Nonlinear maps and tangential cone control

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

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

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

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

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be a nonempty open set, let $F:U\to Y$ be a forward map, let $y\in Y$, and let $x_0\in X$. A point $x^{†}\in U$ is an $x_0$-minimum-norm solution of $F(x)=y$ on $U$ if $F(x^{†})=y$ and

$\mathrm{\parallel }{x}^{†}-x_0\mathrm{\parallel }\le \mathrm{\parallel }x-x_0\mathrm{\parallel }\text{for every }x\in U\text{ with }F(x)=y\mathrm{.}$

This definition asserts no existence or uniqueness; it describes the minimizing property of a supplied exact solution. The minimum is global over the exact solution set on $U$, with distance measured from the specified reference point $x_0$.

Nonlinear forward maps and exact solution sets

Real Hilbert spaces and their induced norm

Christian Clason, Regularization of Inverse Problems, Theorem 9.9

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

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be Fréchet differentiable at $x\in U$. For $\delta >0$ such that the closed ball

${\bar{B}}_{\delta }(x):=\{x+h:h\in X,\mathrm{\parallel }h{\mathrm{\parallel }}_X\le \delta \}$

is contained in $U$, the local tangential cone condition at $x$ on radius $\delta$ with constant $\eta$ holds if $0\le \eta <1$ and

$\mathrm{\parallel }F(x+h)-F(x)-F^{\mathrm{\prime }}(x)h{\mathrm{\parallel }}_Y\le \eta \mathrm{\parallel }F(x+h)-F(x){\mathrm{\parallel }}_Y(\mathrm{\parallel }h{\mathrm{\parallel }}_X\le \delta )\mathrm{.}$

Nonlinear forward maps between real Hilbert spaces

Fréchet differentiability on an open real Banach domain

The norm triangle inequality for real inner-product spaces

Christian Clason, Regularization of Inverse Problems, equation (9.6)

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §2.3

This is the one-basepoint local condition: the derivative is fixed at $x$ while $h$ varies over the ball. Some treatments impose a stronger two-point condition uniformly for every pair of points in a set. The constant restriction $0\le \eta <1$ is part of this definition. Fréchet differentiability alone does not imply this condition.

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be Fréchet differentiable at $x\in U$. Suppose that for some $\delta >0$ with ${\bar{B}}_{\delta }(x)\subseteq U$ and some $0\le \eta <1$, the local tangential cone condition at $x$ holds on radius $\delta$ with constant $\eta$. Then for every $h\in X$ with $\mathrm{\parallel }h{\mathrm{\parallel }}_X\le \delta$,

Local tangential cone condition

Christian Clason, Regularization of Inverse Problems, Theorem 9.8

Barbara Kaltenbacher, Frank Schöpfer, and Thomas Schuster, Iterative methods for nonlinear ill-posed problems in Banach spaces, §2.3

Let $X,Y$ be real Hilbert spaces, let $U\subseteq X$ be open, and let $F:U\to Y$ be Fréchet differentiable. Let $x_0\in X$, $y\in Y$, and let $x^{†}\in U$ be an $x_0$-minimum-norm solution of $F(x)=y$ on $U$. Suppose that for some $\delta >0$ the closed ball ${\bar{B}}_{\delta }(x^{†})$ lies in $U$, the local tangential cone condition holds at $x^{†}$ on radius $\delta$ with a constant $0\le \eta <1$, and

$x^{†}-x_0\in \ker (F^{\mathrm{\prime }}(x^{†}){)}^{\perp }\mathrm{.}$

Then every $x\in {\bar{B}}_{\delta }(x^{†})$ with $F(x)=y$ and $x\mathrm{\ne }{x}^{†}$ satisfies

$\mathrm{\parallel }x-x_0{\mathrm{\parallel }}_X>\mathrm{\parallel }{x}^{†}-x_0{\mathrm{\parallel }}_X\mathrm{.}$

Consequently, $x^{†}$ is the unique $x_0$-minimum-norm solution among the exact solutions in ${\bar{B}}_{\delta }(x^{†})$.

Nonlinear forward maps and exact solutions

The $x_0$-minimum-norm solution condition

Local tangential cone condition

Derivative/data-residual comparison

Fréchet derivative

Kernel of a linear operator

Orthogonal complement

Christian Clason, Regularization of Inverse Problems, Theorem 9.9

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

The conclusion is local minimum-norm uniqueness, not injectivity of $F$: other exact solutions may lie in the ball, but each has strictly larger distance to $x_0$.