An Inverse and Derivative Free Regularized Iterative Scheme for Nonlinear ill-Posed Mootone Operators
DOI:
https://doi.org/10.18311/jims/2024/33334Keywords:
Nonlinear Ill-Posed Problems, Regularization, Iterative Method, Parameter Choice Rule.Abstract
In this paper, we consider an inverse and derivative-free regularized iterative scheme for solving nonlinear ill-posed problems involving monotone operators based on the contraction principle in the Hilbert space. We prove that the scheme converges to the Lavrentiev solution. The salient features of this scheme are: (i.) convergence analysis and desired convergence rate require only weaker assumptions compared to many assumptions used in the standard scheme in literature, (ii.) a parameter choice strategy that gives the same convergence rate as that of an priori method without using the source condition (iii.) computation of an optimal order regularization parameter O(δ 1/2) using the proposed parameter choice rule. Finally, we supply numerical examples to illustrate the proposed scheme. Further, we compare the numerical results of the proposed method with the standard method to demonstrate that our method achieves better computational output.
Downloads
Metrics
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Pradeep D, Shinelal E, Ananthalakshmy V
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2024-04-11
Published 2024-07-01
References
Andrei D. Polyanin, Alexander V. Manzhirov, Handbook of integral equations, Taylor and Francis, 2008.
Bakushinskii, A.B, The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Math. Phys. 32 (1992), 1353–1359.
Blaschke, B., Neubauer, A., Scherzer,O, On convergence rates for the iteratively regularized GaussNewton method. IMA J. Numer. Anal. 17 (1997), 421–436.
K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York, 1985.
S.George and M.T. Nair A derivative-free iterative method for nonlinear ill-posed equations with monotone operators, Journal of Inverse and Ill-Posed Problems, 25 (2017), 543-551.
N.S. Hoang and A.G. Ramm, Dynamical systems gradient method for solving nonlinear equations with onotone Operators, Acta Applicandae Mathematicae, 3 (2009), 473-499.
N.S. Hoang and A.G. Ramm, Dynamical systems method for solving sonlinear quations with monotone operators, Mathematics of Computation, 79 (2009), 239-258.
N.S. Hoang and A.G. Ramm, A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations Applied Mathematis and Computation, 254 (2015), 148–156.
P. Mahale and M.T. Nair, Iterated Lavrentiev regularization for nonlinear ill-posed problems, ANZIAM J. 51 (2009), 191-217.
M.T. Nair, Linear Operator Equations: Approximation And Regularization, World Scientific, 2009.
D. Pradeep and M.P. Rajan, A regularized iterative scheme for solving nonlinear ill-posed problems, Numerical Functional Analysis and Optimization 37 (2016), 342-36.
D. Pradeep and M.P. Rajan, A modified iterative Lavrentiev method for nonlinear monotone ill-posed operators, Indian Journal of Pure and Applied Mathematics 55 (2024), 341–356.
A.G. Ramm, Random Fields Estimation. World Sci., Singapore, 2005.
E.V. Semenova, Lavrentiev regularization and balancing principle for solving ill-posed problems with monotone operators, Computational Methods in Applied Mathematics, 10 (2010), 444–454.
U. Tautenhahn, On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems 18 (2002), 191-207.