In this paper, we propose a new source condition and introduce a new apriori parameter choice strategy for Lavrentiev regularization method for nonlinear ill-posed operator equation involving a monotone operator in the setting of a Hilbert space. Also, a fifth order iterative method is being proposed for approximately solving Lavrentiev regularized equation. A numerical example is illustrated to demonstrate the performance of the method. © 2022, The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics.Item Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2024) Argyros, I.K.; George, S.; Regmi, S.; Argyros, C.I.Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines. © 2024 by the authors.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.