Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa Shobha¹ ; Ioannis K. Argyros² ; Santhosh George¹

¹ Department of Mathematical and Computational Sciences National Institute of Technology Karnataka, India 757 025
² Department of Mathematical Sciences Cameron University Lawton, OK 73505, U.S.A.

Applicationes Mathematicae, Tome 41 (2014) no. 1, pp. 107-129

Résumé

We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations $KF(x)=y.$ It is assumed that the available data is $y^\delta $ with $\| y-y^\delta \| \leq \delta ,$ $ K:Z\rightarrow Y$ is a bounded linear operator and $ F:X\rightarrow Z $ is a nonlinear operator where $X,Y,Z$ are Hilbert spaces. Two cases of $F$ are considered: where $F'(x_0)^{-1}$ exists ($F'(x_0)$ is the Fréchet derivative of $F$ at an initial guess $x_0$) and where $F$ is a monotone operator. The parameter choice using an a priori and an adaptive choice under a general source condition are of optimal order. The computational results provided confirm the reliability and effectiveness of our method.

Détail

DOI : 10.4064/am41-1-9

Keywords: combination modified newton method tikhonov regularization obtain stable approximate solution nonlinear ill posed hammerstein type operator equations assumed available delta y y delta leq delta rightarrow bounded linear operator rightarrow nonlinear operator where hilbert spaces cases considered where exists chet derivative initial guess nbsp where monotone operator parameter choice using priori adaptive choice under general source condition optimal order computational results provided confirm reliability effectiveness method