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Discrete continuous and pde adjoint sensitivities

1 · Prerequisites

2 · Summary

This page separates derivatives of a finite computational graph from derivatives of the exact ODE flow. It gives the continuous and exact finite-graph adjoint formulas. Local value and first-derivative consistency of the actual state and running-cost maps, together with uniform derivative stability and common-tube hypotheses, yield order- $p$ state, tangent, nodal-adjoint and gradient bounds. Differentiating actual explicit Runge–Kutta stages and the matching cost quadrature supplies this local consistency from augmented local order; the stated augmented smoothness and coefficient conditions give concrete instances through order four, including RK4. A separate weighted perturbation theorem retains explicit state, adjoint, derivative-data, endpoint and same-weight quadrature hypotheses. State accuracy alone does not establish gradient accuracy. Implicit Runge–Kutta sensitivities apply to a selected nonsingular algebraic stage branch, not to a stopped stage iteration.

For elliptic equations, the reduced gradient uses the transposed variational adjoint. The finite-element error budget is restricted to conforming state and adjoint approximations and keeps derivative-data defects separate from state and adjoint errors. An $L^{2}$ rate additionally needs dual approximation and $L^{2}$ -bounded parameter derivative maps. The parabolic formula is conditional on differentiable state and adjoint solutions; the finite-element estimate retains the full time-dependent Ritz residual derivative $P_{h}^{'} (t) u (t) + P_{h} (t) u^{'} (t) - u^{'} (t)$ . The QR identity concerns exact finite-time tangent-volume change and does not assert an infinite-time Lyapunov exponent.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Finite parameterized state-step graphs and objectives

Definition

Let $Θ \subseteq R^{m}$ be open, let $N \geq 1$ , and for $0 \leq n \leq N$ let $U_{n} \subseteq R^{d_{n}}$ be open. A finite parameterized state-step graph consists of a $C^{1}$ initial map $x_{0} : Θ \to U_{0}$ and $C^{1}$ step maps $Ψ_{n} : U_{n} \times Θ \to U_{n + 1}$ for $0 \leq n < N$ , such that the recursion $x_{n + 1} (θ) = Ψ_{n} (x_{n} (θ), θ)$ is defined for every $θ$ in the chosen parameter neighborhood. A $C^{1}$ running objective consists of maps $ℓ_{n} : U_{n} \times Θ \to R$ and a terminal objective $ϕ : U_{N} \times Θ \to R$ , all $C^{1}$ . Its reduced objective is $J (θ) = ϕ (x_{N} (θ), θ) + \sum_{n = 0}^{N - 1} ℓ_{n} (x_{n} (θ), θ) .$ The maps and objective define one finite real-arithmetic graph. A gradient of this graph is the derivative of this finite composition; it is not the derivative of an exact implicit solution unless the step map itself is that solution map.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Parameter-dependent coercive elliptic variational states and objectives

Definition

Let $V$ be a real Hilbert space, let $Θ \subseteq R^{m}$ be open, and write $V^{*}$ for the continuous real dual. A parameter-dependent coercive elliptic variational problem is a family of bounded bilinear forms $a_{θ} : V \times V \to R$ and bounded linear functionals $ℓ_{θ} \in V^{*}$ such that, on each parameter neighborhood under discussion, $∥ a_{θ} ∥ \leq M$ and $a_{θ} (v, v) \geq α ∥ v ∥_{V}^{2}$ for fixed $M < \infty$ and $α > 0$ . Operator-norm $C^{1}$ data means that the maps $θ \mapsto A_{θ} \in L (V, V^{*})$ , $A_{θ} u : = a_{θ} (u, \cdot)$ , and $θ \mapsto ℓ_{θ} \in V^{*}$ are continuously Fréchet differentiable. A state at $θ$ is a vector $u_{θ} \in V$ satisfying $a_{θ} (u_{θ}, v) = ℓ_{θ} (v)$ for every $v \in V$ . When a state is specified, a $C^{1}$ objective $J : V \times Θ \to R$ defines the reduced objective $j (θ) = J (u_{θ}, θ)$ . This definition alone asserts neither state existence nor uniqueness of an optimizer.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Parameter-dependent parabolic variational states and objectives

Definition

Let $T > 0$ , let $(V, H, V^{*})$ be the real evolution triple and let $Θ \subseteq R^{m}$ be open. A parameter-dependent parabolic variational problem specifies, for each $θ \in Θ$ , a bilinear form $a_{θ} (t; \cdot, \cdot)$ and data $f_{θ} \in L^{2} (0, T; V^{*})$ , $u_{0, θ} \in H$ . A state is a function $u_{θ} \in W (0, T; V)$ with continuous $H$ trace, $u_{θ} (0) = u_{0, θ}$ , and $⟨ u_{θ}^{'} (t), v ⟩_{V^{*}, V} + a_{θ} (t; u_{θ} (t), v) = ⟨ f_{θ} (t), v ⟩_{V^{*}, V}$ for every $v \in V$ and almost every $t$ . When a state is specified, let $ϕ : H \times Θ \to R$ be differentiable and let $L : (0, T) \times V \times Θ \to R$ be differentiable in $(v, θ)$ for almost every $t$ . Require $t \mapsto L (t, u_{θ} (t), θ)$ to be measurable and to belong to $L^{1} (0, T)$ for every parameter under consideration. Then the finite real number $j (θ) = ϕ (u_{θ} (T), θ) + \int_{0}^{T} L (t, u_{θ} (t), θ) d t$ is its reduced objective. This definition does not assert existence of a state, parameter differentiability of the state or reduced objective, or optimality.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

Parameter-dependent ODE flows have the variational tangent on a common tube

Statement

Let $T \geq 0$ , let $Θ \subseteq R^{m}$ and $D \subseteq R \times R^{d} \times Θ$ be open, and let $f : D \to R^{d}$ be continuously differentiable. Let $V_{θ} \subseteq Θ$ be open and $u_{0} : V_{θ} \to R^{d}$ be $C^{1}$ . For $T > 0$ , assume that for every $θ \in V_{θ}$ the unique solution of $u^{'} = f (t, u, θ)$ , $u (0) = u_{0} (θ)$ , exists on $[0, T]$ ; for $T = 0$ , instead define $u_{θ} (0) = u_{0} (θ)$ . Assume there is a bounded convex open set $O \subset D$ with compact closure in $D$ containing all triples $(t, u_{θ} (t), θ)$ . Then $θ \mapsto u_{θ}$ is $C^{1}$ as a map to $C ([0, T]; R^{d})$ . For any $θ_{0} \in V_{θ}$ and $T > 0$ , its tangent matrix $S (t) = D_{θ} u_{θ} (t) ∣_{θ = θ_{0}}$ is the unique solution of $S^{'} (t) = D_{u} f (t, u_{θ_{0}} (t), θ_{0}) S (t) + D_{θ} f (t, u_{θ_{0}} (t), θ_{0}), S (0) = D u_{0} (θ_{0}) .$ For $T = 0$ , evaluation identifies $C ([0, 0]; R^{d})$ with $R^{d}$ and the parameter derivative is $S (0) = D u_{0} (θ_{0})$ ; no time derivative or differential-equation solution on the singleton is asserted. The result is local to the stated common compact tube and claims no differentiability outside it.

Facts & Assumptions

Given: The open domain, $C^{1}$ field, parameter neighborhood, $C^{1}$ initial map, common bounded convex tube with compact closure in $D$ , and the stated IVP solutions for $T > 0$ or initial-evaluation curves for $T = 0$ .

[F1]

A first-order IVP solution satisfies its Volterra equation (A first-order initial value problem is equivalent to its Volterra integral equation).

[F2]

A uniform bound on the derivative over a convex open set gives the corresponding Lipschitz bound (On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ ).

[F3]

If $g (t) \leq A + L \int_{0}^{t} g (s) d s$ for nonnegative continuous $g$ , then $g (t) \leq A e^{L t}$ (Gronwall's integral inequality with variable and constant coefficients).

[F4]

The norm of a vector-valued integral is at most the integral of its norm (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[F5]

A continuous coefficient matrix on a compact interval gives a unique global solution of its linear matrix IVP (Linear matrix ODEs have unique global solutions on a fixed interval).

[A1]

The parameter neighborhood and convex tube $O$ ensure all secant segments used below remain in $O$ . Continuity of $D f$ on the compact closure $\overline{O}$ makes $D f$ bounded and uniformly continuous there.

Proof

1.1F1F2F3F4A1givenalgebra

If $T = 0$ , then $C ([0, 0]; R^{d})$ is identified with $R^{d}$ by evaluation at $0$ , so $θ \mapsto u_{θ}$ is the given $C^{1}$ map $u_{0}$ and the initial condition determines its parameter tangent at the only time; this branch asserts no time derivative or IVP on the singleton. If $d = 0$ , then $R^{d} = {0}$ , so the state curve and tangent are the unique zero maps. If $m = 0$ , the parameter domain has at most one point; its map into $C ([0, T]; R^{d})$ has the unique derivative from $R^{0}$ , and the sensitivity matrix has no columns. In each case the asserted conclusion follows; hence assume $T > 0$ and $d, m \geq 1$ . Let $δ_{h} (t) = u_{θ_{0} + h} (t) - u_{θ_{0}} (t)$ . Differentiability of $u_{0}$ gives $∥ u_{0} (θ_{0} + h) - u_{0} (θ_{0}) ∥_{2} \leq C_{0} ∥ h ∥_{2}$ for small $h$ . Subtract the Volterra equations [F1] and apply [F2] on the segment between $(u_{θ_{0} + h} (t), θ_{0} + h)$ and $(u_{θ_{0}} (t), θ_{0})$ . Boundedness of $D f$ on $O$ gives $∥ δ_{h} (t) ∥_{2} \leq C_{0} ∥ h ∥_{2} + L \int_{0}^{t} (∥ δ_{h} (s) ∥_{2} + ∥ h ∥_{2}) d s .$ Since $t \leq T$ , [F3] yields $\sup_{t} ∥ δ_{h} (t) ∥_{2} \leq (C_{0} + L T) e^{L T} ∥ h ∥_{2}$ .

1.2A1F5construct

Put $A (t) = D_{u} f (t, u_{θ_{0}} (t), θ_{0})$ and let $B_{j} (t)$ be column $j$ of $D_{θ} f (t, u_{θ_{0}} (t), θ_{0})$ . For each parameter coordinate $j$ , the augmented vector $Z_{j} = (S_{\cdot j}, 1)$ solves the homogeneous linear system $Z_{j}^{'} (t) = (\begin{array}{cc} A (t) & B_{j} (t) \\ 0 & 0 \end{array}) Z_{j} (t), Z_{j} (0) = (\begin{array}{c} D u_{0} (θ_{0})_{\cdot j} \\ 1 \end{array}) .$ Embed $Z_{j} (0)$ as one column of a square $(d + 1) \times (d + 1)$ initial matrix, with all other columns zero. The unique linear matrix solution supplied by [F5] has that vector solution as the selected column. Thus the tangent equation has a unique solution for each of the finitely many columns.

2.1A1F2step 1.1algebra

Put $A (t) = D_{u} f (t, u_{θ_{0}} (t), θ_{0})$ and $B (t) = D_{θ} f (t, u_{θ_{0}} (t), θ_{0})$ , and define $ρ_{h} (t) = f (t, u_{θ_{0}} (t) + δ_{h} (t), θ_{0} + h) - f (t, u_{θ_{0}} (t), θ_{0}) - A (t) δ_{h} (t) - B (t) h$ . Set $z_{0} (t) = (t, u_{θ_{0}} (t), θ_{0})$ and $v_{h} (t) = (0, δ_{h} (t), h)$ , and let $ω (r) = \sup {∥ D f (z) - D f (w) ∥ : z, w \in \overline{O}, ∥ z - w ∥_{2} \leq r}$ . Uniform continuity of $D f$ on $\overline{O}$ gives $ω (r) \to 0$ as $r ↓ 0$ . If $v_{h} (t) \neq 0$ , the set $O \cap B (z_{0} (t), 2 ∥ v_{h} (t) ∥_{2})$ is convex and open and contains both endpoints of the secant. For $R_{t} (z) = f (z) - D f (z_{0} (t)) (z - z_{0} (t))$ , its derivative on this set has norm at most $ω (2 ∥ v_{h} (t) ∥_{2})$ . Applying [F2] there gives $∥ ρ_{h} (t) ∥_{2} \leq ω (2 ∥ v_{h} (t) ∥_{2}) ∥ v_{h} (t) ∥_{2}$ ; if $v_{h} (t) = 0$ , then $ρ_{h} (t) = 0$ . Step 1.1 gives $\sup_{t} ∥ v_{h} (t) ∥_{2} = O (∥ h ∥_{2})$ , so $\sup_{t} ∥ ρ_{h} (t) ∥_{2} = o (∥ h ∥_{2})$ . Also $u_{0} (θ_{0} + h) - u_{0} (θ_{0}) = D u_{0} (θ_{0}) h + o (∥ h ∥_{2})$ .

3.1F1F3F4step 1.2step 2.1

Let $E_{h} (t) = δ_{h} (t) - S (t) h$ , and let $r_{0} (h) = u_{0} (θ_{0} + h) - u_{0} (θ_{0}) - D u_{0} (θ_{0}) h$ . The Volterra equations and the field remainder $ρ_{h}$ from step 2.1 give $E_{h} (t) = r_{0} (h) + \int_{0}^{t} (A (s) E_{h} (s) + ρ_{h} (s)) d s$ , where $A (s) = D_{u} f (s, u_{θ_{0}} (s), θ_{0})$ . By [F4], $∥ E_{h} (t) ∥_{2} \leq ∥ r_{0} (h) ∥_{2} + T \sup_{s} ∥ ρ_{h} (s) ∥_{2} + L \int_{0}^{t} ∥ E_{h} (s) ∥_{2} d s$ . Both first terms are $o (∥ h ∥_{2})$ by step 2.1 and differentiability of $u_{0}$ . Applying [F3] yields $\sup_{t} ∥ E_{h} (t) ∥_{2} = o (∥ h ∥_{2})$ . Thus the Fréchet derivative at $θ_{0}$ exists and equals $S$ .

4.1A1F3step 1.1step 1.2step 3.1∎

For any sequence $θ_{j} \to θ_{0}$ in $V_{θ}$ , step 1.1 gives $u_{θ_{j}} \to u_{θ_{0}}$ uniformly, and uniform continuity of $D f$ on $\overline{O}$ makes the tangent coefficients $A_{j}, B_{j}$ converge uniformly to $A, B$ . Subtracting the tangent integral equations gives $S_{j} (t) - S (t) = D u_{0} (θ_{j}) - D u_{0} (θ_{0}) + \int_{0}^{t} (A_{j} (s) (S_{j} (s) - S (s)) + (A_{j} (s) - A (s)) S (s) + B_{j} (s) - B (s)) d s$ . The initial and coefficient differences tend to zero, while $A_{j}$ are uniformly bounded; [F3] then gives $∥ S_{j} - S ∥_{C ([0, T])} \to 0$ . Since $θ_{0}$ is arbitrary in $V_{θ}$ , the derivative is continuous throughout that set, so the flow map is $C^{1}$ .

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

A tangent and adjoint pairing gives the ODE gradient boundary identity

Statement

Let $T > 0$ , let $S \in C^{1} ([0, T]; R^{d \times m})$ and $p \in C^{1} ([0, T]; R^{d})$ satisfy $S^{'} = A S + B$ and $p^{'} = - A^{T} p - g$ , where $A, B, g$ are continuous and have compatible dimensions. Then $p (T)^{T} S (T) - p (0)^{T} S (0) = \int_{0}^{T} (- g (t)^{T} S (t) + p (t)^{T} B (t)) d t .$

Facts & Assumptions

Given: $T > 0$ , the continuously differentiable matrix and vector functions and their differential equations in the Statement.

[F1]

The Euclidean pairing is the finite coordinate sum $x^{T} y = \sum_{i} x_{i} y_{i}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F2]

The derivative of a product is $(f g)^{'} = f^{'} g + f g^{'}$ (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[F3]

If $G^{'} = q$ and $q$ is integrable on $[0, T]$ , then $\int_{0}^{T} q = G (T) - G (0)$ (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

Proof

1.1givenF1F2algebra

For each column $j$ , write $(p^{T} S)_{j} = \sum_{i = 1}^{d} p_{i} S_{i j}$ . By [F2] coordinatewise, $((p^{T} S)_{j})^{'} = \sum_{i} p_{i}^{'} S_{i j} + p_{i} S_{i j}^{'}$ . Substituting the two equations and collecting the columns gives $(p^{T} S)^{'} = (- A^{T} p - g)^{T} S + p^{T} (A S + B) = - g^{T} S + p^{T} B$ , because $(A^{T} p)^{T} S = p^{T} A S$ .

2.1F3step 1.1∎

For each column $j$ , the scalar function $(p^{T} S)_{j}$ is $C^{1}$ , and the $j$ th component of the right-hand side in step 1.1 is continuous and hence integrable. Apply [F3] to each scalar component on $[0, T]$ ; collecting these identities gives the stated row-vector endpoint identity.

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

The reverse recursion is the exact gradient of a finite state-step objective

Statement

Use the setting of Finite parameterized state-step graphs and objectives. Write $A_{n} = D_{x} Ψ_{n} (x_{n}, θ)$ , $B_{n} = D_{θ} Ψ_{n} (x_{n}, θ)$ , and let $S_{n} = D_{θ} x_{n}$ . Then $S_{0} = D x_{0} (θ), S_{n + 1} = A_{n} S_{n} + B_{n} .$ Define $p_{N} = D_{x} ϕ (x_{N}, θ)^{T}$ and, backward for $n = N - 1, \dots, 0$ , $p_{n} = D_{x} ℓ_{n} (x_{n}, θ)^{T} + A_{n}^{T} p_{n + 1} .$ The exact gradient of the finite objective is $\nabla_{θ} J = D_{θ} ϕ (x_{N}, θ)^{T} + S_{0}^{T} p_{0} + \sum_{n = 0}^{N - 1} (D_{θ} ℓ_{n} (x_{n}, θ)^{T} + B_{n}^{T} p_{n + 1}) .$ The recursion is the reverse derivative of the finite graph in exact real arithmetic; it makes no convergence claim about an iteration used to choose $θ$ .

Facts & Assumptions

Given: The open finite state-step graph, its $C^{1}$ maps, objective, trajectory and parameter in the Statement.

[F1]

Finite parameterized state-step graphs and objectives: $x_{n + 1} = Ψ_{n} (x_{n}, θ)$ and $J = ϕ (x_{N}, θ) + \sum_{n < N} ℓ_{n} (x_{n}, θ)$ with all displayed maps $C^{1}$ .

[F2]

Reverse-mode automatic differentiation for a scalar graph output: reverse mode accumulates each primitive pullback $D_{k} ϕ_{j}^{T} {\overset{ˉ}{z}}_{j}$ at every parent slot in reverse topological order.

[F3]

Reverse mode computes the exact gradient of the real-arithmetic graph: reverse mode returns the exact gradient of the finite $C^{1}$ real-arithmetic graph.

[F4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : the derivative of a composition is the composition of its derivatives.

Proof

1.1F1F4

By the chain rule [F4] applied to $x_{n + 1} (θ) = Ψ_{n} (x_{n} (θ), θ)$ , $S_{n + 1} = A_{n} S_{n} + B_{n}$ ; differentiating the initial map gives $S_{0} = D x_{0} (θ)$ . These are the forward tangent equations for this finite composition.

1.2F1F2F3

Apply the reverse accumulation rule [F2] to the finite graph of states and objective summands in [F1]. The terminal contribution seeds $p_{N} = D_{x} ϕ^{T}$ ; at each state node $x_{n}$ , its local running-cost edge contributes $D_{x} ℓ_{n}^{T}$ , while its state-transition edge contributes $A_{n}^{T} p_{n + 1}$ . Thus the accumulated state cotangent is exactly $p_{n}$ as defined in the Statement, and the direct parameter edges contribute $D_{θ} ϕ^{T}$ and $D_{θ} ℓ_{n}^{T} + B_{n}^{T} p_{n + 1}$ . By [F3], reverse mode returns the exact derivative of this finite C1 graph.

2.1F1F4step 1.1

Differentiate the objective in [F1] using the forward sensitivities in step 1.1. The chain rule gives $\nabla_{θ} J = D_{θ} ϕ^{T} + S_{N}^{T} D_{x} ϕ^{T} + \sum_{n = 0}^{N - 1} (D_{θ} ℓ_{n}^{T} + S_{n}^{T} D_{x} ℓ_{n}^{T}) .$ This is the direct tangent expression, with all derivatives evaluated along the trajectory.

3.1F1step 1.1step 2.1algebra∎

The adjoint equation gives $p_{n}^{T} S_{n} = D_{x} ℓ_{n} S_{n} + p_{n + 1}^{T} A_{n} S_{n}$ . Since $S_{n + 1} = A_{n} S_{n} + B_{n}$ , rearranging yields $D_{x} ℓ_{n} S_{n} = p_{n}^{T} S_{n} - p_{n + 1}^{T} S_{n + 1} + p_{n + 1}^{T} B_{n} .$ Summing over $n$ telescopes; $p_{N} = D_{x} ϕ^{T}$ cancels the terminal tangent term in step 2.1 and leaves $S_{0}^{T} p_{0} + \sum_{n} B_{n}^{T} p_{n + 1}$ . Adding the direct parameter derivatives from step 2.1 gives the displayed adjoint formula.

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

The continuous ODE adjoint gives the parameter gradient with endpoint terms

Statement

Assume $T > 0$ . Let $u_{θ}$ solve the parameter-dependent ODE of Parameter-dependent ODE flows have the variational tangent on a common tube on $[0, T]$ . Let $J (θ) = ϕ (u_{θ} (T), θ) + \int_{0}^{T} ℓ (t, u_{θ} (t), θ) d t,$ where $ϕ$ and $ℓ$ are $C^{1}$ on a neighborhood of the trajectory tube and their first derivatives are bounded and uniformly continuous there. Let $S = D_{θ} u_{θ}$ be the tangent from that lemma. Let $p$ solve the terminal adjoint problem $- p^{'} (t) = D_{u} f (t, u_{θ} (t), θ)^{T} p (t) + D_{u} ℓ (t, u_{θ} (t), θ)^{T}, p (T) = D_{u} ϕ (u_{θ} (T), θ)^{T} .$ Then $\nabla_{θ} J = D_{θ} ϕ (u_{θ} (T), θ)^{T} + D_{θ} u_{0} (θ)^{T} p (0) + \int_{0}^{T} (D_{θ} ℓ (t, u_{θ}, θ)^{T} + D_{θ} f (t, u_{θ}, θ)^{T} p (t)) d t .$ This formula separates initial, terminal, running-cost and vector-field parameter terms. It applies only while the common-tube and differentiability hypotheses hold.

Facts & Assumptions

Given: $T > 0$ , the common-tube ODE, $C^{1}$ initial map, objective, and bounded uniformly continuous first derivatives in the Statement.

[F1]

Parameter-dependent ODE flows have the variational tangent on a common tube: the parameter flow is $C^{1}$ and $S^{'} = D_{u} f S + D_{θ} f$ , $S (0) = D_{θ} u_{0}$ .

[F2]

A tangent and adjoint pairing gives the ODE gradient boundary identity: if $S^{'} = A S + B$ and $p^{'} = - A^{T} p - g$ , then $p (T)^{T} S (T) - p (0)^{T} S (0) = \int_{0}^{T} (- g^{T} S + p^{T} B) d t$ .

[F3]

Linear matrix ODEs have unique global solutions on a fixed interval: a continuous coefficient linear matrix IVP has a unique solution on the full compact interval.

[F4]

The total-derivative chain rule differentiates a composition of $C^{1}$ Euclidean maps (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[A1]

All objective derivatives and ODE derivative coefficients are continuous and bounded on the common compact tube, and their first derivatives have the uniform continuity needed to differentiate the integral objective.

Proof

1.1F3givenconstruct

Fix $θ$ and write $A (t) = D_{u} f (t, u_{θ} (t), θ)$ and $g (t) = D_{u} ℓ (t, u_{θ} (t), θ)^{T}$ . The coefficient matrix $C (t) = (\begin{array}{cc} - A (t)^{T} & - g (t) \\ 0 & 0 \end{array})$ is continuous. Apply [F3] with initial time $T$ to the homogeneous matrix equation $Z^{'} = C (t) Z$ , choosing an initial matrix whose first column is $(D_{u} ϕ (u_{θ} (T), θ)^{T}, 1)^{T}$ and whose other columns are zero. Its first column is $(p (t), 1)^{T}$ and solves the terminal adjoint equation. Any other solution embeds as a column of such a matrix solution, so uniqueness in [F3] gives uniqueness of $p$ .

1.2F1F4A1given

By [F1] and the chain rule [F4], the terminal derivative is $D_{θ} ϕ^{T} + S (T)^{T} D_{u} ϕ^{T}$ . For the running term, a parameter increment $h$ gives $Δ u (t) = S (t) h + o (∥ h ∥)$ uniformly in $t$ by [F1]. Uniform continuity of the objective derivatives on the common compact tube gives a uniform first-order remainder $o (∥ h ∥)$ along the segment from $(u_{θ} (t), θ)$ to $(u_{θ + h} (t), θ + h)$ ; the common integrable majorant permits integration of this remainder. Applying [F4] pointwise and integrating therefore gives the running derivative $\int_{0}^{T} (D_{θ} ℓ^{T} + S^{T} D_{u} ℓ^{T}) d t$ . Consequently $\nabla_{θ} J = D_{θ} ϕ^{T} + S (T)^{T} D_{u} ϕ^{T} + \int_{0}^{T} (D_{θ} ℓ^{T} + S^{T} D_{u} ℓ^{T}) d t$ .

1.3F1F2given

In [F2], set $A = D_{u} f$ , $B = D_{θ} f$ , and $g = D_{u} ℓ^{T}$ . The adjoint equation in the Statement and tangent equation in [F1] satisfy its hypotheses. Transposing that row pairing identity and using $p (T) = D_{u} ϕ^{T}$ gives $S (T)^{T} D_{u} ϕ^{T} - D_{θ} u_{0}^{T} p (0) = \int_{0}^{T} (- S^{T} D_{u} ℓ^{T} + D_{θ} f^{T} p) d t .$

2.1step 1.2step 1.3algebra∎

Substitute step 1.3 into step 1.2; the terms involving $S^{T} D_{u} ℓ^{T}$ cancel, leaving the asserted continuous adjoint gradient formula.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

Local C1 step consistency gives tangent, adjoint and gradient order

Statement

Let $p \geq 1$ be an integer, $T > 0$ , $h = T / N \leq h_{0} \leq 1$ , and $t_{n} = n h$ . The state dimension $d$ and parameter dimension $m$ are fixed. All vector norms are Euclidean and all matrix norms are their induced operator norms; gradients are columns.

Consider the continuous objective $J (θ) = ϕ (u (T, θ), θ) + \int_{0}^{T} ℓ (t, u (t, θ), θ) d t,$ and the actual finite graph $x_{n + 1} = Ψ_{n} (x_{n}, θ), J_{h} (θ) = ϕ_{h} (x_{N}, θ) + \sum_{n = 0}^{N - 1} L_{n} (x_{n}, θ) .$ Assume the hypotheses of the continuous flow-tangent and adjoint-gradient results hold on a common tube. Write $Φ_{n} (x, θ)$ for the exact flow from $t_{n}$ to $t_{n + 1}$ and $K_{n} (x, θ) = \int_{t_{n}}^{t_{n + 1}} ℓ (t, u (t; t_{n}, x, θ), θ) d t .$ These exact maps and the actual numerical maps are $C^{1}$ on open neighborhoods of the admitted state and parameter sets. For every relevant $n$ , all states $x_{n}, u (t_{n})$ and the segment joining them belong to a convex admitted state set. All estimates below hold uniformly on those sets and a fixed parameter neighborhood, independently of $N, n$ .

For $r = e, h$ , define $A_{n}^{e} = D_{x} Φ_{n}, B_{n}^{e} = D_{θ} Φ_{n}, c_{n}^{e} = D_{x} K_{n}^{T}, d_{n}^{e} = D_{θ} K_{n}^{T},$ $A_{n}^{h} = D_{x} Ψ_{n}, B_{n}^{h} = D_{θ} Ψ_{n}, c_{n}^{h} = D_{x} L_{n}^{T}, d_{n}^{h} = D_{θ} L_{n}^{T} .$ Suppose constants $c_{v}, c_{1}, L, M, H \geq 0$ satisfy, at each admitted $(x, θ)$ , $∥ Ψ_{n} - Φ_{n} ∥ \leq c_{v} h^{p + 1},$ $∥ A_{n}^{h} - A_{n}^{e} ∥, ∥ B_{n}^{h} - B_{n}^{e} ∥, ∥ c_{n}^{h} - c_{n}^{e} ∥, ∥ d_{n}^{h} - d_{n}^{e} ∥ \leq c_{1} h^{p + 1},$ $∥ A_{n}^{r} ∥ \leq 1 + L h, ∥ B_{n}^{r} ∥, ∥ c_{n}^{r} ∥, ∥ d_{n}^{r} ∥ \leq M h .$ Each of $A_{n}^{e}, B_{n}^{e}, c_{n}^{e}, d_{n}^{e}$ is Lipschitz in $x$ , with constant at most $H h$ .

Let $S_{0} = D_{θ} u (0, θ)$ and $S_{0}^{h} = D_{θ} x_{0} (θ)$ . Assume $∥ x_{0} - u (0) ∥ \leq c_{0} h^{p}, ∥ S_{0}^{h} - S_{0} ∥ \leq c_{s 0} h^{p}, ∥ S_{0} ∥ \leq M_{0} .$ The terminal costs are $C^{1}$ . The exact columns $D_{x} ϕ^{T}, D_{θ} ϕ^{T}$ have norm at most $M_{ϕ}$ and are Lipschitz in $x$ with constant $H_{ϕ}$ ; at the same admitted $(x, θ)$ , $∥ D_{x} ϕ_{h}^{T} - D_{x} ϕ^{T} ∥, ∥ D_{θ} ϕ_{h}^{T} - D_{θ} ϕ^{T} ∥ \leq C_{ϕ} h^{p} .$ No terminal value-error hypothesis is needed for the gradient conclusion.

Let $S_{n}^{h} = D_{θ} x_{n}$ , and let $p_{n}^{h}$ be the exact discrete adjoint of this numerical graph. Let $S (t) = D_{θ} u (t)$ and let $p (t)$ be the continuous adjoint with terminal value $D_{x} ϕ (u (T), θ)^{T}$ . Define $E = e^{L T}, X = E (c_{0} + T c_{v}), \overline{S} = E (M_{0} + T M), \overline{P} = E (M_{ϕ} + T M),$ $R = c_{1} + H X, C_{S} = E {c_{s 0} + T R (\overline{S} + 1)}, D = C_{ϕ} + H_{ϕ} X,$ $C_{P} = E {D + T R (\overline{P} + 1)},$ $C_{G} = D + c_{s 0} \overline{P} + (M_{0} + c_{s 0}) C_{P} + T {R (1 + \overline{P}) + M C_{P}} .$ Then $\max_{n} ∥ x_{n} - u (t_{n}) ∥ \leq X h^{p}, \max_{n} ∥ S_{n}^{h} - S (t_{n}) ∥ \leq C_{S} h^{p},$ $\max_{n} ∥ p_{n}^{h} - p (t_{n}) ∥ \leq C_{P} h^{p}, ∥ \nabla J_{h} - \nabla J ∥ \leq C_{G} h^{p} .$ These conclusions assume local value and first-derivative consistency, rather than a global adjoint-error estimate. They require no representation by nonnegative quadrature weights.

Facts & Assumptions

Given: The hypotheses and data in the Statement.

[F1]

Parameter-dependent ODE flows have the variational tangent on a common tube, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ : The common-tube flow result gives the actual parameter derivative and its variational equation. The chain rule gives derivatives of compositions of exact or numerical step maps. The vector mean value inequality bounds the endpoint difference along an admitted segment by a uniform bound on the derivative of its segment parameterization; the zero-dimensional endpoint difference is zero.

[F2]

A tangent and adjoint pairing gives the ODE gradient boundary identity, The continuous ODE adjoint gives the parameter gradient with endpoint terms, The reverse recursion is the exact gradient of a finite state-step objective: The continuous adjoint pairing identity integrates on each subinterval. The continuous gradient formula and the finite-graph reverse recursion use column gradients.

[F3]

Finite geometric error bound from a one-step recurrence: A recurrence $e_{n + 1} \leq (1 + L h) e_{n} + C h^{p + 1}$ has a uniform order- $p$ bound. The looser bound $e^{L T} (C_{0} + T C) h^{p}$ is valid also when $L = 0$ . The same argument applies after reversing a finite index sequence.

[A1]

All domains, differentiability, tube, local consistency, stability, Lipschitz, initial-data and terminal-derivative assumptions in the Statement hold uniformly.

Proof

1.1F1F2A1

The exact objective is also the finite graph with maps $Φ_{n}$ , costs $K_{n}$ , initial state $u (0)$ and terminal cost $ϕ$ : the intervals partition the integral. Differentiability of $K_{n}$ follows by differentiating its integral using the common-tube flow derivative and the uniform continuous derivative bounds. In particular the initial-state variational matrix $V$ on interval $n$ has $V (t_{n}) = I$ , $V^{'} = f_{u} V$ , and $c_{n}^{e} = \int_{t_{n}}^{t_{n + 1}} V (t)^{T} ℓ_{u} (t)^{T} d t .$ The pairing identity gives $p (t_{n}) = c_{n}^{e} (u (t_{n}), θ) + A_{n}^{e} (u (t_{n}), θ)^{T} p (t_{n + 1}) .$ Thus the exact-flow graph adjoint equals the continuous adjoint at every node. Its tangent and gradient are also those of the exact continuous objective.

1.2A1F1F3

Write $u_{n} = u (t_{n})$ and $e_{n} = ∥ x_{n} - u_{n} ∥$ . For $d \geq 1$ , parameterize the admitted segment by $s \mapsto u_{n} + s (x_{n} - u_{n})$ , $0 \leq s \leq 1$ . The chain rule and the derivative bound for $Ψ_{n}$ bound the derivative of its composition with this segment by $(1 + L h) ∥ x_{n} - u_{n} ∥$ ; the vector mean value inequality [F1] therefore gives its endpoint Lipschitz bound with constant $1 + L h$ . For $d = 0$ , the state difference and both endpoint values lie in the zero vector space, so the same bound is immediate. Consequently $e_{n + 1} \leq (1 + L h) e_{n} + c_{v} h^{p + 1} .$ By [F3], $e_{n} \leq X h^{p}$ .

2.1A1step 1.2F3

The exact tangent and adjoint recurrences give $∥ S_{n + 1} ∥ \leq (1 + L h) ∥ S_{n} ∥ + M h, ∥ p (t_{n}) ∥ \leq (1 + L h) ∥ p (t_{n + 1}) ∥ + M h .$ Expanding these finite recurrences and using $(1 + L h)^{k} \leq e^{L T}$ shows $∥ S_{n} ∥ \leq \overline{S}$ and $∥ p (t_{n}) ∥ \leq \overline{P}$ . These bounds include $L = 0$ . For any one of the four derivative maps, comparison at numerical and exact nodes gives $∥ F_{n}^{h} (x_{n}, θ) - F_{n}^{e} (u_{n}, θ) ∥ \leq c_{1} h^{p + 1} + H h e_{n} \leq R h^{p + 1} .$

3.1F1A1step 2.1

The actual differentiated numerical graph satisfies $S_{n + 1}^{h} = A_{n}^{h} (x_{n}) S_{n}^{h} + B_{n}^{h} (x_{n})$ . Subtract the exact recurrence and split the matrix product using $A_{n}^{h} (S_{n}^{h} - S_{n}) + (A_{n}^{h} - A_{n}^{e}) S_{n}$ . Hence $∥ S_{n + 1}^{h} - S_{n + 1} ∥ \leq (1 + L h) ∥ S_{n}^{h} - S_{n} ∥ + R (\overline{S} + 1) h^{p + 1} .$ The initial derivative bound and [F3] yield $C_{S} h^{p}$ .

3.2F2A1step 2.1

At the terminal node, the difference of adjoint seeds is at most $D h^{p}$ . Subtract the two backward recurrences, splitting $A_{h}^{T} p_{h} - A_{e}^{T} p_{e} = A_{h}^{T} (p_{h} - p_{e}) + (A_{h} - A_{e})^{T} p_{e}$ . The Euclidean operator norm of a transpose equals that of the matrix. For every finite-dimensional Euclidean vector $v$ , Cauchy–Schwarz gives $∥ v ∥ = \sup_{∥ w ∥ \leq 1} ∣ w^{T} v ∣$ : if $v \neq 0$ , equality is attained at $w = v / ∥ v ∥$ , while for $v = 0$ both sides vanish, including in dimension zero. Applying this closed-unit-ball formula to both sides of the pairing gives the transpose identity in all finite dimensions. Therefore $∥ p_{n}^{h} - p (t_{n}) ∥ \leq (1 + L h) ∥ p_{n + 1}^{h} - p (t_{n + 1}) ∥ + R (\overline{P} + 1) h^{p + 1} .$ Reverse the indices and apply [F3]. This gives $C_{P} h^{p}$ .

4.1F2step 1.1step 2.1step 3.2A1∎

Apply the finite-graph gradient formula to both graphs: $\nabla J_{h} = D_{θ} ϕ_{h} (x_{N})^{T} + (S_{0}^{h})^{T} p_{0}^{h} + \sum_{n < N} {d_{n}^{h} (x_{n}) + (B_{n}^{h} (x_{n}))^{T} p_{n + 1}^{h}},$ $\nabla J = D_{θ} ϕ (u_{N})^{T} + S_{0}^{T} p (0) + \sum_{n < N} {d_{n}^{e} (u_{n}) + (B_{n}^{e} (u_{n}))^{T} p (t_{n + 1})} .$ The terminal direct-derivative difference is at most $D h^{p}$ . The initial contribution is bounded by ${c_{s 0} \overline{P} + (M_{0} + c_{s 0}) C_{P}} h^{p},$ since $h \leq 1$ implies $∥ S_{0}^{h} ∥ \leq M_{0} + c_{s 0}$ . Each running difference is bounded by ${R + R \overline{P} + M C_{P}} h^{p + 1},$ using $B_{h}^{T} (p_{h} - p) + (B_{h} - B_{e})^{T} p$ . Summing $N$ terms and using $N h = T$ proves the stated $C_{G}$ bound.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

Differentiated explicit RK stages give local C1 order

Statement

Use the dimensions, exact interval maps $Φ_{n}, K_{n}$ , terminal and initial data, and Euclidean conventions of Local C1 step consistency gives tangent, adjoint and gradient order. Let $(A, b, c)$ be a fixed explicit $s$ -stage tableau with $c_{i} = \sum_{j < i} a_{i j}$ and $\sum_{i} b_{i} = 1$ . Its weights may be negative or zero. The actual numerical state and cost maps are $Y_{i} = x + h \sum_{j < i} a_{i j} f (t_{n} + c_{j} h, Y_{j}, θ),$ $Ψ_{n} = x + h \sum_{i} b_{i} f (t_{n} + c_{i} h, Y_{i}, θ), L_{n} = h \sum_{i} b_{i} ℓ (t_{n} + c_{i} h, Y_{i}, θ) .$ These are exact finite-stage maps, not outputs of a stopped implicit solver.

Assume $f : D \to R^{d}$ and $ℓ : D \to R$ are $C^{2}$ on an open convex time-state-parameter domain. On a common compact convex subset with positive margin in $D$ , their derivatives through order two are uniformly bounded. All admitted local exact trajectories, all numerical trajectories and stages, and all needed joining segments lie in this domain and common tube, uniformly over the fixed parameter neighborhood and $h \leq h_{0}$ . Stage times outside $[t_{n}, t_{n + 1}]$ are included when the tableau has such abscissae. Unique admitted local flows are assumed as in the flow-tangent supplier.

Put $r = d + m$ , $E = [I_{d} 0] \in R^{d \times r}$ and $E_{θ} = [0 I_{m}] \in R^{m \times r}$ . Consider the autonomous augmented system, with components $t, u, ϑ, C, V, g$ , $t^{'} = 1, ϑ^{'} = 0, u^{'} = f (t, u, ϑ), C^{'} = ℓ (t, u, ϑ),$ $V^{'} = f_{u} (t, u, ϑ) V + f_{θ} (t, u, ϑ) E_{θ},$ $g^{'} = V^{T} ℓ_{u} (t, u, ϑ)^{T} + E_{θ}^{T} ℓ_{θ} (t, u, ϑ)^{T} .$ For every admitted local start use $(t_{n}, x, θ, 0, E, 0)$ . Suppose application of the same tableau to this augmented system has uniform exact-start defect at most $C_{a} h^{p + 1}$ , for an integer $p \geq 1$ , on a common compact convex augmented tube with positive margin. This is a local augmented-order hypothesis, not a global tangent, adjoint or gradient-error assumption.

Then the actual maps $Ψ_{n}, L_{n}$ satisfy all local value, first-derivative, $1 + L h$ derivative-stability, $M h$ derivative-size, and $H h$ exact-derivative Lipschitz hypotheses of Local C1 step consistency gives tangent, adjoint and gradient order, with constants independent of $n, N$ . Subject to that lemma's initial and terminal hypotheses, their actual differentiated graph tangents, exact reverse adjoints, and reduced gradients consequently have order- $p$ global error.

A concrete sufficient instantiation for $1 \leq p \leq 4$ is that this augmented vector field is $C^{4}$ with bounded derivatives through order four on its admitted compact convex tube and the tableau satisfies the coefficient equations through order $p$ in Sufficient explicit Runge–Kutta order conditions through order four. Jointly $C^{5}$ base data $f, ℓ$ with bounded derivatives through order five on the base tube are sufficient for this augmented $C^{4}$ condition. The classical RK4 tableau $a_{21} = a_{32} = \frac{1}{2}, a_{43} = 1, b = (\frac{1}{6}, \frac{1}{3}, \frac{1}{3}, \frac{1}{6}), c = (0, \frac{1}{2}, \frac{1}{2}, 1),$ with all other $a_{i j} = 0$ , therefore gives the claimed order-four tangent, nodal-adjoint and gradient approximation under these hypotheses. For general $p$ , retain the explicit local augmented-defect hypothesis above; no higher-order coefficient theorem is being asserted.

Facts & Assumptions

Given: The hypotheses and data in the Statement.

[F1]

Explicit Runge–Kutta tableaus, stage maps, and failure status, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Chain sum product and composition rules for Banach derivatives: Explicit RK stages are finite successive compositions. The ordinary chain and product rules differentiate their actual maps.

[F2]

Parameter-dependent ODE flows have the variational tangent on a common tube, Gronwall's integral inequality with variable and constant coefficients: The common-tube flow derivative solves the variational equation. Grönwall bounds its solutions uniformly on a fixed finite time interval.

[F3]

Sufficient explicit Runge–Kutta order conditions through order four: The order-conditions supplier establishes local order through four for a $C^{4}$ autonomous augmented field on its stated compact convex tube. The time equation produces the actual stage times because $c_{i} = \sum_{j < i} a_{i j}$ .

[F4]

Global error from uniform one-step defect and stability, Local C1 step consistency gives tangent, adjoint and gradient order: Uniform local defect and one-step Lipschitz stability imply global error for the same augmented method. The preceding local-C1 lemma independently derives the backward nodal-adjoint estimate and the graph-gradient estimate.

[A1]

The domain, local flow, derivative, tube, local augmented-defect, initial-data and terminal-data assumptions in the Statement hold. The augmented local tube includes the exact local augmented paths and actual augmented RK stages for all the specified seeds. No tube containment is inferred from a formal order condition alone.

Proof

1.1F2A1

Set $ξ = (x, θ)$ . The exact local solution has $V (t) = D_{ξ} u (t)$ because $V (t_{n}) = E$ and its displayed equation is the variational equation, with $D_{ξ} θ = E_{θ}$ . Differentiating the cost integral gives $g (t) = D_{ξ} C (t)^{T}$ . Thus at the local endpoint, $V^{+} = [D_{x} Φ_{n} D_{θ} Φ_{n}], g^{+} = (\begin{array}{c} D_{x} K_{n}^{T} \\ D_{θ} K_{n}^{T} \end{array}) .$ The flow result and bounded continuous derivatives justify these differentiations uniformly.

2.1F1A1step 1.1

Differentiate each actual stage, in its explicit order. If $V_{i} = D_{ξ} Y_{i}$ , then $V_{i} = E + h \sum_{j < i} a_{i j} (f_{u, j} V_{j} + f_{θ, j} E_{θ}),$ $D_{ξ} Ψ_{n} = E + h \sum_{i} b_{i} (f_{u, i} V_{i} + f_{θ, i} E_{θ}),$ $D_{ξ} L_{n}^{T} = h \sum_{i} b_{i} (V_{i}^{T} ℓ_{u, i}^{T} + E_{θ}^{T} ℓ_{θ, i}^{T}) .$ These are exactly the $V$ and $g$ outputs obtained by applying the same RK tableau to the augmented system. Its $C$ output is $L_{n}$ and its $u$ output is $Ψ_{n}$ . Consequently the assumed augmented local defect bounds $Ψ_{n} - Φ_{n}$ , $L_{n} - K_{n}$ , and all four first-derivative differences by constants times $h^{p + 1}$ . Projection to a component does not increase the Euclidean norm; a matrix Frobenius bound also bounds its operator norm.

3.1F1F2A1

For a fixed finite tableau, the stage derivative recurrences in step 2.1 give uniform bounds on $D_{x} Y_{i}, D_{θ} Y_{i}$ by induction, using $h \leq h_{0}$ and the bounded first derivatives of $f$ . Here $D_{x} Y_{i} = I + O (h)$ and $D_{θ} Y_{i} = O (h)$ . Substitution into the output derivatives gives $D_{x} Ψ_{n} = I + O (h), D_{θ} Ψ_{n} = O (h), D_{x} L_{n}^{T} = O (h), D_{θ} L_{n}^{T} = O (h),$ uniformly. In particular $∥ D_{x} Ψ_{n} ∥ \leq 1 + L_{h} h$ for a fixed constant $L_{h}$ . The exact variational integral equation gives the same bounds for the exact maps, with $∥ D_{x} Φ_{n} ∥ \leq e^{K h} \leq 1 + K e^{K h_{0}} h$ . Enlarging the constants gives the common $L, M$ required by the preceding lemma.

3.2F3A1step 2.1

For $p \leq 4$ , apply [F3] to the displayed autonomous augmented field itself. Its derivatives through order four exist and are uniformly bounded under the asserted augmented $C^{4}$ hypothesis. Joint base $C^{5}$ regularity is sufficient because its variational components use first derivatives of the base fields. All its exact paths and stages are admitted by [A1]. The source proposition therefore supplies the exact local augmented-defect premise used in step 2.1. This application concerns the differentiated augmented method, rather than just its state projection.

4.1F1F2A1step 3.1

The exact maps have the required second-derivative bounds of size $O (h)$ . To justify differentiating their variational equations, apply the common-tube $C^{1}$ flow result to the first variational system: its field is $C^{1}$ since $f$ is $C^{2}$ , its initial derivative seed is constant, and its variational components remain bounded by [F2]. For coordinate directions $a, b$ of $ξ$ , put $W_{a b} = D_{ξ_{a}} D_{ξ_{b}} u$ . Then $W_{a b}^{'} = f_{u} W_{a b} + D_{(u, θ)}^{2} f [(V_{a}, E_{θ, a}), (V_{b}, E_{θ, b})], W_{a b} (t_{n}) = 0.$ The forcing is uniformly bounded, so Grönwall on an interval of length $h$ gives $∥ W_{a b} ∥ \leq C h$ . Differentiating the cost integral twice produces the integrand $D_{(u, θ)}^{2} ℓ [(V_{a}, E_{θ, a}), (V_{b}, E_{θ, b})] + ℓ_{u} W_{a b},$ which is uniformly bounded, and hence $D_{ξ}^{2} K_{n} = O (h)$ . Thus $D_{ξ}^{2} Φ_{n} = O (h)$ and $D_{ξ}^{2} K_{n} = O (h)$ . Applying the one-dimensional mean value inequality along each admitted state segment gives Lipschitz constants $H h$ for $A_{n}^{e}, B_{n}^{e}, c_{n}^{e}, d_{n}^{e}$ .

4.2F1F2A1step 2.1step 3.1

There is also a direct augmented global-error interpretation. Along a full trajectory append $C^{'} = ℓ$ and the column $G^{'} = S^{T} ℓ_{u}^{T} + ℓ_{θ}^{T}$ , with $C (0) = 0, G (0) = 0$ and $S^{'} = f_{u} S + f_{θ}$ . Differentiating the actual stages shows that its numerical step is exactly $S^{+} = A_{n}^{h} S + B_{n}^{h}, C^{+} = C + L_{n}, G^{+} = G + S^{T} c_{n}^{h} + d_{n}^{h} .$ The analogous exact step uses the exact maps. On bounded $S$ sets, step 2.1 therefore supplies a uniform $O (h^{p + 1})$ local defect for $(u, S, C, G)$ . Exact $S$ is bounded by Grönwall; numerical $S$ is bounded by the $1 + L h, M h$ recurrence from step 3.1. The sums defining $C, G$ then bound them uniformly as well. Their stage values are bounded by the same finite-stage induction.

5.1F4step 2.1step 3.1step 4.1

Steps 2.1–4.1 establish precisely the local hypotheses of the preceding lemma. That lemma now proves the global state, actual graph-tangent, backward nodal-adjoint and gradient bounds. In particular the adjoint conclusion is derived; it is not inserted as an additional hypothesis.

6.1F4A1step 3.1step 4.2

On a common compact convex augmented tube containing these values and joining stage segments, the $C^{1}$ augmented field has a derivative bound $K$ . For two starting augmented states with difference $δ$ , explicit stage differences obey $δ_{i} \leq δ + h K \sum_{j < i} ∣ a_{i j} ∣ δ_{j}$ . Induction gives $δ_{i} \leq D_{i} δ$ , where $D_{1} = 1$ and $D_{i} = 1 + h_{0} K \sum_{j < i} ∣ a_{i j} ∣ D_{j}$ . The output difference is at most $(1 + h K \sum_{i} ∣ b_{i} ∣ D_{i}) δ$ . Thus the actual augmented method has one-step stability $1 + Λ h$ . The global-error supplier applies, giving order- $p$ error in the actual tangent and accumulated cost derivative as well. This independently exhibits the augmented local-to-global route; step 5.1 additionally supplies the nodal adjoint.

7.1F3step 3.2algebra∎

For the displayed RK4 tableau the eight coefficient sums, in the order stated in the source proposition, are $1, 1 / 2, 1 / 3, 1 / 6, 1 / 4, 1 / 12, 1 / 8, 1 / 24$ . For example $A c = (0, 0, 1 / 4, 1 / 2)$ , $A c^{[2]} = (0, 0, 1 / 8, 1 / 4)$ , and $A^{2} c = (0, 0, 0, 1 / 4)$ , which verify the mixed sums directly. Hence step 3.2 applies with $p = 4$ . No division by an individual $b_{i}$ or internal-stage adjoint order condition occurs in this argument.

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

Discrete and continuous ODE gradients agree to order p under augmented error bounds

Statement

Let $T > 0$ , let $N \geq 1$ and $p \geq 1$ be integers, and set $h = T / N$ and $t_{n} = n h$ for $0 \leq n \leq N$ . Let the finite graph defining $J_{h}$ have $N$ transitions with node states $x_{0}, \dots, x_{N}$ . Let $J$ and $J_{h}$ be the continuous and finite state-step objectives of The continuous ODE adjoint gives the parameter gradient with endpoint terms and The reverse recursion is the exact gradient of a finite state-step objective, respectively. Write $λ (t) : = p (t)$ for the continuous adjoint in The continuous ODE adjoint gives the parameter gradient with endpoint terms, and write $x_{n}$ and $λ_{n}^{h}$ for the discrete state and adjoint. Set $u_{n} : = u (t_{n})$ and $λ_{n} : = λ (t_{n})$ for the exact state and adjoint values. Let $q (t, u, λ) = D_{θ} ℓ (t, u, θ)^{T} + D_{θ} f (t, u, θ)^{T} λ$ be the continuous running-gradient vector, evaluated exactly as $q (t) : = q (t, u (t), λ (t))$ . Suppose the exact and numerical state/adjoint pairs lie in a common tube and $\max_{n} (∥ x_{n} - u_{n} ∥_{2} + ∥ λ_{n}^{h} - λ_{n} ∥_{2}) \leq C_{x} h^{p} .$ For the state part of this bound, it is sufficient that the hypotheses of Global error from uniform one-step defect and stability hold with a uniform $O (h^{p + 1})$ exact-start defect; the coefficient conditions of Sufficient explicit Runge–Kutta order conditions through order four supply that defect for an explicit Runge--Kutta method of order $p \leq 4$ under that item's $C^{4}$ and common-tube assumptions. No cited state-error result supplies the adjoint part, which is an independent hypothesis here. Write $b$ and $b_{h}$ for the column initial/terminal contribution vectors in the continuous and discrete gradient formulas. Assume the discrete running-gradient terms admit vectors ${\hat{q}}_{n}$ and nonnegative weights $w_{n}$ , $0 \leq n \leq N$ , with $\nabla J_{h} = b_{h} + \sum_{n = 0}^{N} w_{n} {\hat{q}}_{n}, \nabla J = b + \int_{0}^{T} q (t) d t, \sum_{n = 0}^{N} w_{n} = T .$ The weighted discrete representation is an explicit premise; it is not implied by the generic finite state-step adjoint recursion. Suppose $∥ {\hat{q}}_{n} - q (t_{n}, x_{n}, λ_{n}^{h}) ∥_{2} \leq C_{d} h^{p}, ∥ b_{h} - b ∥_{2} \leq C_{b} h^{p},$ and that the same weights give the quadrature $Q_{h} (q) = \sum_{n = 0}^{N} w_{n} q (t_{n})$ satisfying ${∥ Q_{h} (q) - \int_{0}^{T} q (t) d t ∥}_{2} \leq C_{Q} h^{p} .$ If $q$ is Lipschitz in $(u, λ)$ on the common tube with constant $L_{q}$ in the sum norm, then $∥ \nabla J_{h} - \nabla J ∥_{2} \leq (C_{b} + T (C_{d} + L_{q} C_{x}) + C_{Q}) h^{p} .$ The adjoint, derivative-data and quadrature hypotheses are separate from state accuracy; without them the state estimate alone gives no gradient error conclusion.

There is also the following independent method-consistency conclusion. Under the local $C^{1}$ step-and-cost consistency, derivative stability, derivative Lipschitz, initial-data, terminal-derivative and tube hypotheses of Local C1 step consistency gives tangent, adjoint and gradient order, the actual numerical graph satisfies $\max_{n} ∥ D_{θ} x_{n} - D_{θ} u (t_{n}) ∥_{2} \leq C_{S} h^{p}, \max_{n} ∥ λ_{n}^{h} - λ (t_{n}) ∥_{2} \leq C_{P} h^{p},$ $∥ \nabla J_{h} - \nabla J ∥_{2} \leq C_{G} h^{p},$ with that lemma's explicit constants, together with its state bound. These conclusions derive the global tangent and adjoint errors from local map errors. They do not assume the preceding global state/adjoint bound or weighted representation.

In particular, Differentiated explicit RK stages give local C1 order supplies these local hypotheses for the actual explicit RK state map and the cost using the same tableau, under its common-tube, base $C^{2}$ , and local augmented-order hypotheses. For $p \leq 4$ , its augmented $C^{4}$ hypothesis and the existing coefficient conditions suffice; jointly $C^{5}$ base data are an explicit sufficient smoothness condition, and its displayed classical RK4 tableau gives a concrete order-four instance. Generic $p$ is retained through its genuine local augmented-order hypothesis. This branch permits negative or zero RK weights and makes no internal-stage adjoint accuracy claim.

Facts & Assumptions

Given: $T > 0$ , integers $N \geq 1$ and $p \geq 1$ , the mesh $h = T / N$ , the stated continuous and discrete objectives, exact adjoint formulas, uniform state/adjoint error bound, derivative-data perturbation bound, quadrature bound, and common compact tube. For the method-consistency conclusion, the actual step and cost maps and the separate local hypotheses of the two cited local lemmas are also given.

[F1]

The continuous ODE adjoint gives the parameter gradient with endpoint terms: the continuous gradient is the initial/terminal contribution plus $\int_{0}^{T} q (t) d t$ .

[F2]

The reverse recursion is the exact gradient of a finite state-step objective: the discrete gradient is the corresponding initial/terminal contribution plus the reverse-mode sum of the discrete running derivative terms.

[F3]

Global error from uniform one-step defect and stability: the common-tube, uniform local-defect, one-step stability and initial-error hypotheses imply the stated uniform state-error bound.

[F4]

Sufficient explicit Runge–Kutta order conditions through order four: for an explicit tableau satisfying the stated coefficient equations through $p \leq 4$ , a $C^{4}$ vector field on the common compact tube has uniform exact-start defect $O (h^{p + 1})$ . It supplies a state local-defect premise only when applied just to the state field; it makes no adjoint or tangent error claim by itself.

[F5]

Local C1 step consistency gives tangent, adjoint and gradient order derives global state, actual tangent, nodal-adjoint and gradient errors by comparing the exact-flow and numerical graphs under its local value/first-derivative hypotheses.

[F6]

Differentiated explicit RK stages give local C1 order identifies derivatives of actual RK stages and matching cost quadrature with the same augmented method, verifies the local hypotheses of [F5], and instantiates local augmented order through four using [F4] on the augmented field.

[A1]

For the weighted conclusion, the constants $C_{b}, C_{x}, C_{d}, C_{Q}, L_{q}$ are independent of $h$ , and all error, quadrature and derivative-data hypotheses in that part of the Statement hold.

[A2]

For the independent method-consistency conclusion, every hypothesis of [F5] holds, or is supplied by [F6] together with the initial and terminal hypotheses of [F5]. No global adjoint-error estimate is included in [A2].

Proof

1.1F3F4A1

If the stated explicit Runge--Kutta conditions are used for the state field, [F4] supplies the exact-start defect required by [F3]. Under the stated stability, common-tube and initial-error hypotheses, [F3] gives the state part of the asserted $C_{x} h^{p}$ bound; the adjoint part in the weighted conclusion remains the separately assumed bound.

1.2A2F5

Independently assume [A2]. Apply [F5] to the actual maps defining $J_{h}$ and the exact interval flow and running-cost maps defining $J$ . Its forward recurrences give state and tangent errors, its backward recurrence gives the nodal-adjoint error, and its graph-gradient comparison gives the stated $C_{G} h^{p}$ bound. This argument uses no weighted decomposition.

2.1A1F1F2step 1.1algebra

By [F1], [F2], and the explicit weighted representation in the Statement, the two gradients are their respective endpoint terms plus, respectively, the weighted sum of ${\hat{q}}_{n}$ and the integral of $q$ . At each node, insert and subtract $q (t_{n}, x_{n}, λ_{n}^{h})$ . The derivative-data assumption and Lipschitz bound give $∥ {\hat{q}}_{n} - q (t_{n}) ∥_{2} \leq (C_{d} + L_{q} C_{x}) h^{p}$ .

3.1A1F1F2step 2.1algebra

Since $w_{n} \geq 0$ and $\sum_{n} w_{n} = T$ , the node-data perturbation contributes at most $T (C_{d} + L_{q} C_{x}) h^{p}$ . The exact-integrand quadrature error contributes at most $C_{Q} h^{p}$ , and the endpoint difference at most $C_{b} h^{p}$ . The Euclidean triangle inequality gives the displayed weighted bound.

4.1A2F4F6step 1.2∎

For the RK specialization, [F6] differentiates the actual finite stages, proves the local derivative-size and Lipschitz bounds, and obtains the needed local $C^{1}$ consistency from augmented local order. Its application of [F4] to the augmented field verifies this order for $p \leq 4$ under the actual augmented smoothness and tube assumptions. Its RK4 coefficient calculation supplies the concrete fourth-order instance. Therefore step 1.2 applies.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

A contractive implicit Runge-Kutta stage branch has a nonsingular residual Jacobian

Statement

Use the implicit Runge–Kutta residual $G (Y) = Y - 1 \otimes y - h (A \otimes I_{d}) F (Y)$ from Implicit Runge–Kutta stage equations and update, where $y$ is the input state in that definition. Let $B = {Y : ∥ Y - 1 \otimes y ∥_{\infty, 2} \leq R}$ , with $R > 0$ , be a closed product ball in the admissible stage set. Suppose $F$ is $C^{1}$ on a convex open neighborhood of $B$ , and constants $M, L \geq 0$ satisfy $∥ F (Y) ∥_{\infty, 2} \leq M, ∥ D_{Y} F (Y) Z ∥_{\infty, 2} \leq L ∥ Z ∥_{\infty, 2} (Y \in B) .$ Assume $h ∥ A ∥_{\infty} M \leq R$ and $q : = h ∥ A ∥_{\infty} L < 1$ . Then Small-step implicit Runge–Kutta stages have a unique branch and controlled residual error gives a unique stage root in $B$ , and $D_{Y} G (Y)$ is invertible at every point of $B$ . This lemma supplies nonsingularity for implicit differentiation.

Facts & Assumptions

Given: The finite-dimensional stage space, implicit Runge–Kutta residual, selected invariant ball, $C^{1}$ map $F$ , derivative bound $L$ on the ball, and $q < 1$ .

[F1]

The stage residual is $G (Y) = Y - 1 \otimes y - h (A \otimes I_{d}) F (Y)$ , where $y$ is the input state; a selected stage root satisfies $G (Y) = 0$ (Implicit Runge–Kutta stage equations and update).

[F2]

On a closed product ball of radius $R$ centered at $c$ , bounds $∥ F ∥_{\infty, 2} \leq M$ and $∥ F (Y) - F (Z) ∥_{\infty, 2} \leq L ∥ Y - Z ∥_{\infty, 2}$ with $h ∥ A ∥_{\infty} M \leq R$ and $h ∥ A ∥_{\infty} L < 1$ give a unique root (Small-step implicit Runge–Kutta stages have a unique branch and controlled residual error).

[F3]

A vector-valued function continuous on $[0, 1]$ , differentiable on $(0, 1)$ , and with Euclidean derivative norm at most $M$ has endpoint difference at most $M$ (The mean value inequality: if $f : [a, b] \to R^{m}$ is continuous and differentiable on $(a, b)$ with $∥ f^{'} ∥_{2} \leq M$ , then $∥ f (b) - f (a) ∥_{2} \leq M (b - a)$ ). The chain rule differentiates a $C^{1}$ map along a segment (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

For a linear map on a finite-dimensional vector space, $\dim V = \dim \ker T + \dim im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

Proof

1.1givenF2F3

Let $Y, W \in B$ . Its convexity puts $Y + t (W - Y)$ in $B$ for $0 \leq t \leq 1$ . For each block $i$ , the curve $γ_{i} (t) = F_{i} (Y + t (W - Y))$ is continuous on $[0, 1]$ and differentiable inside because $F$ is $C^{1}$ on an open neighborhood. Its derivative obeys $∥ γ_{i}^{'} (t) ∥_{2} \leq ∥ D_{Y} F (Y + t (W - Y)) (W - Y) ∥_{\infty, 2} \leq L ∥ W - Y ∥_{\infty, 2} .$ Apply [F3] to each block and take the maximum to obtain $∥ F (W) - F (Y) ∥_{\infty, 2} \leq L ∥ W - Y ∥_{\infty, 2}$ . The fixed-point map $Φ (Y) = 1 \otimes y + h (A \otimes I_{d}) F (Y)$ satisfies $∥ Φ (Y) - 1 \otimes y ∥_{\infty, 2} \leq h ∥ A ∥_{\infty} M \leq R$ , so it maps $B$ into itself. Thus [F2] supplies the unique stage root in $B$ .

1.2F1givenalgebra

Differentiating [F1] with respect to $Y$ gives $D_{Y} G = I - h (A \otimes I_{d}) D_{Y} F$ . If $D_{Y} G Z = 0$ , the block row-sum norm and the derivative bound imply $∥ Z ∥_{\infty, 2} \leq h ∥ A ∥_{\infty} L ∥ Z ∥_{\infty, 2} = q ∥ Z ∥_{\infty, 2} .$ Since $q < 1$ , this forces $Z = 0$ ; thus the square finite-dimensional map $D_{Y} G$ has trivial kernel.

2.1F4step 1.2∎

By [F4], a linear map on the stage space has rank equal to its dimension when its kernel has dimension zero. Therefore $D_{Y} G$ is onto as well as one-to-one, hence invertible. This holds at every point of the prescribed branch ball.

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

An implicit Runge-Kutta stage has the stated parameter sensitivity on its selected branch

Statement

Let $u = u (θ)$ be $C^{1}$ and consider a selected $s$ -stage Runge–Kutta branch for a $C^{1}$ parameter-dependent vector field $f (t, y, θ)$ . Let $F (Y, u, θ)$ be the stack of its stage evaluations and $G (Y, u, θ) = Y - 1 \otimes u - h (A \otimes I_{d}) F (Y, u, θ) .$ Assume the branch is the unique root in the invariant ball of Small-step implicit Runge–Kutta stages have a unique branch and controlled residual error, and its stage derivative $G_{Y}$ obeys the hypotheses of A contractive implicit Runge-Kutta stage branch has a nonsingular residual Jacobian. Then the branch $Y (θ)$ is locally $C^{1}$ and $D_{θ} Y = - G_{Y}^{- 1} (G_{u} D_{θ} u + G_{θ}) .$ For the tableau weights $b = (b_{1}, \dots, b_{s})^{T}$ and update $Ψ_{h} (u, θ) = u + h (b^{T} \otimes I_{d}) F (Y (θ), u (θ), θ)$ , $D_{θ} Ψ_{h} = D_{θ} u + h (b^{T} \otimes I_{d}) (F_{Y} D_{θ} Y + F_{u} D_{θ} u + F_{θ}) .$ These are exact derivatives of the selected algebraic stage branch. A finite-budget stage iteration is a different map and is not covered by this formula.

Facts & Assumptions

Given: The selected implicit Runge–Kutta tableau, the $C^{1}$ field, the initial state map, the invariant stage branch and the stated Jacobian nonsingularity.

[F1]

The stage residual is $G (Y, u, θ) = Y - 1 \otimes u - h (A \otimes I_{d}) F (Y, u, θ)$ , and the update uses the weighted stage evaluations (Implicit Runge–Kutta stage equations and update).

[F2]

The selected branch is uniquely defined in its invariant ball when the contraction and ball-invariance hypotheses hold (Small-step implicit Runge–Kutta stages have a unique branch and controlled residual error).

[F3]

If a $C^{1}$ residual has a zero and an invertible state-block Jacobian, it has a local $C^{1}$ solution branch with derivative $- R_{z}^{- 1} R_{θ}$ (Implicit differentiation and adjoint formulas under a nonsingular residual Jacobian).

[F4]

The residual Jacobian is invertible under the stated contraction derivative bound (A contractive implicit Runge-Kutta stage branch has a nonsingular residual Jacobian).

[F5]

The derivative of a composition is given by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[A1]

The field and stage maps are $C^{1}$ on an open neighborhood of the selected branch, and $G_{Y}$ is evaluated on that branch.

Proof

1.1F1F2F3F4A1

By [F2], the stated root exists on the selected ball, and [F4] makes $G_{Y}$ invertible there. Regard $R (θ, Y) = G (Y, u (θ), θ)$ . Its parameter derivative is $R_{θ} = G_{u} D_{θ} u + G_{θ}$ . Apply [F3] to obtain the local branch and the displayed formula for $D_{θ} Y$ .

2.1F1F5step 1.1algebra

The update is the finite composition of $u (θ)$ , $Y (θ)$ and the $C^{1}$ stage evaluation map in [F1]. Apply [F5] and the product rule for the linear output map $h (b^{T} \otimes I_{d})$ to get the displayed derivative.

3.1F1step 1.1step 2.1∎

The calculation differentiates the exact equation $G (Y, u (θ), θ) = 0$ on its selected branch. It contains no update of a finite stage solver, so no derivative of a stopped iteration is asserted.

LemmaStatement: AI-adaptedProof: AI-generatedOpen item page →

A uniformly coercive elliptic variational state is C1 in its parameter

Statement

Assume Countable Choice and the setting of Parameter-dependent coercive elliptic variational states and objectives. On an open neighborhood $Θ_{0} \subseteq Θ$ , suppose the same constants $M < \infty$ and $α > 0$ give boundedness and coercivity for every $a_{θ}$ , and $θ \mapsto A_{θ}$ and $θ \mapsto ℓ_{θ}$ are operator-norm $C^{1}$ . Then each state $u_{θ}$ exists uniquely, the map $θ \mapsto u_{θ} \in V$ is $C^{1}$ , and for each direction $η \in R^{m}$ its derivative $w_{η} = D_{θ} u_{θ} [η]$ is the unique solution of $a_{θ} (w_{η}, v) = D_{θ} ℓ_{θ} [η] (v) - D_{θ} a_{θ} [η] (u_{θ}, v) (v \in V) .$ Uniform coercivity is required on the parameter neighborhood; no differentiability conclusion is asserted at a loss of coercivity.

Facts & Assumptions

Given: Countable Choice, a real Hilbert space, the uniformly bounded/coercive family, operator-norm $C^{1}$ data, and the variational state definition in Parameter-dependent coercive elliptic variational states and objectives.

[F1]

Lax–Milgram gives a unique variational solution and $α ∥ u ∥_{V} \leq ∥ F ∥_{V^{*}}$ for each bounded linear datum (The Lax--Milgram theorem, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

Operator-norm $C^{1}$ data means the increments of $A_{θ}$ and $ℓ_{θ}$ equal their first derivatives plus remainders whose norms are $o (∥ h ∥_{2})$ (Parameter-dependent coercive elliptic variational states and objectives).

[A1]

The constants $M, α$ are uniform on the open parameter neighborhood $Θ_{0}$ , with $α > 0$ .

Proof

1.1F1A1given

For each $θ \in Θ_{0}$ , apply [F1] to $a_{θ}$ and $ℓ_{θ}$ ; this gives a unique $u_{θ}$ and the bound $∥ u_{θ} ∥_{V} \leq α^{- 1} ∥ ℓ_{θ} ∥_{V^{*}}$ . Since $ℓ_{θ}$ is continuous, these state norms are locally bounded in $θ$ .

2.1A1F2step 1.1algebra

For sufficiently small $h$ with $θ + h \in Θ_{0}$ , put $δ u = u_{θ + h} - u_{θ}$ , $δ A = A_{θ + h} - A_{θ}$ , and $δ ℓ = ℓ_{θ + h} - ℓ_{θ}$ . Subtract the two variational equations to get $a_{θ} (δ u, v) = δ ℓ (v) - (δ A u_{θ + h}) (v)$ . Taking $v = δ u$ , using coercivity and the dual norm, and dividing by $∥ δ u ∥_{V}$ when it is nonzero gives $∥ δ u ∥_{V} \leq α^{- 1} (∥ δ ℓ ∥_{V^{*}} + ∥ δ A ∥ ∥ u_{θ + h} ∥_{V}) = O (∥ h ∥_{2})$ . The same bound is immediate when $δ u = 0$ . Thus the state map is locally Lipschitz.

2.2F1F2step 1.1given

For each direction $h \in R^{m}$ , define $w_{h}$ by $a_{θ} (w_{h}, v) = D ℓ_{θ} [h] (v) - (D A_{θ} [h] u_{θ}) (v)$ . The right side is a bounded linear functional, so [F1] gives a unique $w_{h}$ . Uniqueness and linearity of the right side in $h$ show $h \mapsto w_{h}$ is linear.

3.1A1F1F2step 2.1step 2.2algebra

Write $δ ℓ = D ℓ_{θ} [h] + r_{ℓ}$ and $δ A = D A_{θ} [h] + r_{A}$ , where $∥ r_{ℓ} ∥ + ∥ r_{A} ∥ = o (∥ h ∥_{2})$ by [F2]. Subtract the equation for $w_{h}$ in step 2.2 from the state-difference equation in step 2.1. With $e_{h} = δ u - w_{h}$ , this gives $a_{θ} (e_{h}, v) = r_{ℓ} (v) - (r_{A} u_{θ}) (v) - (D A_{θ} [h] δ u) (v) - (r_{A} δ u) (v)$ . The first two terms have dual norm $o (∥ h ∥_{2})$ ; the last two have that order because $∥ D A_{θ} [h] ∥ = O (∥ h ∥_{2})$ , $∥ r_{A} ∥ = o (∥ h ∥_{2})$ , and $∥ δ u ∥_{V} = O (∥ h ∥_{2})$ by step 2.1. Coercivity yields $∥ e_{h} ∥_{V} = o (∥ h ∥_{2})$ , proving Fréchet differentiability and the stated derivative equation.

4.1A1F1F2step 2.1step 2.2algebra∎

The solution operator $A_{θ}^{- 1} : V^{*} \to V$ has norm at most $α^{- 1}$ . For nearby $θ, \tilde{θ}$ , the identity $A_{\tilde{θ}}^{- 1} - A_{θ}^{- 1} = A_{\tilde{θ}}^{- 1} (A_{θ} - A_{\tilde{θ}}) A_{θ}^{- 1}$ shows its operator-norm continuity. The formula in step 2.2, together with continuity of $u_{θ}$ , $D A_{θ}$ , and $D ℓ_{θ}$ , then shows $D_{θ} u_{θ}$ is continuous. Hence the state map is $C^{1}$ .

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

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.

TheoremStatement: AI-adaptedProof: AI-generatedOpen item page →

Conforming elliptic finite-element reduced gradients have an energy-norm error budget

Statement

Assume Countable Choice and the elliptic state/adjoint setting of The elliptic adjoint gives the reduced parameter gradient. Let $V_{h} \subseteq V$ be a finite-dimensional conforming space. Let $a_{θ, h}$ and $ℓ_{θ, h}$ be operator-norm $C^{1}$ in $θ$ , uniformly coercive on $V_{h}$ , and define the discrete state by $a_{θ, h} (u_{h}, v_{h}) = ℓ_{θ, h} (v_{h})$ . Let $J_{h} : V_{h} \times Θ \to R$ be $C^{1}$ , and let $p_{h} \in V_{h}$ solve $a_{θ, h} (v_{h}, p_{h}) = D_{u} J_{h} (u_{h}, θ) [v_{h}]$ . Suppose the parameter derivatives of the discrete forms and loads have specified bounded extensions to $V$ that agree with their values on $V_{h}$ . Let $u, p$ be the exact state and adjoint, and let $u_{h}, p_{h} \in V_{h}$ be their discrete counterparts, with certified energy bounds $∥ u - u_{h} ∥_{V} \leq E_{u}, ∥ p - p_{h} ∥_{V} \leq E_{p} .$ For an exact conforming Galerkin solve these bounds may be supplied by Céa quasi-optimality for coercive conforming Galerkin problems; for a conforming perturbed solve they may be supplied by First Strang estimate for conforming perturbed Galerkin problems, with all load, form, geometry and quadrature defects retained as in Variational-crime settings and consistency defects. Define exact and discrete reduced gradients by $g [η] = D_{θ} J (u, θ) [η] + ℓ_{θ}^{'} [η] (p) - a_{θ}^{'} [η] (u, p), g_{h} [η] = D_{θ} J_{h} (u_{h}, θ) [η] + ℓ_{θ, h}^{'} [η] (p_{h}) - a_{θ, h}^{'} [η] (u_{h}, p_{h}) .$ Assume for $∥ η ∥_{2} \leq 1$ that $∥ ℓ_{θ}^{'} [η] ∥_{V^{*}} \leq M_{ℓ}$ , $∥ a_{θ}^{'} [η] ∥_{L (V, V^{*})} \leq M_{a}$ , the direct-objective defect satisfies $∣ D_{θ} J_{h} (u_{h}, θ) [η] - D_{θ} J (u, θ) [η] ∣ \leq ε_{J}$ , and the derivative-data defects obey $∥ ℓ_{θ, h}^{'} [η] - ℓ_{θ}^{'} [η] ∥_{V^{*}} \leq ε_{ℓ}, ∥ a_{θ, h}^{'} [η] - a_{θ}^{'} [η] ∥_{L (V, V^{*})} \leq ε_{a} .$ Put $U_{*} = \max (∥ u ∥_{V}, ∥ u_{h} ∥_{V})$ and $P_{*} = \max (∥ p ∥_{V}, ∥ p_{h} ∥_{V})$ . Then $∥ g_{h} - g ∥_{Θ^{*}} \leq ε_{J} + M_{ℓ} E_{p} + ε_{ℓ} P_{*} + M_{a} (E_{u} P_{*} + U_{*} E_{p}) + ε_{a} U_{*} P_{*} .$ The derivative-data defects are separate obligations: ordinary state and adjoint Strang bounds do not by themselves bound derivative defects.

Facts & Assumptions

Given: The exact elliptic reduced-gradient formula, the conforming discrete state and adjoint equations, their energy error bounds $E_{u}, E_{p}$ , and the derivative-data bounds in the Statement.

[F1]

The reduced gradient is $J_{θ} [η] + ℓ_{θ}^{'} [η] (p) - a_{θ}^{'} [η] (u, p)$ (The elliptic adjoint gives the reduced parameter gradient).

[F2]

Céa's estimate bounds a conforming Galerkin error by $M / α$ times its best energy approximation (Céa quasi-optimality for coercive conforming Galerkin problems).

[F3]

For conforming perturbed forms, the first Strang estimate includes best approximation and form/load defects; the variational-crime framework identifies geometry, data and quadrature contributions to those defects (First Strang estimate for conforming perturbed Galerkin problems, Variational-crime settings and consistency defects).

[F4]

The derivative of a C1 composition is given by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[A1]

Both state and adjoint equations satisfy the hypotheses of either [F2] or [F3], with their own right-hand sides, defects, and coercivity constants. Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is used through those suppliers.

Proof

1.1A1F2F3

Apply [F2] to an exact conforming Galerkin state and its adjoint, or [F3] to each conforming perturbed equation, using the transposed form $a^{†} (v, w) = a (w, v)$ for the adjoint. Transposition preserves boundedness and coercivity constants. Under [A1], the resulting two bounds are $E_{u}, E_{p}$ , with all applicable form and load defects, including geometry and quadrature contributions, retained as specified by [F3].

1.2F1F4givenalgebra

Choose a basis of $V_{h}$ . Coercivity makes the discrete state matrix $K_{h} (θ)$ invertible, and its entries and the load coordinates are $C^{1}$ . Uniform coercivity makes $K_{h} (θ + z)^{- 1}$ locally bounded. The identity $K_{h} (θ + z)^{- 1} - K_{h} (θ)^{- 1} = - K_{h} (θ + z)^{- 1} (K_{h} (θ + z) - K_{h} (θ)) K_{h} (θ)^{- 1}$ first gives continuity of the inverse. Using $K_{h} (θ + z) - K_{h} (θ) = D K_{h} (θ) [z] + o (∥ z ∥_{2})$ in that identity gives $D (K_{h}^{- 1}) (θ) [z] = - K_{h} (θ)^{- 1} D K_{h} (θ) [z] K_{h} (θ)^{- 1}$ ; this derivative is continuous, so $K_{h}^{- 1}$ and hence $u_{h} (θ)$ are $C^{1}$ . Differentiating the state equation gives $a_{θ, h} (w_{h}, v_{h}) = ℓ_{θ, h}^{'} [η] (v_{h}) - a_{θ, h}^{'} [η] (u_{h}, v_{h})$ . By [F4], $D j_{h} [η] = J_{θ, h}^{'} [η] + D_{u} J_{h} [w_{h}]$ ; use the discrete adjoint equation with $v_{h} = w_{h}$ , then the differentiated state equation with $v_{h} = p_{h}$ , to obtain the formula for $g_{h}$ in the Statement. The continuous formula is supplied by [F1].

2.1F1step 1.2givenalgebra

Fix $η$ with $∥ η ∥_{2} \leq 1$ . Subtract the continuous and discrete formulas. The direct-objective term is bounded by $ε_{J}$ . Add and subtract $ℓ_{θ}^{'} [η] (p_{h})$ ; then $∣ ℓ_{θ, h}^{'} [η] (p_{h}) - ℓ_{θ}^{'} [η] (p) ∣ \leq ε_{ℓ} ∥ p_{h} ∥_{V} + M_{ℓ} ∥ p_{h} - p ∥_{V} \leq ε_{ℓ} P_{*} + M_{ℓ} E_{p}$ .

3.1givenstep 2.1algebra

Add and subtract $a_{θ}^{'} [η] (u_{h}, p_{h})$ . The derivative-form data defect contributes at most $ε_{a} U_{*} P_{*}$ . The remaining perturbation equals $a_{θ}^{'} [η] (u - u_{h}, p_{h}) + a_{θ}^{'} [η] (u, p - p_{h})$ up to an overall sign, so $∣ a_{θ}^{'} [η] (u - u_{h}, p_{h}) + a_{θ}^{'} [η] (u, p - p_{h}) ∣ \leq M_{a} (E_{u} P_{*} + U_{*} E_{p}) .$ Thus for every unit $η$ , $∣ g_{h} [η] - g [η] ∣$ is bounded by the displayed constant.

4.1step 3.1algebra∎

Taking the supremum over all $∥ η ∥_{2} \leq 1$ gives the asserted dual norm bound. The separate $ε$ terms show why state error alone does not certify a reduced-gradient error.

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An elliptic reduced-gradient L2 rate follows under dual approximation

Statement

Assume Countable Choice and the Galerkin and adjoint hypotheses of Conforming elliptic finite-element reduced gradients have an energy-norm error budget. Suppose the adjoint regularity and dual approximation assumptions of The h^(r+s) L² rate under primal and dual approximation hold separately for the state and adjoint equations, giving $∥ u - u_{h} ∥_{L^{2}} \leq C_{u} h^{r + s}, ∥ p - p_{h} ∥_{L^{2}} \leq C_{p} h^{r + s}$ for $0 < h \leq 1$ , $r \geq 1$ , and $s \in (0, 1]$ . Assume the parameter derivative operators satisfy $∣ ℓ_{θ}^{'} [η] (v) ∣ \leq M_{ℓ}^{0} ∥ η ∥_{2} ∥ v ∥_{L^{2}}, ∣ a_{θ}^{'} [η] (v, w) ∣ \leq M_{a}^{0} ∥ η ∥_{2} ∥ v ∥_{L^{2}} ∥ w ∥_{L^{2}} .$ Assume the discrete derivative maps in Conforming elliptic finite-element reduced gradients have an energy-norm error budget have compatible $L^{2}$ extensions and constants $C_{J}, C_{ℓ}, C_{a}$ independent of $h$ give, for every $∥ η ∥_{2} \leq 1$ , a direct-objective derivative defect at most $C_{J} h^{r + s}$ , an $L^{2}$ -dual load-derivative defect at most $C_{ℓ} h^{r + s}$ , and a bilinear derivative defect at most $C_{a} h^{r + s} ∥ v ∥_{L^{2}} ∥ w ∥_{L^{2}}$ . Then the reduced-gradient error is $O (h^{r + s})$ . This rate requires those extra $L^{2}$ -bounded parameter derivative maps; the energy-norm result alone does not imply it.

Facts & Assumptions

Given: The energy-norm reduced-gradient setting, the stated dual regularity bounds for the state and adjoint, Countable Choice, and the displayed $L^{2}$ derivative bounds.

[F1]

The Aubin–Nitsche estimate gives an $h^{r + s}$ $L^{2}$ error under its stated primal and dual approximation hypotheses (The h^(r+s) L² rate under primal and dual approximation).

[F2]

The energy-norm reduced-gradient bound separates state error, adjoint error, and derivative-data defects (Conforming elliptic finite-element reduced gradients have an energy-norm error budget).

[A1]

Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and the dual regularity and interpolation hypotheses hold separately for the continuous and adjoint Galerkin equations, with the constants displayed in the Statement; derivative-data and direct-objective defects are $O (h^{r + s})$ in the stated $L^{2}$ operator norms.

Proof

1.1F1A1given

Apply [F1] to the state equation and to the adjoint equation with the transposed coercive form. Under [A1], this yields the two displayed $L^{2}$ error bounds. Countable Choice is required by [F1].

2.1givenstep 1.1algebra

The parameter-derivative bounds in the Statement give $∣ ℓ_{θ}^{'} [η] (p_{h} - p) ∣ \leq M_{ℓ}^{0} ∥ η ∥_{2} C_{p} h^{r + s}$ and, by adding and subtracting $a_{θ}^{'} [η] (u, p_{h})$ , $∣ a_{θ}^{'} [η] (u_{h}, p_{h}) - a_{θ}^{'} [η] (u, p) ∣ \leq M_{a}^{0} ∥ η ∥_{2} (C_{u} h^{r + s} ∥ p_{h} ∥_{L^{2}} + ∥ u ∥_{L^{2}} C_{p} h^{r + s})$ . The $L^{2}$ norms of $u$ and $p_{h}$ are bounded by the stated state bounds and $∥ p_{h} ∥_{L^{2}} \leq ∥ p ∥_{L^{2}} + C_{p} h^{r + s}$ .

3.1A1F2step 2.1∎

Apply the exact gradient subtraction in [F2]. The direct-objective and derivative-data defects contribute their assumed $O (h^{r + s})$ bounds; step 2.1 bounds the state- and adjoint-dependent terms by constants times $h^{r + s}$ . Taking the supremum over $∥ η ∥_{2} \leq 1$ proves the asserted rate. If the $L^{2}$ derivative bounds or either dual approximation hypothesis fails, this rate has not been proved.

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A parabolic adjoint gives the gradient when the state map is differentiable

Statement

Assume Countable Choice and $T > 0$ , and use the real evolution triple $(V, H, V^{*})$ of Parameter-dependent parabolic variational states and objectives. On an open parameter set $Θ \subseteq R^{m}$ , assume the state map $θ \mapsto u_{θ}$ is Fr'echet differentiable into $W (0, T; V)$ , with derivative $w_{η}$ in direction $η \in R^{m}$ . For each fixed $θ$ and direction $η$ , assume there is a null set $N_{θ, η} \subset (0, T)$ such that the linearized equation below holds for every $t \notin N_{θ, η}$ and simultaneously every $v \in V$ , using representatives of its data: $⟨ w_{η}^{'} (t), v ⟩ + a_{θ} (t; w_{η} (t), v) = ⟨ f_{θ}^{'} [η] (t), v ⟩ - a_{θ}^{'} [η] (t; u_{θ} (t), v), w_{η} (0) = u_{0, θ}^{'} [η] .$ Let $ϕ : H \times Θ \to R$ be $C^{1}$ , and let $L : [0, T] \times V \times Θ \to R$ be measurable in time and $C^{1}$ in $(u, θ)$ for almost every time, with a common integrable majorant that justifies differentiating its time integral. Assume $D_{u} ϕ$ is represented by an element of $H$ and $D_{u} L (t, u_{θ}, θ) \in V^{*}$ is square-integrable in time. Suppose the adjoint $p \in W (0, T; V)$ satisfies the following equation simultaneously for every $v \in V$ outside one null set $M_{θ} \subset (0, T)$ , and has the stated terminal trace: $- ⟨ p^{'} (t), v ⟩ + a_{θ} (t; v, p (t)) = D_{u} L (t, u_{θ} (t), θ) [v], p (T) = \nabla_{H} ϕ (u_{θ} (T), θ) .$ Then for $j (θ) = ϕ (u_{θ} (T), θ) + \int_{0}^{T} L (t, u_{θ} (t), θ) d t$ , $D j (θ) [η] = D_{θ} ϕ (u_{θ} (T), θ) [η] + (p (0), u_{0, θ}^{'} [η])_{H} + \int_{0}^{T} (D_{θ} L (t, u_{θ}, θ) [η] + ⟨ f_{θ}^{'} [η] (t), p (t) ⟩ - a_{θ}^{'} [η] (t; u_{θ} (t), p (t))) d t .$ The exceptional sets are independent of the test $v$ ; they need not be common to all parameters or directions. The forcing/form-derivative difference in the integrand is interpreted together. The hypotheses imply it is measurable and integrable: almost everywhere it equals $⟨ w_{η}^{'}, p ⟩ + ⟨ p^{'}, w_{η} ⟩ + D_{u} L [w_{η}]$ . No global boundedness or coercivity of $a_{θ}$ , or separability of $V$ , is required for this conditional identity.

This is conditional on the displayed differentiable state and adjoint; it does not prove either exists. The adjoint form is transposed as shown, so symmetry of $a_{θ}$ is not required.

Facts & Assumptions

Given: $T > 0$ , the real evolution triple, differentiable parameterized state and objective, its linearized equation, and a solution of the stated terminal adjoint equation, with both displayed variational equations holding simultaneously for all tests outside their specified null sets.

[F1]

The parameterized parabolic state and reduced objective use the variational equation on $W (0, T; V)$ with $H$ traces (Parameter-dependent parabolic variational states and objectives).

[F2]

For $w, p \in W (0, T; V)$ , the evolution-space pairing identity is $(p (T), w (T))_{H} - (p (0), w (0))_{H} = \int_{0}^{T} (⟨ p^{'}, w ⟩ + ⟨ w^{'}, p ⟩) d t$ (The evolution-space energy identity and continuous H trace).

[F3]

Differentiable compositions and sums obey the Banach-space chain rule (Chain sum product and composition rules for Banach derivatives).

[A1]

Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is required by the pairing supplier (The evolution-space energy identity and continuous H trace). No use is made of a parabolic well-posedness theorem to infer state or adjoint existence.

[A2]

For each fixed parameter and direction, the linearized and adjoint equations hold outside null sets independent of $v$ . Their union, together with the null sets on which the $W$ representatives are undefined, still has measure zero. This is an explicit hypothesis, not a consequence of the fixed-test state convention in [F1].

Proof

1.1F1F3given

Apply [F3] to the terminal term and the running integrand. The common integrable majorant permits differentiation under the integral, so $D j [η] = D_{θ} ϕ [η] + (\nabla_{H} ϕ, w_{η} (T))_{H} + \int_{0}^{T} (D_{θ} L [η] + D_{u} L [w_{η}]) d t .$ The terminal condition replaces $\nabla_{H} ϕ$ by $p (T)$ .

1.2A1A2F2givenalgebra

Fix $θ, η$ and discard the union of the null sets in [A2]. At every remaining time, $w_{η} (t), p (t) \in V$ and both equations hold for all tests, so the choices $v = w_{η} (t)$ in the adjoint equation and $v = p (t)$ in the linearized equation are legitimate. Adding them and cancelling the two occurrences of $a_{θ} (t; w_{η} (t), p (t))$ gives $⟨ f_{θ}^{'} [η], p ⟩ - a_{θ}^{'} [η] (u_{θ}, p) = ⟨ w_{η}^{'}, p ⟩ + ⟨ p^{'}, w_{η} ⟩ + D_{u} L [w_{η}] .$ Each term on the right belongs to $L^{1} (0, T)$ by the $L^{2} (V^{*})$ and $L^{2} (V)$ hypotheses and Cauchy--Schwarz. Thus the difference on the left has a measurable integrable representative without assuming either summand separately integrable. Apply [F2] to $w_{η}, p$ ; Countable Choice enters through this pairing supplier.

2.1F1step 1.1step 1.2algebra∎

The pairing identity now gives $(p (T), w_{η} (T))_{H} = (p (0), u_{0, θ}^{'} [η])_{H} + \int_{0}^{T} (- D_{u} L [w_{η}] + ⟨ f_{θ}^{'} [η], p ⟩ - a_{θ}^{'} [η] (u_{θ}, p)) d t$ . Substitute this into step 1.1; the terms $D_{u} L [w_{η}]$ cancel and yield the displayed formula.

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Parabolic finite-element gradient errors include the time-dependent Ritz residual

Statement

Assume Countable Choice and the differentiable continuous state/adjoint setting of A parabolic adjoint gives the gradient when the state map is differentiable. Let $u, p$ and $u_{h}, p_{h}$ denote the continuous and semidiscrete state/adjoint pairs. Write $r [η] = D_{θ} ϕ (u (T), θ) [η] + \int_{0}^{T} D_{θ} L (t, u (t), θ) [η] d t, r_{h} [η] = D_{θ} ϕ_{h} (u_{h} (T), θ) [η] + \int_{0}^{T} D_{θ} L_{h} (t, u_{h} (t), θ) [η] d t .$ Assume their gradients $g, g_{h} \in (R^{m})^{*}$ have the representations, for every $η \in R^{m}$ , $g [η] = r [η] + (p (0), D_{θ} u_{0} [η])_{H} + \int_{0}^{T} (⟨ D_{θ} f [η], p ⟩ - D_{θ} a [η] (u, p)) d t,$ $g_{h} [η] = r_{h} [η] + (p_{h} (0), D_{θ} u_{0, h} [η])_{H} + \int_{0}^{T} (⟨ D_{θ} f_{h} [η], p_{h} ⟩ - D_{θ} a_{h} [η] (u_{h}, p_{h})) d t .$ The continuous representation is supplied by the cited adjoint theorem; the semidiscrete representation is an explicit hypothesis here. Assume $u, p \in H^{1} (0, T; V)$ , the parabolic form and its parameter derivative are uniformly bounded, and the form is coercive with constant $α > 0$ . Suppose the forward state and time-reversed adjoint both satisfy the hypotheses of Semidiscrete Ritz error splitting and energy estimate, with operator-norm $C^{1}$ comparison maps $P_{u, h} (t), P_{p, h} (t) : V \to V_{h}$ . For the state comparison map, set $ρ_{u} = P_{u, h} u - u$ , $ϑ_{u} = u_{h} - P_{u, h} u$ , and $R_{u} (t) (v_{h}) = (ρ_{u}^{'} (t), v_{h})_{H} + a_{θ} (t; ρ_{u} (t), v_{h}), ρ_{u}^{'} = P_{u, h}^{'} (t) u + P_{u, h} (t) u^{'} - u^{'} .$ In reverse time $s = T - t$ , put $q (s) = p (T - s)$ and ${\tilde{P}}_{p, h} (s) = P_{p, h} (T - s)$ ; define $ρ_{p} (s) = {\tilde{P}}_{p, h} (s) q (s) - q (s)$ , $ϑ_{p} (s) = p_{h} (T - s) - {\tilde{P}}_{p, h} (s) q (s)$ , and $R_{p} (s) (v_{h}) = (ρ_{p}^{'} (s), v_{h})_{H} + a_{θ} (T - s; v_{h}, ρ_{p} (s))$ . The projection product rule gives $ρ_{p}^{'} = {\tilde{P}}_{p, h}^{'} q + {\tilde{P}}_{p, h} q^{'} - q^{'}$ , with ${\tilde{P}}_{p, h}^{'} (s) = - P_{p, h}^{'} (T - s)$ . The Ritz supplier gives, for $z = u, p$ , $∥ z - z_{h} ∥_{L^{2} (0, T; V)} \leq E_{z} : = ∥ ρ_{z} ∥_{L^{2} (0, T; V)} + α^{- 1 / 2} {(∥ ϑ_{z} (0) ∥_{H}^{2} + α^{- 1} ∥ R_{z} ∥_{L^{2} (0, T; V_{h}^{*})}^{2})}^{1 / 2},$ where the adjoint's initial trace means its terminal trace in original time. In particular, $R_{z}$ contains the derivative of the time-dependent projection; it cannot be dropped from this hypothesis.

For $∥ η ∥_{2} \leq 1$ , assume $∥ D_{θ} f [η] ∥_{L^{2} (V^{*})} \leq M_{f}$ , $∥ D_{θ} a (t) [η] ∥_{L (V, V^{*})} \leq M_{a}$ a.e., and define $U_{*} : = \max (∥ u ∥_{L^{2} (V)}, ∥ u_{h} ∥_{L^{2} (V)})$ , $P_{*} : = \max (∥ p ∥_{L^{2} (V)}, ∥ p_{h} ∥_{L^{2} (V)})$ , $P_{0, *} : = \max (∥ p (0) ∥_{H}, ∥ p_{h} (0) ∥_{H})$ , and $U_{0}^{'} : = ∥ D_{θ} u_{0} ∥_{L (R^{m}, H)}$ . Suppose $∥ r_{h} - r ∥_{(R^{m})^{*}} \leq ε_{J}$ , $∥ D_{θ} u_{0, h} - D_{θ} u_{0} ∥_{L (R^{m}, H)} \leq ε_{0}$ , $∥ D_{θ} f_{h} - D_{θ} f ∥_{L (R^{m}, L^{2} (V^{*}))} \leq ε_{f}$ , and $∥ D_{θ} a_{h} - D_{θ} a ∥_{L^{\infty} (0, T; L (R^{m}, L (V, V^{*})))} \leq ε_{a}$ . Put $E_{p, 0} = ∥ p_{h} (0) - p (0) ∥_{H}$ . Then $∥ g_{h} - g ∥_{(R^{m})^{*}} \leq ε_{J} + U_{0}^{'} E_{p, 0} + P_{0, *} ε_{0} + M_{f} E_{p} + ε_{f} P_{*} + M_{a} (E_{u} P_{*} + U_{*} E_{p}) + ε_{a} U_{*} P_{*} .$ The derivative-data and trace defects are additional assumptions; an energy estimate for the state alone does not imply this gradient bound.

Facts & Assumptions

Given: The continuous and semidiscrete differentiable state/adjoint pairs, their objective-gradient formulas, the bounded coercive form, parameter derivative bounds, and the time-dependent Ritz comparison maps and residuals.

[F1]

The continuous parabolic adjoint formula expresses the gradient through the initial trace, forcing derivative, bilinear-form derivative and direct objective terms (A parabolic adjoint gives the gradient when the state map is differentiable). The semidiscrete representation is assumed explicitly in the Statement.

[F2]

The Ritz splitting obeys the residual identity and energy estimate, and for a time-dependent comparison map its residual uses $ρ^{'} = P_{h}^{'} u + P_{h} u^{'} - u^{'}$ (Semidiscrete Ritz error splitting and energy estimate).

[F3]

The weak time product rule yields $(P_{h} (t) u (t))^{'} = P_{h}^{'} (t) u (t) + P_{h} (t) u^{'} (t)$ for an operator-norm $C^{1}$ family (Time derivatives of fixed and differentiable comparison projections).

[A1]

Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) is required by the cited parabolic energy/pairing results. Both state and time-reversed transposed adjoint estimates, all traces, and all displayed derivative-data defects are assumed to satisfy their stated hypotheses (Semidiscrete Ritz error splitting and energy estimate, The evolution-space energy identity and continuous H trace).

Proof

1.1A1F2F3givenalgebra

For the forward state, the product rule [F3] gives $ρ_{u}^{'} = P_{u, h}^{'} u + P_{u, h} u^{'} - u^{'}$ , so the residual in [F2] is exactly the one displayed in the Statement. Apply the energy estimate in [F2] to $ϑ_{u}$ ; it gives $∥ ϑ_{u} ∥_{L^{2} (V)} \leq α^{- 1 / 2} (∥ ϑ_{u} (0) ∥_{H}^{2} + α^{- 1} ∥ R_{u} ∥_{L^{2} (V_{h}^{*})}^{2})^{1 / 2}$ . Since $u - u_{h} = - ρ_{u} - ϑ_{u}$ , the triangle inequality proves the stated $E_{u}$ bound.

2.1A1F2F3step 1.1construct

Set $s = T - t$ and $q (s) = p (T - s)$ . The backward adjoint equation becomes a forward parabolic variational equation with transposed form $a_{θ} (T - s; v, w)$ , which has the same boundedness and coercivity constants. The comparison ${\tilde{P}}_{p, h} (s) = P_{p, h} (T - s)$ is operator-norm $C^{1}$ and ${\tilde{P}}_{p, h}^{'} (s) = - P_{p, h}^{'} (T - s)$ . By [F3], its residual contains ${\tilde{P}}_{p, h}^{'} q + {\tilde{P}}_{p, h} q^{'} - q^{'}$ ; apply the energy estimate in [F2] and the splitting $p - p_{h} = - ρ_{p} - ϑ_{p}$ to get the stated $E_{p}$ bound by the same triangle inequality as step 1.1.

3.1A1F1step 2.1givenalgebra

Subtract the continuous and semidiscrete representations assumed in the Statement; [F1] supplies the continuous one. The direct term contributes $ε_{J}$ . For the initial pairing, add and subtract $(p_{h} (0), D_{θ} u_{0})$ ; Cauchy--Schwarz bounds the difference by $U_{0}^{'} E_{p, 0} + P_{0, *} ε_{0}$ . In the forcing term, add and subtract $D_{θ} f [p_{h}]$ ; the derivative-data defect is at most $ε_{f} P_{*}$ , and the exact derivative contributes at most $M_{f} E_{p}$ by step 2.1. Thus the direct, initial, and forcing contributions are bounded by the first four terms in the stated gradient estimate.

4.1A1F1step 1.1step 2.1step 3.1givenalgebra∎

For the form term, add and subtract $D_{θ} a [u_{h}, p_{h}]$ . Its derivative-data defect is at most $ε_{a} U_{*} P_{*}$ . Split the exact derivative difference as $D_{θ} a [u_{h} - u, p_{h}] + D_{θ} a [u, p_{h} - p]$ ; Cauchy--Schwarz in time, the bound $∥ D_{θ} a [η] ∥ \leq M_{a}$ , and steps 1.1 and 2.1 give $M_{a} (E_{u} P_{*} + U_{*} E_{p})$ . Combine this with step 3.1 and take the supremum over $∥ η ∥_{2} \leq 1$ to obtain the displayed dual norm bound.

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Positive-diagonal QR factors give the exact finite-time tangent-volume change

Statement

Let $A_{1}, \dots, A_{N} \in R^{n \times n}$ be invertible, with $n, N \geq 1$ , and let $Q_{0}$ be orthogonal. For $1 \leq k \leq N$ , form the unique positive-diagonal QR factorization $A_{k} Q_{k - 1} = Q_{k} R_{k},$ where $Q_{k}$ is orthogonal and $R_{k}$ is upper triangular with positive diagonal $r_{k, i i}$ . Put $P_{N} = A_{N} \dots A_{1}$ and define the exact finite-time log-volume change by $\log ∣ \det P_{N} ∣$ . Then $\log ∣ \det P_{N} ∣ = \sum_{k = 1}^{N} \sum_{i = 1}^{n} \log r_{k, i i} .$ No infinite-time Lyapunov exponent or shadowing conclusion follows.

For a numerical log record, define $σ_{\min} (A) : = \inf_{∥ x ∥_{2} = 1} ∥ A x ∥_{2}$ . If $σ_{\min} (A_{k}) \geq σ > 0$ , each exact $r_{k, i i} \geq σ$ . If positive computed diagonals ${\tilde{r}}_{k, i i}$ satisfy $∣ {\tilde{r}}_{k, i i} - r_{k, i i} ∣ \leq ε_{k, i} \leq σ / 2$ , and each computed logarithm ${\hat{l}}_{k, i}$ satisfies $∣ {\hat{l}}_{k, i} - \log {\tilde{r}}_{k, i i} ∣ \leq δ_{k, i}$ , then $∣ \sum_{k, i} {\hat{l}}_{k, i} - \log ∣ \det P_{N} ∣ ∣ \leq \sum_{k, i} (\frac{2 ε_{k, i}}{σ} + δ_{k, i}) .$ If any factor loses rank, this log-volume branch returns failure.

Facts & Assumptions

Given: The finite list of invertible tangent matrices, an orthogonal $Q_{0}$ , and the stated factorizations. The numerical bound additionally assumes the displayed certified singular-value and log-evaluation errors.

[F1]

Every invertible real square matrix has a unique factorization $A = Q R$ with $Q$ orthogonal and $R$ upper triangular with positive diagonal (Every invertible real or complex square matrix has a unique factorisation $A = Q R$ with $Q$ orthogonal or unitary and $R$ upper triangular with positive real diagonal).

[F2]

Determinants multiply under finite square matrix products (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ).

[F3]

The determinant of an upper triangular matrix is the product of its diagonal entries (The determinant of a triangular matrix is the product of its diagonal entries).

[F4]

An orthogonal matrix has determinant modulus one (Orthogonal and unitary operators form groups, and their determinants have modulus one).

[F5]

The logarithm difference is the integral of $1 / t$ between positive endpoints (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[A1]

For the numerical bound, $σ_{\min} (A_{k}) \geq σ > 0$ , $∣ {\tilde{r}}_{k, i i} - r_{k, i i} ∣ \leq ε_{k, i} \leq σ / 2$ , and the computed logarithm errors are bounded by $δ_{k, i}$ .

Proof

1.1givenalgebrainduction

Induction on $k$ using $A_{k} Q_{k - 1} = Q_{k} R_{k}$ gives $P_{k} Q_{0} = Q_{k} R_{k} R_{k - 1} \dots R_{1}, P_{k} : = A_{k} \dots A_{1} .$ For $k = 1$ this is the defining factorization; multiplying the identity at $k$ on the left by $A_{k + 1}$ and substituting $A_{k + 1} Q_{k} = Q_{k + 1} R_{k + 1}$ proves the next case.

1.2F1givenalgebra

Orthogonal factors preserve Euclidean norms, so $σ_{\min} (R_{k}) = σ_{\min} (Q_{k} R_{k}) = σ_{\min} (A_{k} Q_{k - 1}) = σ_{\min} (A_{k}) \geq σ$ . For a fixed $i$ , the leading $(i - 1) \times (i - 1)$ block of $R_{k}$ is invertible. Choose $x \in R^{i - 1}$ solving $R_{1 : i - 1, 1 : i - 1} x = - R_{1 : i - 1, i}$ and set $z = (x, 1, 0, \dots, 0)^{T}$ . Upper triangularity gives $R_{k} z = r_{k, i i} e_{i}$ and $∥ z ∥_{2} \geq 1$ , so $σ_{\min} (R_{k}) \leq ∥ R_{k} z ∥_{2} / ∥ z ∥_{2} \leq r_{k, i i}$ . For $i = 1$ , take $z = e_{1}$ . Hence $r_{k, i i} \geq σ$ .

2.1F2F3F4step 1.1algebra

Take determinants in step 1.1. By [F2], $\det (P_{N}) \det (Q_{0}) = \det (Q_{N}) \prod_{k = 1}^{N} \det (R_{k})$ . By [F3] each $\det (R_{k}) = \prod_{i} r_{k, i i} > 0$ , and by [F4] $∣ \det (Q_{0}) ∣ = ∣ \det (Q_{N}) ∣ = 1$ . Thus $∣ \det (P_{N}) ∣ = \prod_{k, i} r_{k, i i}$ . Taking logarithms of positive numbers proves the exact identity.

3.1A1F5step 1.2algebra∎

Both $r_{k, i i}$ and ${\tilde{r}}_{k, i i}$ are at least $σ / 2$ . By [F5], $∣ \log {\tilde{r}}_{k, i i} - \log r_{k, i i} ∣ \leq (2 / σ) ∣ {\tilde{r}}_{k, i i} - r_{k, i i} ∣$ . Adding the computation error $δ_{k, i}$ and summing the scalar bounds by the triangle inequality proves the stated numerical log bound. The lower singular-value bound is essential to keep the log derivative bounded; if any factor is singular its determinant is zero and its log-volume is undefined.

5 · Examples, counterexamples and false statements

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Discrete continuous and pde adjoint sensitivities

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2 · Summary

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Positive-diagonal QR factors give the exact finite-time tangent-volume change

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5 · Examples, counterexamples and false statements

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