Finite Dimensional Realization of a Guass-Newton Method for Ill-Posed Hammerstein Type Operator Equations

2012 ◽  
pp. 293-301
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George
Keyword(s):  
Newton Method ◽  
Type Operator ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

scholarly journals Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations

2004 ◽  
Vol 2004 (37) ◽  
pp. 1973-1996 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Wide Class ◽  
Noise Level ◽  
A Priori ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.

scholarly journals A two step newton type iteration for ill-posed Hammerstein type operator equations in Hilbert scales

2014 ◽  
Vol 26 (1) ◽  
pp. 91-116
Author(s):  
Monnanda Erappa Shobha ◽  
Santhosh George ◽  
M. Kunhanandan
Keyword(s):  
Type Operator ◽  
Operator Equations ◽  
Ill Posed

Simplified Generalized Gauss–Newton Method for Nonlinear Ill-Posed Operator Equations in Hilbert Scales

2018 ◽  
Vol 18 (4) ◽  
pp. 687-702 ◽  
Author(s):  
Pallavi Mahale ◽  
Pradeep Kumar Dadsena
Keyword(s):  
Approximate Solution ◽  
Error Estimate ◽  
Newton Method ◽  
Operator Equation ◽  
Hilbert Spaces ◽  
Nonlinear Operator ◽  
Optimal Error Estimate ◽  
Operator Equations ◽  
Optimal Error ◽  
Ill Posed

AbstractIn this paper, we study the simplified generalized Gauss–Newton method in a Hilbert scale setting to get an approximate solution of the ill-posed operator equation of the form {F(x)=y} where {F:D(F)\subseteq X\to Y} is a nonlinear operator between Hilbert spaces X and Y. Under suitable nonlinearly conditions on F, we obtain an order optimal error estimate under the Morozov type stopping rule.

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