TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Vanishing Tikhonov bias and controlled noise amplification imply convergence

Statement

Assume Countable Choice. Let $A : X \to Y$ be compact between real or complex Hilbert spaces, $x^{†} ⊥ \ker A$ , and $y = A x^{†}$ . For a sequence $δ_{j} \geq 0$ , data $y_{j} = y + e_{j}$ with $∥ e_{j} ∥ \leq δ_{j}$ , and parameters $λ_{j} > 0$ , suppose $λ_{j} \to 0, \frac{δ_{j}}{\sqrt{λ_{j}}} \to 0.$ Then the Tikhonov solutions $x_{j} = (A^{*} A + λ_{j} I)^{- 1} A^{*} y_{j}$ converge in norm to $x^{†}$ . No source condition on $x^{†}$ is required.

Facts & Assumptions

Given: Countable Choice, compact $A : X \to Y$ , the minimum-norm solution $x^{†} ⊥ \ker A$ of $A x = y$ , noisy data $y_{j} = y + e_{j}$ with $∥ e_{j} ∥ \leq δ_{j}$ , and positive parameters $λ_{j}$ satisfying the stated limits.

[F1]

Compact-operator Tikhonov bias vanishes on the minimum-norm solution: For exact data $y = A x^{†}$ , the bias $∥ x^{†} - (A^{*} A + λ I)^{- 1} A^{*} y ∥$ tends to zero as $λ ↓ 0$ .

[F2]

Hilbert-space Tikhonov amplifies data by at most $1 / (2 \sqrt{λ})$ : For bounded $A$ , $R_{λ} = (A^{*} A + λ I)^{- 1} A^{*}$ satisfies $∥ R_{λ} e ∥ \leq ∥ e ∥ / (2 \sqrt{λ})$ .

[F3]

Compact spectral filters, source conditions, and qualification: The regularized solution is $R_{λ} y$ and the parameter $λ$ is the coefficient of the squared norm penalty.

Proof

1.1givenF1F2F3algebra

Let $x_{λ} = R_{λ} y$ be the exact-data solution. Linearity gives $x_{j} - x^{†} = (x_{λ_{j}} - x^{†}) + R_{λ_{j}} e_{j}$ . By [F1] and [F2], $∥ x_{j} - x^{†} ∥ \leq ∥ x_{λ_{j}} - x^{†} ∥ + \frac{δ_{j}}{2 \sqrt{λ_{j}}} .$

2.1givenstep 1.1algebraF1∎

The first term on the right tends to zero because $λ_{j} \to 0$ and [F1]; the second tends to zero by the assumed ratio. Thus $∥ x_{j} - x^{†} ∥ \to 0$ .

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Sources

Guido Clason, Regularization of Inverse Problems, full lecture notes (standard reference, not scraped)
Per Christian Hansen, Introduction to Inverse Problems, Chapter 5: Regularization Parameter Choice (standard reference, not scraped)