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Derivative of the nonlinear shooting residual

Statement

Let $Ψ : U \to R^{d}$ be a $C^{1}$ endpoint map (total differentiability with continuous derivative entries) on an open set $U \subseteq R^{d}$ , and let $B$ be $C^{1}$ on an open neighborhood of ${(s, Ψ (s)) : s \in U}$ . Define $R_{Ψ} (s) = B (s, Ψ (s))$ . Then $R_{Ψ}$ is $C^{1}$ and

$D R_{Ψ} (s) = D_{1} B (s, Ψ (s)) + D_{2} B (s, Ψ (s)) D Ψ (s) .$

This applies to the exact flow endpoint map under Initial-data sensitivity of an ODE flow on a common compact tube and to the approximate endpoint map in Certified differentiable IVP approximation interface for shooting; in the latter case the derivative is exact for the numerical map, not the exact flow.

Facts & Assumptions

Given: $U$ , $Ψ$ , $B$ , and their stated $C^{1}$ properties.

[F1]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : for totally differentiable maps $g, f$ , $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ .

[F2]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the Jacobian matrix represents the total derivative in standard coordinates.

[F3]

Initial-data sensitivity of an ODE flow on a common compact tube: the exact endpoint map $s \mapsto ϕ (b; s)$ is $C^{1}$ on its open initial-state domain.

[F4]

Certified differentiable IVP approximation interface for shooting: the approximate endpoint map is $C^{1}$ and its supplied derivative is its derivative as a map on initial states.

Proof

1.1givenF2algebra

Define $G : U \to R^{d} \times R^{d}$ by $G (s) = (s, Ψ (s))$ . The coordinate product map is totally differentiable and $D G (s) v = (v, D Ψ (s) v)$ .

2.1givenF1step 1.1algebra

The residual is the composition $R_{Ψ} = B \circ G$ . Applying [F1] at each $s$ gives $D R_{Ψ} (s) v = D_{1} B (s, Ψ (s)) v + D_{2} B (s, Ψ (s)) D Ψ (s) v$ for every $v \in R^{d}$ , hence the displayed matrix identity.

3.1F1F3F4step 2.1∎

Since both component maps and their derivatives are continuous, the derivative in step 2.1 varies continuously with $s$ ; therefore $R_{Ψ}$ is $C^{1}$ . Facts [F3] and [F4] verify the two stated choices of $Ψ$ under their respective hypotheses.

Depends on

Used by

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Sources

R. Verfürth, Numerical Methods and Stochastics, Part I: Numerical Methods, Lecture Notes (standard reference, not scraped)
T. A. Driscoll and R. J. Braun, Fundamentals of Numerical Computation, Chapter 10 (standard reference, not scraped)