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The finite-element discrete adjoint gives the reduced gradient

Statement

Let $O \subseteq R^{m}$ be open and $V = R^{n}$ have its standard Euclidean inner product. Let $R : O \times V \to V$ and $G : O \times V \to Y$ be continuously differentiable, where $Y$ is a finite-dimensional real Euclidean space. Suppose $S : O \to V$ is continuously differentiable, $R (q, S (q)) = 0$ for every $q \in O$ , and the matrix $D_{u} R (q, S (q))$ is invertible at a fixed $q \in O$ . Let $y \in Y$ , $q_{0} \in R^{m}$ , and $α > 0$ . Define

J (q, u) : = \frac{1}{2} ∥ G (q, u) - y ∥_{Y}^{2} + \frac{α}{2} ∥ q - q_{0} ∥_{2}^{2}, Φ (q) : = J (q, S (q)) .

Let $p \in V$ be the unique solution of

(D_{u} R (q, S (q)))^{T} p = \nabla_{u} J (q, S (q)),

where $\nabla_{q} J$ and $\nabla_{u} J$ are the Euclidean representatives of the two partial derivative functionals. Then for every $h \in R^{m}$ ,

D Φ (q) [h] = {⟨ \nabla_{q} J (q, S (q)) - (D_{q} R (q, S (q)))^{T} p, h ⟩}_{R^{m}} .

Consequently $\nabla Φ (q) = \nabla_{q} J (q, S (q)) - (D_{q} R (q, S (q)))^{T} p$ . This is the exact derivative of the specified finite-dimensional model; it asserts no convergence of an iteration that changes $q$ .

Statement references

Finite-element state, residual, and discrete-adjoint contract

Euclidean inner products, norms, and matrix transposes

Finite-dimensional Riesz representation

Chain rule for total derivatives in Euclidean spaces

Hannes Meinlschmidt, Optimization with Partial Differential Equations, §3.2

Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, §§1–3

Facts & Assumptions

Given: The open set, Euclidean spaces, continuously differentiable residual, observation, and state maps, state identity, invertible state Jacobian, data, reference parameter, positive penalty, and adjoint equation in the Statement.

[F1]

The hypotheses give the state identity $R (q, S (q)) = 0$ on $O$ .

[F2]

The vector $p$ is defined in the Statement by $(D_{u} R (q, S (q)))^{T} p = \nabla_{u} J (q, S (q))$ . The assumed invertibility of $D_{u} R (q, S (q))$ makes this solution unique.

[F3]

For continuously differentiable maps between finite-dimensional real Euclidean spaces, the derivative of a composition is the composition of the derivatives (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

Every linear functional on a finite-dimensional real inner-product space has a unique representing vector (Finite-dimensional Riesz representation: every functional is uniquely $v \mapsto ⟨ v, w ⟩$ ).

[F5]

The Euclidean pairing is bilinear and satisfies $⟨ A^{T} p, v ⟩ = ⟨ p, A v ⟩$ for every real matrix $A$ of compatible dimensions (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

1.1givenF1F3algebra

Fix an arbitrary direction $h \in R^{m}$ . Differentiate [F1] at $q$ using [F3]. The chain rule gives $D_{q} R (q, S (q)) h + D_{u} R (q, S (q)) D S (q) [h] = 0$ . Since $D_{u} R (q, S (q))$ is invertible by the Statement, this identity determines $D S (q) [h]$ .

1.2givenF3F5algebra

Apply [F3] to $Φ (q) = J (q, S (q))$ . For Euclidean vectors $z, w$ , bilinearity [F5] gives $\frac{1}{2} ∥ z + t w ∥_{2}^{2} - \frac{1}{2} ∥ z ∥_{2}^{2} = t ⟨ z, w ⟩ + \frac{t^{2}}{2} ∥ w ∥_{2}^{2}$ , so the derivative of a squared-norm term is its residual paired with the derivative of its argument. Applying the chain rule to the data and penalty terms in $J$ gives $D Φ (q) [h] = ⟨ \nabla_{q} J (q, S (q)), h ⟩_{R^{m}} + ⟨ \nabla_{u} J (q, S (q)), D S (q) [h] ⟩_{V}$ .

2.1F2F5step 1.1step 1.2algebra

Substitute [F2] in the second term of step 1.2. The transpose identity [F5] gives $⟨ (D_{u} R)^{T} p, D S [h] ⟩_{V} = ⟨ p, D_{u} R D S [h] ⟩_{V}$ . By step 1.1, $D_{u} R D S [h] = - D_{q} R h$ , so $D Φ (q) [h] = {⟨ \nabla_{q} J (q, S (q)) - (D_{q} R (q, S (q)))^{T} p, h ⟩}_{R^{m}}$ .

3.1F4step 2.1∎

The expression in step 2.1 represents the derivative functional on every $h$ . By [F4] its representing vector is unique, so that vector is $\nabla Φ (q)$ and has the stated formula.

Remarks

The transpose convention is the Euclidean coefficient-space convention fixed in the definition. A continuous PDE adjoint or an adjoint for another coefficient inner product must be derived using that model's own operator pairing.

Depends on

Used by

Finite-budget finite-element PDE Tikhonov inversion Algorithm
An unchecked under-solved adjoint can reverse the reduced gradient Counterexample
A zero algebraic solve residual leaves a separate finite-element model error Example
Certified discrete-adjoint residuals give sound finite statuses Proposition

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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)