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The elliptic adjoint gives the reduced parameter gradient

Statement

Assume Countable Choice and the elliptic state setting of Parameter-dependent coercive elliptic variational states and objectives and A uniformly coercive elliptic variational state is C1 in its parameter. Let $J : V \times Θ \to R$ be $C^{1}$ . For a parameter $θ$ , let $p_{θ} \in V$ solve the adjoint variational problem $a_{θ} (v, p_{θ}) = D_{u} J (u_{θ}, θ) [v] (v \in V) .$ Then the reduced objective $j (θ) = J (u_{θ}, θ)$ is $C^{1}$ and $D_{θ} j (θ) [η] = D_{θ} J (u_{θ}, θ) [η] + D_{θ} ℓ_{θ} [η] (p_{θ}) - D_{θ} a_{θ} [η] (u_{θ}, p_{θ}) .$ No symmetry of $a_{θ}$ is required; the adjoint uses the transposed variational form displayed above.

Facts & Assumptions

Given: The uniformly coercive real variational state, its $C^{1}$ dependence, the $C^{1}$ objective, and Countable Choice.

[F1]

The state derivative $w_{η}$ is the unique solution of $a_{θ} (w_{η}, v) = D_{θ} ℓ_{θ} [η] (v) - D_{θ} a_{θ} [η] (u_{θ}, v)$ (A uniformly coercive elliptic variational state is C1 in its parameter).

[F2]

A bounded linear functional has a unique adjoint variational solution with $a_{θ} (v, p) = G (v)$ under Countable Choice (Well-posedness of the adjoint variational problem, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F3]

The chain rule for Fr'echet differentiable maps between real Banach spaces gives $D (J \circ Γ) = D J (Γ) D Γ$ (Chain sum product and composition rules for Banach derivatives).

[A1]

$J$ and its first derivatives are continuous on the state/parameter neighborhood.

Proof

1.1F2A1given

The adjoint datum $D_{u} J (u_{θ}, θ)$ is a bounded linear functional on $V$ . By [F2], the adjoint $p_{θ}$ exists uniquely, with the equation in the Statement.

1.2F1F2F3given

Let $Γ (θ) = (u_{θ}, θ)$ . The state differentiability in [F1] gives $D Γ (θ) [η] = (w_{η}, η)$ . Apply the Banach-space chain rule [F3] to $j = J \circ Γ$ to obtain $D_{θ} j [η] = D_{θ} J (u_{θ}, θ) [η] + D_{u} J (u_{θ}, θ) [w_{η}] .$ Since $J$ and the state map are $C^{1}$ , this derivative is continuous, so $j$ is $C^{1}$ . Use the adjoint equation with $v = w_{η}$ to replace the second term by $a_{θ} (w_{η}, p_{θ})$ .

2.1F1step 1.2

Use the state-sensitivity equation [F1] with test function $v = p_{θ}$ . It gives $a_{θ} (w_{η}, p_{θ}) = D_{θ} ℓ_{θ} [η] (p_{θ}) - D_{θ} a_{θ} [η] (u_{θ}, p_{θ})$ .

3.1step 1.2step 2.1algebra∎

Substitute step 2.1 into step 1.2 to obtain the formula. The argument uses only the displayed adjoint equation, so it remains valid for a nonsymmetric coercive form.

Depends on

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Sources

Hannes Meinlschmidt, Optimization with Partial Differential Equations, complete lecture notes, §§1.6 and 3.2 (standard reference, not scraped)
Michael B. Giles and Niles A. Pierce, An Introduction to the Adjoint Approach to Design, §§2.4–3 (standard reference, not scraped)