DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Nonlinear forward maps between Hilbert spaces

Definition

Let $X$ and $Y$ be real Hilbert spaces and let $U \subseteq X$ be a nonempty open set in the norm topology. A nonlinear forward model on Hilbert spaces is a map $F : U \to Y$ ; no linearity is assumed, and linear maps are allowed as special cases. For $y \in Y$ , its exact solution set is

$S_{F} (y) : = {x \in U : F (x) = y} .$

A point $x^{†} \in U$ generates the exact datum $y = F (x^{†})$ . When $F$ is Fréchet differentiable at $x \in U$ , write $F^{'} (x) : X \to Y$ for its unique bounded real-linear derivative.

Definition references

General forward-map and exact-solution interface

Real Hilbert spaces

Fréchet derivative on an open real Banach domain

Uniqueness of the Fréchet derivative

Christian Clason, Regularization of Inverse Problems, Chapter 9

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §1

Depends on

Used by

Finite-budget regularized gradient on a Stiefel constraint Algorithm
Nonlinear Landweber iteration with discrepancy stopping Algorithm
Regularized nonlinear Levenberg–Marquardt with discrepancy stopping Algorithm
A smooth map can fail the tangential cone condition Counterexample
Finite-element PDE solver admissibility and certificate contract Definition
Local tangential cone condition Definition
Product-type tangential cone condition Definition
x₀-minimum-norm solutions of nonlinear equations Definition
A Bochner-integral forward map with weakly closed graph Example
A nonlinear Tikhonov functional can have two global minimizers Example
A smooth map with a local tangential cone neighborhood Example
Product condition gives the LM range-approximation bound Lemma
Local regularized LM discrepancy stopping has logarithmic iteration count Theorem
Local uniqueness of an x₀-minimum-norm solution under TCC Theorem
Nonlinear Landweber Fejér decrease and discrepancy stopping Theorem
Variational source conditions yield nonlinear regularization rates Theorem

Dependency tree · two levels

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Sources

Christian Clason, Regularization of Inverse Problems, Chapter 9 (standard reference, not scraped)
Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §1 (standard reference, not scraped)