Simplified Generalized Gauss-Newton Iterative Method under Morozove Type Stopping Rule

Abstract

Recently, Mahale and Nair considered a simplified generalized Gauss-Newton iterative method for getting an approximate solution for the nonlinear ill-posed operator equation under the modified general source condition. The advantage of this method and the source condition over the classical Gauss-Newton iterative method is that the iterations and source condition involve calculation of the Fréchet derivative only at the point x ₀, i.e., at the initial approximation for the exact solution x ^† of the nonlinear ill-posed operator equation F(x) = y. Motivated by the work of Qinian Jin and Tautenhan, error analysis of the simplified Gauss-Newton iterative method is done in this article under a Morozove-type stopping rule, which is much simpler than the stopping rule considered in the article of Mahale and Nair. An order optimal error estimate is obtained under a modified general source condition which also involves calculation of the Fréchet derivative at the point x ₀.

Keywords:

• General source conditions
• Iterative regularization methods
• Morozove discrepancy principle
• Nonlinear ill-posed equations
• Regularization parameter
• Stopping index

Mathematics Subject Classification:

• 47A52
• 65F22
• 65J15
• 65J22
• 65M30

Notes

As we know that r _α0 (0) = 1 which implies such M always exist.