Two-Step Newton-Tikhonov Method for Hammerstein-Type Equations: Finite-Dimensional Realization

Santhosh George,Monnanda Erappa Shobha

doi:10.5402/2012/783579

Santhosh George, Monnanda Erappa Shobha

Open Access

https://doi.org/10.5402/2012/783579

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Abstract

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)⊆X→X is nonlinear monotone operator, K:X→Y 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 x0 of the the solution x̂: balancing principle of Pereverzev and Schock (2005) 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.

Highlights

Tikhonov’s regularization e.g., 1 method has been used extensively to stabilize the approximate solution of nonlinear ill-posed problems
In recent years, increased emphasis has been placed on iterative regularization procedures 2, 3 for obtaining the approximate solution of such problems
We examine the use of iterative regularization procedures for Hammerstein-type 4, 5 equations of the form

Summary

Introduction

Tikhonov’s regularization e.g., 1 method has been used extensively to stabilize the approximate solution of nonlinear ill-posed problems. In recent years, increased emphasis has been placed on iterative regularization procedures 2, 3 for obtaining the approximate solution of such problems. In 4 , z is approximated with zδα where zδα K∗K αI −1K∗f δ, α > 0, δ > 0, 1.6 and solve 1.5 iteratively using the following Newton-type procedure: xnδ 1,α xnδ,α − F x0 −1 F xnδ,α − zδα. Motivated by TSDNM, in 11 , we propose a Two-Step Newton-Tikhonov Methods TSNTM for solving 1.1. In this paper we consider the finitedimensional realization of the second case, that is, F is monotone. The finite-dimensional realization of the method and associated algorithm are proposed for which local-cubic convergence is established theoretically and validated numerically.

Discretized Tikhonov Regularization

A Priori Choice of the Parameter

An Adaptive Choice of the Parameter

Algorithm

Example

Numerical Example