A Global Newton-Type Scheme Based on a Simplified Newton-Type Approach

TL;DR

This paper introduces a globalization method for Newton-type iterations that transitions to a simplified Newton scheme when the residual is small, ensuring convergence to a zero without adaptive step size control.

Contribution

It proposes a new globalization approach for Newton-type methods based on residual size, backed by Banach's fixed-point theorem, and demonstrates its effectiveness through theoretical analysis and a low-dimensional example.

Findings

The scheme guarantees convergence to a zero within a neighborhood of a suitable iterate.

It simplifies the Newton iteration once the residual is sufficiently small.

Demonstrated advantages include reduced chaotic behavior and improved convergence control.

Abstract

Globalization concepts for Newton-type iteration schemes are widely used when solving nonlinear problems numerically. Most of these schemes are based on a predictor/corrector step size methodology with the aim of steering an initial guess to a zero of $f$ without switching between different attractors. In doing so, one is typically able to reduce the chaotic behavior of the classical Newton-type iteration scheme. In this note we propose a globalization methodology for general Newton-type iteration concepts which changes into a simplified Newton iteration as soon as the transformed residual of the underlying function is small enough. Based on Banach's fixed-point theorem, we show that there exists a neighborhood around a suitable iterate $x_{n}$ such that we can steer the iterates---without any adaptive step size control but using a simplified Newton-type iteration within this neighborhood---arbitrarily close to an exact zero of $f$. We further exemplify the theoretical result within a global Newton-type iteration procedure and discuss further an algorithmic realization. Our proposed scheme will be demonstrated on a low-dimensional example thereby emphasizing the advantage of this new solution procedure.