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Certified inverse bounds turn finite-element solve residuals into state and adjoint error bounds

Statement

Let $A \in R^{n \times n}$ and $B \in R^{r \times r}$ be invertible finite-element state and adjoint system matrices. Let $u = A^{- 1} b$ and $p = B^{- 1} c$ be the exact solutions, and let $\tilde{u}, \tilde{p}$ be computed vectors. Suppose certified residual upper bounds $ρ_{u}, ρ_{p} \geq 0$ and certified operator bounds $K_{u}, K_{p} \geq 0$ satisfy

∥ A \tilde{u} - b ∥_{2} \leq ρ_{u}, ∥ B \tilde{p} - c ∥_{2} \leq ρ_{p}, ∥ A^{- 1} ∥ \leq K_{u}, ∥ B^{- 1} ∥ \leq K_{p} .

Then the exact state and adjoint errors satisfy

∥ u - \tilde{u} ∥_{2} \leq K_{u} ρ_{u}, ∥ p - \tilde{p} ∥_{2} \leq K_{p} ρ_{p} .

The inverse-norm bounds are separate inputs; a residual bound alone gives no state or adjoint error estimate.

Statement references

Certified state and adjoint residual interface

Induced operator norm and its least-bound property

Euclidean coefficient norms

Fernando Gaspoz, Christian Kreuzer, Andreas Veeser, and Winnifried Wollner, A Posteriori Error Analysis for Optimization with PDE Constraints, §3.1

Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, §§2–3

Facts & Assumptions

Given: Invertible state and adjoint matrices, exact and computed vectors, certified residual upper bounds, and certified upper bounds for both inverse norms.

[F1]

The state and adjoint residual upper bounds are the certified norms of the finite-dimensional equations defined by the model contract (Finite-element PDE solver admissibility and certificate contract).

[F2]

The operator norm satisfies $∥ T x ∥_{2} \leq ∥ T ∥ ∥ x ∥_{2}$ for each bounded linear map $T$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Proof

1.1givenF1F2algebra

Since $A u = b$ , we have $A (u - \tilde{u}) = b - A \tilde{u}$ . Apply $A^{- 1}$ and then [F2] to obtain $∥ u - \tilde{u} ∥_{2} = ∥ A^{- 1} (b - A \tilde{u}) ∥_{2} \leq ∥ A^{- 1} ∥ ∥ b - A \tilde{u} ∥_{2} \leq K_{u} ρ_{u}$ .

2.1givenF1F2step 1.1algebra∎

Likewise, $B (p - \tilde{p}) = c - B \tilde{p}$ . Applying $B^{- 1}$ and [F2] gives $∥ p - \tilde{p} ∥_{2} = ∥ B^{- 1} (c - B \tilde{p}) ∥_{2} \leq ∥ B^{- 1} ∥ ∥ c - B \tilde{p} ∥_{2} \leq K_{p} ρ_{p}$ . Together with step 1.1, this proves both bounds.

Remarks

This finite-dimensional estimate is choice-free. It does not provide $K_{u}$ or $K_{p}$ ; the model must certify them separately. For a nonlinear state equation or a continuous-to-discrete comparison, another stability argument is required.

Depends on

Used by

Certified discrete-adjoint residuals give sound finite statuses Proposition

Dependency tree · two levels

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Sources

Fernando Gaspoz, Christian Kreuzer, Andreas Veeser, and Winnifried Wollner, A Posteriori Error Analysis for Optimization with PDE Constraints, §3.1 (standard reference, not scraped)
Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, §§2–3 (standard reference, not scraped)