Nonlinear Tikhonov existence and stability

Let $X$ and $Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty, and let $F:D\to Y$ be a forward map. The domain $D$ is weakly closed when it is closed in the weak topology $\sigma (X,X^{\ast })$, the coarsest topology making every bounded real-linear functional on $X$ continuous. The graph of $F$ is sequentially weak-to-weak closed when, for every sequence $(x_n)$ in $D$ and every $x\in X$ and $z\in Y$, the conditions $x_n\rightharpoonup x$ and $F(x_n)\rightharpoonup z$ imply $x\in D$ and $F(x)=z$; here weak convergence means convergence under every bounded real-linear functional on the corresponding Hilbert space. We call $F$ a weakly closed forward map when its domain is weakly closed and its graph is sequentially weak-to-weak closed. No linearity, norm continuity, Fréchet differentiability, or convexity is assumed; linear maps are allowed as special cases.

General forward maps on an arbitrary domain

Real Hilbert spaces

The weak topology

Weak sequence convergence

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Chapter 10

Let $X$ and $Y$ be real Hilbert spaces, let $D\subseteq X$ be a nonempty domain, and let $F:D\to Y$ be a forward map. For a regularization parameter $\alpha >0$, reference element $x_0\in X$, and datum $y\in Y$, the nonlinear Tikhonov functional is the map $J_{\alpha ,y,x_0}:D\to [0,\mathrm{\infty })$ defined by

A global minimizer is a point $x_{\alpha }\in D$ such that $J_{\alpha ,y,x_0}(x_{\alpha })\le J_{\alpha ,y,x_0}(x)$ for every $x\in D$. This definition asserts neither existence nor uniqueness.

Forward maps and their domains

Real Hilbert spaces and their norms

Christian Clason, Regularization of Inverse Problems, equation (10.1)

Benedikt Wirth, Inverse Problems, Definition 83

Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed, meaning closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy this graph condition: whenever $x_n\in D$, $x_n\rightharpoonup x$, and $F(x_n)\rightharpoonup z$, one has $x\in D$ and $F(x)=z$; weak convergence means convergence under every bounded real-linear functional. For $x_0\in X$ and $y\in Y$, a point $x^{†}\in D$ is an $x_0$-minimum-norm solution of $F(x)=y$ when $F(x^{†})=y$ and

This definition asserts neither existence nor uniqueness. The minimum is global over the exact solution set in $D$ and uses distance from the specified reference point $x_0$.

Weakly closed forward maps and their domains

The open-domain version of the same distance condition

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Definition 83 and Theorem 85

Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed, meaning closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy this graph condition: whenever $x_n\in D$, $x_n\rightharpoonup x$, and $F(x_n)\rightharpoonup z$, one has $x\in D$ and $F(x)=z$. Weak convergence means convergence under every bounded real-linear functional. For $y\in Y$, the exact solution set $S_F(y):=\{x\in D:F(x)=y\}$ is weakly sequentially closed in $X$: every sequence in $S_F(y)$ that converges weakly to $x$ has $x\in S_F(y)$.

Weakly closed domain and sequential graph closure

Weak sequence convergence

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Chapter 10

Assume the ultrafilter lemma, dependent choice, and the real Hahn–Banach dominated-extension principle. Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed, and let $F:D\to Y$ be a map such that for every sequence $(x_n)$ in $D$ and every $x\in X,z\in Y$, $x_n\rightharpoonup x$ and $F(x_n)\rightharpoonup z$ imply $x\in D$ and $F(x)=z$. Here $D$ is closed in the coarsest topology making every bounded real-linear functional continuous, and weak convergence means convergence under every such functional. For any $\alpha >0$, $x_0\in X$, and $y\in Y$, the nonlinear Tikhonov functional

attains a global minimum on $D$; that is, there exists $x_{\alpha }\in D$ such that $J_{\alpha ,y,x_0}(x_{\alpha })\le J_{\alpha ,y,x_0}(x)$ for every $x\in D$. No exact solution of $F(x)=y$ is required.

Weakly closed domains and sequential weak-to-weak graph closure

The Tikhonov functional and global minimizer

Christian Clason, Regularization of Inverse Problems, Theorem 10.1

Benedikt Wirth, Inverse Problems, Theorem 85(a)

The theorem guarantees global minimizers, not uniqueness. The sequential weak-to-weak graph condition is needed to identify the weak limit of the data sequence; weak-to-strong closedness alone would not identify a merely weak data limit. The direct method makes no claim that a numerical algorithm finds a global minimizer; nonlinear objectives may also have local minima.

Assume the ultrafilter lemma, dependent choice, and HB. Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy: for every sequence $x_n\in D$ and $x\in X,z\in Y$, if $x_n\rightharpoonup x$ and $F(x_n)\rightharpoonup z$, then $x\in D$ and $F(x)=z$; weak convergence means convergence under every bounded real-linear functional. If $S_F(y):=\{x\in D:F(x)=y\}$ is nonempty for some $y\in Y$, then for every $x_0\in X$ there exists $x^{†}\in S_F(y)$ such that $\mathrm{\parallel }{x}^{†}-x_0{\mathrm{\parallel }}_X\le \mathrm{\parallel }x-x_0{\mathrm{\parallel }}_X$ for every $x\in S_F(y)$.

Weakly closed domains and sequential graph closure

The reference-point minimum-norm solution condition

The direct method

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Theorem 85(b)

The theorem gives existence, not uniqueness: a nonlinear exact solution set need not have a unique point at minimum distance from $x_0$. The use of DC also supplies the countable choice assumed by the Hilbert reflexivity supplier; no full axiom of choice is used.

Assume the ultrafilter lemma, dependent choice, and HB. Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy: whenever $x_n\in D$, $x_n\rightharpoonup x$, and $F(x_n)\rightharpoonup z$, one has $x\in D$ and $F(x)=z$; weak convergence means convergence under every bounded real-linear functional. Fix $\alpha >0$, $x_0\in X$, and $y\in Y$, and let data $y_n\in Y$ satisfy $\mathrm{\parallel }{y}_n-y{\mathrm{\parallel }}_Y\to 0$. There exists a sequence $(x_n)$ of global minimizers of $J_{\alpha ,y_n,x_0}$ on $D$. Every such sequence has a subsequence $x_{n_k}$ with $x_{n_k}\rightharpoonup x$, and every such weak limit $x$ minimizes $J_{\alpha ,y,x_0}$ on $D$. No uniqueness of the limiting minimizer is asserted.

Weakly closed domain and sequential graph closure

The Tikhonov functional

Existence for each perturbed datum

Christian Clason, Regularization of Inverse Problems, Theorem 10.2

Benedikt Wirth, Inverse Problems, Theorem 86

The conclusion is weak-cluster stability at fixed positive $\alpha$. Nonlinearity can prevent uniqueness, so the theorem does not assert convergence of the full sequence or strong convergence.

Assume the ultrafilter lemma, dependent choice, and HB. Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy: whenever $x_n\in D$, $x_n\rightharpoonup x$, and $F(x_n)\rightharpoonup z$, one has $x\in D$ and $F(x)=z$; weak convergence means convergence under every bounded real-linear functional. Fix $x_0\in X$ and suppose $S_F(y^{†}):=\{x\in D:F(x)=y^{†}\}$ is nonempty. Let ${\delta }_n\ge 0$, ${\delta }_n\to 0$, ${\alpha }_n>0$, ${\alpha }_n\to 0$, and ${\delta }_n^2\mathrm{/}{\alpha }_n\to 0$, and let $y_n\in Y$ satisfy $\mathrm{\parallel }{y}_n-y^{†}{\mathrm{\parallel }}_Y\le {\delta }_n$. There exists a sequence $(x_n)$ of global minimizers of $J_{{\alpha }_n,y_n,x_0}$ on $D$; every such sequence has a subsequence $x_{n_k}\rightharpoonup x$, and every such weak limit $x$ is an $x_0$-minimum-norm solution of $F(x)=y^{†}$ on $D$. No uniqueness or full-sequence convergence is asserted.

Weakly closed domains and sequential graph closure

The nonlinear Tikhonov functional

The reference-point minimum-norm exact solution condition

Existence for every positive parameter and datum

Existence of a reference-point minimum-norm solution

Christian Clason, Regularization of Inverse Problems, Theorem 10.3

Benedikt Wirth, Inverse Problems, Theorem 87

The fixed-positive-$\alpha$ stability theorem concerns minimizers of one limiting functional. This vanishing-noise theorem has the separate assumptions of exact solvability, ${\alpha }_n\to 0$, and ${\delta }_n^2\mathrm{/}{\alpha }_n\to 0$. It identifies weak cluster points as minimum-norm exact solutions but does not assert a unique limit.