Two-step Newton-Tikhonov method for Hammerstein-type equations: finite-dimensional realization
Summary: Finite-dimensional realization of a two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equations \(KF(x) = f\). Here \(F : D(F) \subseteq X \to X\) is nonlinear monotone operator, \(K : X \to X\) is a bounded linear operator, \(X\) is a real Hilbert space, and \(Y\) is a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operator \(K\) and the second one involves the Fréchet derivative of \(F\) at an initial approximation \(x_0\) of the the solution \(\hat{x}\): balancing principle of \textit{S. Pereverzev} and \textit{E. Schock} [SIAM J. Numer. Anal. 43, No.~5, 2060--2076 (2005; Zbl 1103.65058)] is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.
nonlinear ill-posed Hammerstein-type operator equations
scientific article; zbMATH DE number 6172986
scientific article