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

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Certified discrete-adjoint residuals give sound finite statuses

Statement

Let $Q_{h} \subseteq R^{m}$ be a nonempty closed convex set and let $q \in Q_{h}$ . Write $u = S_{h} (q)$ , $p = p_{h} (q)$ , and $g_{h} (q) = \nabla Φ_{h} (q)$ for the exact finite-element state, exact discrete adjoint, and exact discrete-objective gradient. Suppose a solver supplies $\tilde{u}, \tilde{p}$ , a forward output ${\tilde{F}}_{h} = G_{h} (q, \tilde{u})$ , and a computed gradient ${\tilde{g}}_{h} = \nabla_{q} J_{h} (q, \tilde{u}) - (D_{q} R_{h} (q, \tilde{u}))^{T} \tilde{p}$ . It also supplies finite nonnegative bounds $η_{u}, η_{p}, P, L_{J, u}, L_{R, u}, M_{q}, L_{G, u}$ satisfying

∥ u - \tilde{u} ∥_{V_{h}} \leq η_{u}, ∥ p - \tilde{p} ∥_{V_{h}} \leq η_{p}, ∥ p ∥_{V_{h}} \leq P,

∥ \nabla_{q} J_{h} (q, u) - \nabla_{q} J_{h} (q, \tilde{u}) ∥ \leq L_{J, u} η_{u},

∥ (D_{q} R_{h} (q, u))^{T} - (D_{q} R_{h} (q, \tilde{u}))^{T} ∥ \leq L_{R, u} η_{u}, ∥ (D_{q} R_{h} (q, \tilde{u}))^{T} ∥ \leq M_{q},

and $∥ G_{h} (q, u) - G_{h} (q, \tilde{u}) ∥_{Y_{h}} \leq L_{G, u} η_{u}$ . Define

ε_{g} : = (L_{J, u} + L_{R, u} P) η_{u} + M_{q} η_{p}, ε_{F} : = L_{G, u} η_{u} .

Then $∥ {\tilde{g}}_{h} - g_{h} (q) ∥ \leq ε_{g}$ and $∥ {\tilde{F}}_{h} - F_{h} (q) ∥_{Y_{h}} \leq ε_{F}$ . For $L > 0$ , define

G_{L} (q) : = L (q - P_{Q_{h}} (q - g_{h} (q) / L)), {\tilde{G}}_{L} (q) : = L (q - P_{Q_{h}} (q - {\tilde{g}}_{h} / L)) .

Then $∥ G_{L} (q) ∥_{2} \leq ∥ {\tilde{G}}_{L} (q) ∥_{2} + ε_{g}, ∥ F_{h} (q) - y^{δ} ∥_{Y_{h}} \leq ∥ {\tilde{F}}_{h} - y^{δ} ∥_{Y_{h}} + ε_{F} .$ Thus the finite-element algorithm may certify discrete projected stationarity and discrete discrepancy from these bounds. The state and adjoint residuals must first be converted to valid $η_{u}, η_{p}$ bounds under the chosen model's hypotheses; raw residuals alone do not imply the asserted gradient or mesh error.

Statement references

Finite-element state, adjoint, and error-bound contract

Exact reduced gradient from the finite-element discrete adjoint

Linear finite-element residual-to-error certificate

Existence and uniqueness of Euclidean nearest-point projection

Nonexpansiveness of Euclidean projection

Induced operator norm bound

Euclidean norm and triangle inequality

Hannes Meinlschmidt, Optimization with Partial Differential Equations, complete lecture notes

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-element model, $q \in Q_{h}$ , exact state $u = S_{h} (q)$ , exact adjoint $p = p_{h} (q)$ , the solver estimates and constants in the Statement, data $y^{δ}$ , and $L > 0$ .

[A1]

The model contract supplies $∥ u - \tilde{u} ∥ \leq η_{u}$ , $∥ p - \tilde{p} ∥ \leq η_{p}$ , $∥ p ∥ \leq P$ , the derivative bounds $L_{J, u}, L_{R, u}, M_{q}$ , and the observation bound $L_{G, u} η_{u}$ (Finite-element PDE solver admissibility and certificate contract).

[F1]

The exact reduced gradient is $g_{h} (q) = \nabla_{q} J_{h} (q, u) - (D_{q} R_{h} (q, u))^{T} p$ (The finite-element discrete adjoint gives the reduced gradient).

[F2]

Every point in a nonempty closed convex subset of finite-dimensional Euclidean space has a unique nearest point (Every point has a unique nearest point in a nonempty closed Euclidean convex set).

[F3]

Euclidean projection onto a nonempty closed convex set is nonexpansive (Projection onto a closed convex set is nonexpansive).

[F4]

For a bounded linear map $T$ , the induced operator norm gives $∥ T x ∥ \leq ∥ T ∥ ∥ x ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[F5]

Euclidean norms satisfy the triangle inequality (Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation).

[F6]

For linear finite-element state and adjoint systems, residual upper bounds and separately certified inverse-norm bounds imply the error estimates $η_{u}, η_{p}$ ; nonlinear systems require a separate model-specific stability argument (Certified inverse bounds turn finite-element solve residuals into state and adjoint error bounds).

Proof

1.1A1F1F4F6algebra

By [F1], the exact discrete gradient has the stated adjoint formula. In a linear finite-element state/adjoint branch, [F6] supplies one residual-to-error path for the contract's $η_{u}, η_{p}$ values when inverse-norm bounds are certified; a nonlinear model needs its own stability argument. Using the contract estimates in [A1], $g_{h} (q) - {\tilde{g}}_{h} = \nabla_{q} J_{h} (q, u) - \nabla_{q} J_{h} (q, \tilde{u}) - (D_{q} R_{h} (q, u))^{T} p + (D_{q} R_{h} (q, \tilde{u}))^{T} \tilde{p}$ . Add and subtract $(D_{q} R_{h} (q, \tilde{u}))^{T} p$ . The first objective-gradient difference has norm at most $L_{J, u} η_{u}$ by [A1]. The derivative-matrix difference term has norm at most $L_{R, u} η_{u} ∥ p ∥ \leq L_{R, u} P η_{u}$ by [A1] and [F4]. The remaining term has norm at most $M_{q} ∥ p - \tilde{p} ∥ \leq M_{q} η_{p}$ by [A1] and [F4]. Summing these three bounds gives $∥ {\tilde{g}}_{h} - g_{h} (q) ∥ \leq ε_{g}$ . The observation certificate in [A1] directly gives $∥ {\tilde{F}}_{h} - F_{h} (q) ∥ \leq ε_{F}$ .

2.1F2F3F5step 1.1algebra

By [F2], the projections in $G_{L} (q)$ and ${\tilde{G}}_{L} (q)$ exist and are unique. Put $x = q - {\tilde{g}}_{h} / L$ and $z = q - g_{h} (q) / L$ . Nonexpansiveness [F3] and step 1.1 give $∥ {\tilde{G}}_{L} (q) - G_{L} (q) ∥_{2} = L ∥ P_{Q_{h}} (x) - P_{Q_{h}} (z) ∥_{2} \leq L ∥ x - z ∥_{2} = ∥ {\tilde{g}}_{h} - g_{h} (q) ∥_{2} \leq ε_{g}$ . The triangle inequality [F5] now gives $∥ G_{L} (q) ∥_{2} \leq ∥ {\tilde{G}}_{L} (q) ∥_{2} + ε_{g}$ .

3.1F5step 1.1step 2.1algebra∎

The triangle inequality [F5] and step 1.1 give $∥ F_{h} (q) - y^{δ} ∥_{Y_{h}} \leq ∥ F_{h} (q) - {\tilde{F}}_{h} ∥_{Y_{h}} + ∥ {\tilde{F}}_{h} - y^{δ} ∥_{Y_{h}} \leq ε_{F} + ∥ {\tilde{F}}_{h} - y^{δ} ∥_{Y_{h}}$ . Together with step 2.1, this proves both certificate inequalities.

Remarks

The certificates $η_{u}, η_{p}$ are model-specific consequences of residuals and stability constants; they must be justified before this proposition is applied. The published linear elliptic finite-element residual result has its own continuous-objective and full Strang hypotheses; it is not substituted for the discrete target error bounds above.

Depends on

Used by

Finite-budget finite-element PDE Tikhonov inversion Algorithm
An unchecked under-solved adjoint can reverse the reduced gradient Counterexample
Finite-element PDE Tikhonov statuses are sound and terminate Theorem

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Sources

Hannes Meinlschmidt, Optimization with Partial Differential Equations, complete lecture notes, §3.2 (standard reference, not scraped)
Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, §§1–3 (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)