Adaptive Newton-Type Schemes Based on Projections

TL;DR

This paper introduces an adaptive Newton-type method based on projections that uses ideas from differential equations to improve convergence and robustness in solving nonlinear equations.

Contribution

It proposes a novel adaptive step size control for Newton schemes inspired by ODE numerical methods, maintaining quadratic convergence near solutions.

Findings

The adaptive scheme effectively solves nonlinear equations in low-dimensional examples.

Performance data shows improved convergence behavior over classical Newton methods.

The approach successfully mimics the continuous problem, ensuring stability and efficiency.

Abstract

In this work we present and discuss a possible globalization concept for Newton-type methods. We consider nonlinear problems $f(x)=0$ in $\mathbb{R}^{n}$ using the concepts from ordinary differential equations as a basis for the proposed numerical solution procedure. Thus, the starting point of our approach is within the framework of solving ordinary differential equations numerically. Accordingly, we are able to reformulate general Newton-type iteration schemes using an adaptive step size control procedure. In doing so, we derive and discuss a discrete adaptive solution scheme thereby trying to mimic the underlying continuous problem numerically without losing the famous quadratic convergence regime of the classical Newton method in a vicinity of a regular solution. The derivation of the proposed adaptive iteration scheme relies on a simple orthogonal projection argument taking into account that, sufficiently close to regular solutions, the vector field corresponding to the Newton scheme is approximately linear. We test and exemplify our adaptive root-finding scheme using a few low-dimensional examples. Based on the presented examples, we finally show some performance data.