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

Abstract

Abstract In 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 {F(x)=y} where F : D ⁢ ( F ) ⊆ X → Y {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.