TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Finite-element PDE Tikhonov statuses are sound and terminate

Statement

For any finite-element PDE model satisfying its solver and admissibility contract, the finite-budget Tikhonov algorithm returns after finitely many operations one of converged, discrepancy-stopped, linear-solve-failed, inadmissible, or budget-exhausted. A converged return satisfies the exact discrete projected-gradient bound $∥ G_{L} (q) ∥_{2} \leq τ_{g}$ ; a discrepancy-stopped return satisfies $∥ F_{h} (q) - y^{δ} ∥_{Y_{h}} \leq κ δ$ . Every accepted safeguarded step has nonincreasing exact discrete objective $Φ_{h}$ . inadmissible and linear-solve-failed are returned only on the corresponding failed predicate or solver-certificate checks. budget-exhausted asserts no optimizer or continuous-model accuracy.

Statement references

Finite-budget finite-element PDE Tikhonov inversion

Gradient and discrepancy certificate inequalities

Finite solver and objective-interval return contract

Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case

Fernando Gaspoz, Christian Kreuzer, Andreas Veeser, and Winnifried Wollner, A Posteriori Error Analysis for Optimization with PDE Constraints

Facts & Assumptions

Given: The finite budgets, a current coefficient in the admissible set, finite solver and objective-evaluation routines, valid residual/error certificates, and the exact algorithm instructions in the Statement references.

[A1]

The admissibility predicate decides in finite time; projection failure means its call budget is exhausted; objective evaluation fails only on a required state/adjoint solve-contract failure or its call-budget exhaustion; and all routines return in their declared finite budgets (Finite-element PDE solver admissibility and certificate contract).

[F1]

The algorithm performs at most $N_{i t e r} + 1$ outer validations and at most $N_{i t e r} N_{b a c k t r a c k}$ candidate trials, subject to the global finite solve/evaluation budget (Finite-budget finite-element PDE Tikhonov inversion).

[F2]

The solver's gradient-error certificate bounds the exact discrete projected-gradient residual, and its forward-error certificate bounds the exact discrete data residual (Certified discrete-adjoint residuals give sound finite statuses).

[F3]

A returned objective interval $[ℓ_{h} (q), u_{h} (q)]$ contains the exact value $Φ_{h} (q)$ (Finite-element PDE solver admissibility and certificate contract).

Proof

1.1F1F3givenalgebra

Consider any accepted line-search trial $q_{j}$ . By [F3], $Φ_{h} (q_{j}) \leq u_{h} (q_{j})$ and $ℓ_{h} (q) \leq Φ_{h} (q)$ . The algorithm accepts only when $u_{h} (q_{j}) \leq ℓ_{h} (q)$ , so $Φ_{h} (q_{j}) \leq Φ_{h} (q)$ . Every accepted update therefore makes the exact discrete objective nonincreasing.

1.2F2givenalgebra

If the algorithm returns converged, its checked inequality is $∥ {\tilde{G}}_{L} (q) ∥_{2} + ε_{g} \leq τ_{g}$ . By [F2], $∥ G_{L} (q) ∥_{2} \leq ∥ {\tilde{G}}_{L} (q) ∥_{2} + ε_{g}$ , hence $∥ G_{L} (q) ∥_{2} \leq τ_{g}$ . If it returns discrepancy-stopped, its checked inequality is $∥ {\tilde{F}}_{h} - y^{δ} ∥ + ε_{F} \leq κ δ$ ; [F2] gives $∥ F_{h} (q) - y^{δ} ∥ \leq κ δ$ .

1.3A1F1given

The outer counter ranges only over $0, \dots, N_{i t e r}$ , each line-search counter over $0, \dots, N_{b a c k t r a c k} - 1$ , and all solver or objective evaluations return within a finite budget or report failure by [A1]. Therefore each loop either returns one of its named statuses, takes a bounded next step, or exhausts a finite budget. The algorithm cannot run indefinitely.

1.4A1F1given

By its initial and per-iterate branches, inadmissible is returned only when the coefficient is outside $Q_{h}$ or the declared admissibility predicate is false. By its solve-validation branch, linear-solve-failed is returned only when a required solver fails or its residual/error certificate is missing or invalid. A budget-exhausted return follows a depleted global or routine-call budget, a failed finite backtracking search, or the outer-iteration limit; none of these branches claims a minimizer.

2.1step 1.1step 1.2step 1.3step 1.4given∎

Steps 1.1–1.4 establish the stated objective decrease, status bounds, status meanings, and finite termination. The proof refers only to the discrete objective named by the model contract, so it supplies no mesh-convergence or true-parameter conclusion.

Depends on

Used by

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Sources

Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case (standard reference, not scraped)
Fernando Gaspoz, Christian Kreuzer, Andreas Veeser, and Winnifried Wollner, A Posteriori Error Analysis for Optimization with PDE Constraints (standard reference, not scraped)