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Weak-cluster stability for fixed-parameter nonlinear Tikhonov regularization

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 $α > 0$ , $x_{0} \in X$ , and $y \in Y$ , and let data $y_{n} \in Y$ satisfy $∥ y_{n} - y ∥_{Y} \to 0$ . There exists a sequence $(x_{n})$ of global minimizers of $J_{α, y_{n}, x_{0}}$ on $D$ . Every such sequence has a subsequence $x_{n_{k}}$ with $x_{n_{k}} ⇀ x$ , and every such weak limit $x$ minimizes $J_{α, y, x_{0}}$ on $D$ . No uniqueness of the limiting minimizer is asserted.

Statement references

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

Facts & Assumptions

Given: The ultrafilter lemma, DC, and HB; a weakly closed forward map $F : D \to Y$ between real Hilbert spaces; $α > 0$ , $x_{0} \in X$ ; and data $y_{n} \to y$ in norm.

[A1]

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

[A2]

DC supplies an infinite 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]

For each datum in a real Hilbert space and each positive parameter, the Tikhonov functional has a global minimizer under the stated weak-closedness and axiom hypotheses (Existence of a global nonlinear Tikhonov minimizer).

[F2]

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 ( ${A C}_{ω}$ )).

[F3]

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

[F4]

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).

[F5]

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

[F6]

The graph condition says $x_{n} \in D$ , $x_{n} ⇀ x$ , and $F (x_{n}) ⇀ z$ imply $x \in D$ and $F (x) = z$ (Weakly closed nonlinear forward maps on Hilbert spaces).

[F7]

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

Proof

Given: The assumptions above.

1.1A2F1F2choose

For each $n$ , [F1] makes the set $M_{n}$ of global minimizers of $J_{α, y_{n}, x_{0}}$ on $D$ nonempty. Since the family $(M_{n})$ is countable and DC implies countable choice by [F2], choose $x_{n} \in M_{n}$ for every $n$ .

2.1givenF7step 1.1choosealgebra

Fix $v \in D$ , possible because $D \neq \emptyset$ . Strong convergence $y_{n} \to y$ bounds $J_{α, y_{n}, x_{0}} (v)$ by some $M > 0$ , and minimality gives $J_{α, y_{n}, x_{0}} (x_{n}) \leq M$ . By [F7] and $α > 0$ , this bounds $∥ x_{n} - x_{0} ∥_{X}$ and $∥ F (x_{n}) - y_{n} ∥_{Y}$ ; the convergent sequence $(y_{n})$ is bounded, so both $(x_{n})$ and $(F (x_{n}))$ are bounded.

3.1A1A2A3F3F4step 2.1construct

By [F3], [F4], and the bounds in step 2.1, pass to a subsequence with $x_{n_{k}} ⇀ x$ and then a further subsequence with $F (x_{n_{k}}) ⇀ z$ . The further subsequence still converges weakly to $x$ .

4.1F6step 3.1

By [F6], $x \in D$ and $F (x) = z$ . Since $y_{n_{k}} \to y$ strongly, $F (x_{n_{k}}) - y_{n_{k}} ⇀ F (x) - y$ , and $x_{n_{k}} - x_{0} ⇀ x - x_{0}$ .

5.1A3F5F7step 3.1step 4.1algebra∎

By [F5], each norm in [F7] is weakly lower semicontinuous; composition with the increasing continuous square function on nonnegative reals preserves this property. For any $ε > 0$ , the two summand sequences are eventually at least their respective limit values minus $ε / 2$ , so their sum is eventually at least $J_{α, y, x_{0}} (x) - ε$ . Thus $J_{α, y, x_{0}} (x) \leq {lim inf}_{k} J_{α, y_{n_{k}}, x_{0}} (x_{n_{k}})$ . For any $w \in D$ , minimality gives $J_{α, y_{n_{k}}, x_{0}} (x_{n_{k}}) \leq J_{α, y_{n_{k}}, x_{0}} (w)$ , and the right side converges to $J_{α, y, x_{0}} (w)$ because $F (w)$ is fixed and $y_{n_{k}} \to y$ . Hence $J_{α, y, x_{0}} (x) \leq {lim inf}_{k} J_{α, y_{n_{k}}, x_{0}} (x_{n_{k}}) \leq {lim sup}_{k} J_{α, y_{n_{k}}, x_{0}} (x_{n_{k}}) \leq J_{α, y, x_{0}} (w)$ . This holds for every $w \in D$ , so $x$ is a global minimizer. Applying the same argument to any weakly convergent subsequence proves the claim for every weak cluster point.

Remarks

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

Depends on

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Sources

Christian Clason, Regularization of Inverse Problems, Theorem 10.2 (standard reference, not scraped)
Benedikt Wirth, Inverse Problems, Theorem 86 (standard reference, not scraped)