Residual growth control for general maps and an approximate inverse function result

TL;DR

This paper develops a method to control the residuals of nonlinear functions along specific curves in their domain, extending existing results and introducing an approximate inverse function related to residual control in nonlinear analysis.

Contribution

It extends an existing residual control result for nonlinear maps by proving the existence of a residual-controlling curve under a generalized derivative assumption, and introduces an approximate inverse function result.

Findings

Existence of a residual-controlling curve for nonlinear maps.

Extension of continuous Newton method results.

Introduction of an approximate inverse function concept.

Abstract

The need to control the residual of a potentially nonlinear function $\mathcal{F}$ arises in several situations in mathematics. For example, computing the zeros of a given map, or the reduction of some cost function during an optimization process are such situations. In this note, we discuss the existence of a curve $t\mapsto x(t)$ in the domain of the nonlinear map $\mathcal{F}$ leading from some initial value $x_0$ to a value $u$ such that we are able to control the residual $\mathcal{F}(x(t))$ based on the value $\mathcal{F}(x_0)$. More precisely, we slightly extend an existing result from J.W. Neuberger by proving the existence of such a curve, assuming that the directional derivative of $\mathcal{F}$ can be represented by $x \mapsto \mathcal{A}(x)\mathcal{F}(x_0)$, where $\mathcal{A}$ is a suitable defined operator. The presented approach covers, in case of $\mathcal{A}(x) = -\mathsf{Id}$, some well known results from the theory of so-called continuous Newton methods. Moreover, based on the presented results, we discover an approximate inverse function result.