On the feasibility and convergence of the inexact Newton method under minor conditions on the error terms

TL;DR

This paper establishes semi-local convergence results for the inexact Newton method with minimal error assumptions, demonstrating its applicability to boundary value problems and the Cahn-Hilliard equation.

Contribution

It introduces a semi-local theorem for the inexact Newton method under minor error conditions, extending its theoretical understanding and practical applications.

Findings

Feasibility proven under bounded error assumptions

Convergence of the method established with minimal error constraints

Application demonstrated to boundary value problems and Cahn-Hilliard equation

Abstract

In this paper we introduce a semi-local theorem for the feasibility and convergence of the inexact Newton method, regarding the sequence $x_{k+1} = x_k - Df(x_k)^{-1}f(x_k) + r_k$, where $r_k$ represents the error in each step. Unlike the previous results of this type in the literature, we prove the feasibility of the inexact Newton method under the minor hypothesis that the error $r_k$ is bounded by a small constant to be computed, and moreover we prove results concerning the convergence of the sequence $x_k$ to the solution under this hypothesis. Moreover, we show how to apply this this method to compute rigorously zeros for two-point boundary value problems of Neumann type. Finally, we apply it to a version of the Cahn-Hilliard equation.