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

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Forward discretization enters before inversion and controls the Tikhonov parameter coupling

Statement

Assume Countable Choice. Let $A : X \to Y$ be compact and $A_{h} : X \to Y$ be bounded and finite rank, between real or complex Hilbert spaces. Suppose $x^{†} ⊥ \ker A$ , $y = A x^{†}$ , $y_{δ} = y + e$ with $∥ e ∥ \leq δ$ , and $∥ A - A_{h} ∥ \leq η$ . Form the discretized datum $y_{h, δ} = A_{h} x^{†} + e$ before inversion. For $λ > 0$ , define $R_{λ} = (A^{*} A + λ I)^{- 1} A^{*}, R_{λ, h} = (A_{h}^{*} A_{h} + λ I)^{- 1} A_{h}^{*},$ and for a computed output ${\tilde{x}}_{h}$ define its exact normal-equation residual $r_{h, h} = (A_{h}^{*} A_{h} + λ I) {\tilde{x}}_{h} - A_{h}^{*} y_{h, δ} .$ Then $\begin{aligned} ∥ x^{†} - {\tilde{x}}_{h} ∥ \leq & ∥ x^{†} - R_{λ} y ∥ + \frac{δ + η ∥ x^{†} ∥}{2 \sqrt{λ}} + ∥ (R_{λ} - R_{λ, h}) y_{h, δ} ∥ + \frac{∥ r_{h, h} ∥}{λ} . \end{aligned}$ The forward-model term $η ∥ x^{†} ∥$ is counted before inversion and is amplified by $1 / (2 \sqrt{λ})$ . The regularized-operator difference is kept as a separate term; the sequence conclusion below explicitly assumes that this term tends to zero.

For a sequence of discretizations and outputs, if $λ_{j} \to 0$ , $\frac{δ_{j} + η_{j} ∥ x^{†} ∥}{\sqrt{λ_{j}}} \to 0, ∥ (R_{λ_{j}} - R_{λ_{j}, h_{j}}) y_{h_{j}, δ_{j}} ∥ \to 0, \frac{∥ r_{h_{j}, h_{j}} ∥}{λ_{j}} \to 0,$ then ${\tilde{x}}_{h_{j}} \to x^{†}$ . If a certified residual evaluation satisfies $∥ r_{h, h} - {\hat{r}}_{h, h} ∥ \leq ρ_{h, h}$ , its term may be bounded by $(∥ {\hat{r}}_{h, h} ∥ + ρ_{h, h}) / λ$ .

Facts & Assumptions

Given: Countable Choice, compact $A$ , bounded finite-rank $A_{h}$ , Hilbert spaces $X, Y$ , $x^{†} ⊥ \ker A$ , $y = A x^{†}$ , $e$ with $∥ e ∥ \leq δ$ , $∥ A - A_{h} ∥ \leq η$ , $λ > 0$ , the pre-inversion datum $y_{h, δ} = A_{h} x^{†} + e$ , and an output ${\tilde{x}}_{h}$ with its exact residual $r_{h, h}$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is the hypothesis needed for the compact spectral theorem; no full Choice is used.

[F1]

Tikhonov error separates bias, data noise, discretization and solve residual: With $d_{h} = (A - A_{h}) x^{†} = y_{δ} - y_{h, δ}$ , the exact error chain is $(x^{†} - R_{λ} y) + (R_{λ} y - R_{λ} y_{δ}) + R_{λ} d_{h} + (R_{λ} - R_{λ, h}) y_{h, δ} + (R_{λ, h} y_{h, δ} - {\tilde{x}}_{h}) .$ Its last term has norm at most $∥ r_{h, h} ∥ / λ$ .

[F2]

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

[F3]

The shifted Hilbert-space normal operator is invertible with inverse norm at most $1 / λ$ : The inverse of $A_{h}^{*} A_{h} + λ I$ has norm at most $1 / λ$ .

[F4]

Compact-operator Tikhonov bias vanishes on the minimum-norm solution: For compact $A$ and $x^{†} ⊥ \ker A$ , $∥ x^{†} - R_{λ} A x^{†} ∥ \to 0$ as $λ ↓ 0$ .

[F5]

The operator norm as the least bound and as the unit-sphere or unit-ball supremum: $∥ (A - A_{h}) x ∥ \leq ∥ A - A_{h} ∥ ∥ x ∥$ .

Proof

1.1givenF1algebraA1F3

Both shifted normal operators are invertible by [F3], since $A$ and $A_{h}$ are bounded; hence $R_{λ}$ and $R_{λ, h}$ are well-defined. Set $d_{h} = (A - A_{h}) x^{†}$ . Then $y_{δ} - y_{h, δ} = (A x^{†} + e) - (A_{h} x^{†} + e) = d_{h}$ . Substituting this equality into the five-term identity [F1] gives the exact error decomposition for the pre-inversion discretized datum.

2.1F1F2F5step 1.1algebraA1

By [F5], $∥ d_{h} ∥ \leq η ∥ x^{†} ∥$ . By [F2], the data-noise term is at most $δ / (2 \sqrt{λ})$ and the pre-inversion term $∥ R_{λ} d_{h} ∥$ is at most $η ∥ x^{†} ∥ / (2 \sqrt{λ})$ . The last term is at most $∥ r_{h, h} ∥ / λ$ by [F1]. Applying the triangle inequality to the decomposition in step 1.1 yields the displayed error bound, retaining the regularized-operator difference as a separate term.

3.1givenF1F4step 2.1algebraA1∎

Along the stated sequence, the bias term tends to zero by [F4]. The combined data and forward-discretization term tends to zero by the first parameter condition; the regularized-operator difference and normal-residual terms tend to zero by the other two conditions. The bound in the Statement therefore gives $∥ x^{†} - {\tilde{x}}_{h_{j}} ∥ \to 0$ . The residual-evaluation variant follows from $∥ r_{h, h} ∥ \leq ∥ {\hat{r}}_{h, h} ∥ + ρ_{h, h}$ by the triangle inequality.

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Sources

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