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Vanishing-noise convergence to an $x_{0}$ -minimum-norm exact solution

Statement

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 $x_{n} \in D$ , $x_{n} ⇀ x$ , and $F (x_{n}) ⇀ z$ , 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 $δ_{n} \geq 0$ , $δ_{n} \to 0$ , $α_{n} > 0$ , $α_{n} \to 0$ , and $δ_{n}^{2} / α_{n} \to 0$ , and let $y_{n} \in Y$ satisfy $∥ y_{n} - y^{†} ∥_{Y} \leq δ_{n}$ . There exists a sequence $(x_{n})$ of global minimizers of $J_{α_{n}, y_{n}, x_{0}}$ on $D$ ; every such sequence has a subsequence $x_{n_{k}} ⇀ x$ , 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.

Statement references

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

Facts & Assumptions

Given: The ultrafilter lemma, DC, and HB; a weakly closed forward map $F : D \to Y$ between real Hilbert spaces; $x_{0} \in X$ ; a nonempty exact solution set for $y^{†}$ ; and sequences $δ_{n}, α_{n}, y_{n}$ satisfying the stated noise and parameter conditions.

[A1]

The ultrafilter lemma extends every filter to an ultrafilter (The ultrafilter extension principle (UL/BPI)).

[A2]

DC supplies a sequence following an everywhere-extendible relation (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[A3]

HB is the real dominated-extension principle (The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

DC implies countable choice, the principle that every countable family of nonempty sets admits a choice function (Dependent choice implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

Under countable choice, real Hilbert spaces are reflexive (Hilbert spaces are reflexive by Riesz representation).

[F3]

Under the ultrafilter lemma, DC, and HB, bounded sequences in real reflexive Banach spaces have weakly convergent subsequences (A bounded sequence in a reflexive Banach space has a weakly convergent subsequence).

[F4]

The exact solution set contains an $x_{0}$ -minimum-norm solution under these hypotheses (Existence of an $x_{0}$ -minimum-norm exact solution).

[F5]

For each $α_{n} > 0$ and $y_{n} \in Y$ , the Tikhonov functional has a global minimizer under these hypotheses (Existence of a global nonlinear Tikhonov minimizer).

[F6]

Under HB, weak convergence implies lower semicontinuity of the norm (Weak convergence implies lower semicontinuity of the norm).

[F7]

A sequentially weak-to-weak closed graph maps weak limits $(x_{n}, F (x_{n})) ⇀ (x, z)$ to $x \in D$ and $F (x) = z$ (Weakly closed nonlinear forward maps on Hilbert spaces).

[F8]

$J_{α, y, x_{0}} (x) = \frac{1}{2} ∥ F (x) - y ∥_{Y}^{2} + \frac{α}{2} ∥ x - x_{0} ∥_{X}^{2}$ (The nonlinear Tikhonov functional).

[F9]

An $x_{0}$ -minimum-norm solution is an exact solution with least distance to $x_{0}$ among all exact solutions in $D$ ( $x_{0}$ -minimum-norm solutions on weakly closed domains).

Proof

Given: The assumptions above.

1.1A2F1F4F5choose

By [F4], choose an $x_{0}$ -minimum-norm exact solution $\hat{x} \in D$ and put $R : = ∥ \hat{x} - x_{0} ∥_{X}$ . For each $n$ , [F5] makes the set of global minimizers of $J_{α_{n}, y_{n}, x_{0}}$ nonempty; by [F1] and the assumed DC, choose one $x_{n}$ from each set.

2.1givenF5F8step 1.1algebra

Minimality, [F8], exactness of $\hat{x}$ , and $∥ y_{n} - y^{†} ∥_{Y} \leq δ_{n}$ give $\frac{1}{2} ∥ F (x_{n}) - y_{n} ∥_{Y}^{2} + \frac{α_{n}}{2} ∥ x_{n} - x_{0} ∥_{X}^{2} \leq J_{α_{n}, y_{n}, x_{0}} (\hat{x}) \leq \frac{δ_{n}^{2}}{2} + \frac{α_{n}}{2} R^{2}$ . Consequently $∥ x_{n} - x_{0} ∥_{X}^{2} \leq δ_{n}^{2} / α_{n} + R^{2}$ , so $(x_{n})$ is bounded, and $∥ F (x_{n}) - y_{n} ∥_{Y}^{2} \leq δ_{n}^{2} + α_{n} R^{2} \to 0$ .

3.1A1A2A3F2F3F7step 2.1constructalgebra

By [F2] and [F3], a subsequence satisfies $x_{n_{k}} ⇀ x \in X$ . The triangle inequality and step 2.1 give $∥ F (x_{n_{k}}) - y^{†} ∥_{Y} \leq ∥ F (x_{n_{k}}) - y_{n_{k}} ∥_{Y} + δ_{n_{k}} \leq \sqrt{δ_{n_{k}}^{2} + α_{n_{k}} R^{2}} + δ_{n_{k}} \to 0$ . Thus $F (x_{n_{k}}) \to y^{†}$ strongly, hence weakly. Applying [F7] gives $x \in D$ and $F (x) = y^{†}$ .

4.1A3F6F9step 1.1step 2.1step 3.1algebra∎

By [F6], $∥ x - x_{0} ∥_{X}^{2} \leq {lim inf}_{k} ∥ x_{n_{k}} - x_{0} ∥_{X}^{2}$ . The bound in step 2.1 gives ${lim sup}_{k} ∥ x_{n_{k}} - x_{0} ∥_{X}^{2} \leq \lim_{k} (δ_{n_{k}}^{2} / α_{n_{k}} + R^{2}) = R^{2}$ , so $∥ x - x_{0} ∥_{X}^{2} \leq {lim inf}_{k} ∥ x_{n_{k}} - x_{0} ∥_{X}^{2} \leq {lim sup}_{k} ∥ x_{n_{k}} - x_{0} ∥_{X}^{2} \leq R^{2}$ . Since $x$ is an exact solution and $\hat{x}$ minimizes distance among exact solutions by [F9], $R^{2} \leq ∥ x - x_{0} ∥_{X}^{2}$ ; equality follows, so $x$ is an $x_{0}$ -minimum-norm exact solution. The same reasoning applies to any weakly convergent subsequence.

Remarks

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

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Sources

Christian Clason, Regularization of Inverse Problems, Theorem 10.3 (standard reference, not scraped)
Benedikt Wirth, Inverse Problems, Theorem 87 (standard reference, not scraped)

Vanishing-noise convergence to an x0x0​-minimum-norm exact solution