Modified Minimal Error Method for Nonlinear Ill-Posed Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 313-321
Author(s):  
M. Sabari ◽  
Santhosh George
Keyword(s):  
Error Estimate ◽  
Noisy Data ◽  
Optimal Order ◽  
Order Error ◽  
Source Condition ◽  
Minimal Error ◽  
Ill Posed ◽  
Error Method

AbstractAn error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate.

scholarly journals Newton Type Iteration for Tikhonov Regularization of Nonlinear Ill-Posed Problems in Hilbert Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George
Keyword(s):  
Integral Equation ◽  
Iterative Method ◽  
Iterative Scheme ◽  
Initial Guess ◽  
Optimal Order ◽  
Source Condition ◽  
Adaptive Scheme ◽  
Actual Solution ◽  
Ill Posed ◽  
General Source

Recently, Vasin and George (2013) considered an iterative scheme for approximately solving an ill-posed operator equationF(x)=y. In order to improve the error estimate available by Vasin and George (2013), in the present paper we extend the iterative method considered by Vasin and George (2013), in the setting of Hilbert scales. The error estimates obtained under a general source condition onx0-x^(x0is the initial guess andx^is the actual solution), using the adaptive scheme proposed by Pereverzev and Schock (2005), are of optimal order. The algorithm is applied to numerical solution of an integral equation in Numerical Example section.

A Quadratic Convergence Yielding Iterative Method for Nonlinear Ill-posed Operator Equations

2012 ◽  
Vol 12 (1) ◽  
pp. 32-45 ◽  
Author(s):  
Santhosh George ◽  
Atef Ibrahim Elmahdy
Keyword(s):  
Iterative Method ◽  
Operator Equation ◽  
Quadratic Convergence ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Majorizing Sequence ◽  
Operator Equations ◽  
Order Error ◽  
Ill Posed ◽  
The Given

AbstractIn this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y. The iteration procedure converges quadratically to the unique solution of the equation for the regularized approximation. It is known that (Tautanhahn (2002)) this solution converges to the solution of the given ill-posed operator equation. The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.

scholarly journals An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems

2011 ◽  
Vol 15 (2) ◽  
pp. 325-341 ◽  
Author(s):  
Huan-Zhen Chen ◽  
◽  
Zhao-Jie Zhou ◽  
Hong Wang ◽  
Hong-Ying Man ◽  
...  
Keyword(s):  
Mixed Methods ◽  
Error Estimate ◽  
Optimal Order ◽  
Order Error ◽  
Convection Diffusion ◽  
Diffusion Problems

scholarly journals Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ailing Zhu
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Optimal Order ◽  
Triangular Meshes ◽  
Element Space ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Mixed Element

The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

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