Year: 2024
Type: article
Abstract: The need to control the residual of a potentially nonlinear function $$\mathcal {F}$$ 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 $$t\mapsto x(t)$$ t ↦ x ( t ) in the domain of the nonlinear map $$\mathcal {F}$$ F leading from some initial value $$x_0$$ x 0 to a value u such that we are able to control the residual $$\mathcal {F}(x(t))$$ F ( x ( t ) ) based on the value $$\mathcal {F}(x_0)$$ 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 $$\mathcal {F}$$ F can be represented by $$x \mapsto \mathcal {A}(x)\mathcal {F}(x_0)$$ x ↦ A ( x ) F ( x 0 ) , where $$\mathcal {A}$$ A is a suitable defined operator. The presented approach covers, in case of $$\mathcal {A}(x) = -\textsf{Id}$$ A ( x ) = - 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.
Source: Archiv der Mathematik
Author: Mario Amrein
Institution: ZHAW Zurich University of Applied Sciences
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
OpenAlex ID: W4402037611