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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.

Depends on

Used by

Backward Euler differentiates a stiff scalar decay fit on its exact branch Example

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Sources

William W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system, §§2–3 (standard reference, not scraped)
John C. Butcher, Implicit Runge-Kutta Methods, tutorial, §§1–2 (standard reference, not scraped)