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The multiple-shooting Jacobian is bordered block-bidiagonal

Statement

Fix a mesh $a = t_{0} < \dots < t_{N} = b$ , $N \geq 1$ , and an open product domain of segment starts $(s_{0}, \dots, s_{N})$ on which $B$ and each exact or certified approximate segment endpoint map $Ψ_{i} : U_{i} \to R^{d}$ are $C^{1}$ . Here $C^{1}$ means total differentiability with continuous derivative matrix entries. Define $F$ by rows $F_{i} = Ψ_{i} (s_{i}) - s_{i + 1}$ for $0 \leq i < N$ and $F_{N} = B (s_{0}, s_{N})$ . Its Jacobian block row $i < N$ has $D Ψ_{i} (s_{i})$ in column $i$ , $- I_{d}$ in column $i + 1$ , and zero in every other column. Its boundary row has $D_{1} B$ and $D_{2} B$ in columns $0$ and $N$ and zero elsewhere. When $N = 1$ , columns $0$ and $N$ are distinct.

Facts & Assumptions

Given: The mesh, open product domain, $C^{1}$ maps $B, Ψ_{i}$ , and residual $F$ .

[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]

Initial-data sensitivity of an ODE flow on a common compact tube: an exact segment endpoint map is $C^{1}$ when its solutions remain in a common compact convex tube.

[F3]

Certified differentiable IVP approximation interface for shooting: a certified approximate segment endpoint map is $C^{1}$ and supplies its own exact derivative.

[F4]

Square nonlinear two-point boundary-value problems and endpoint residuals: the exact multiple-shooting rows are the segment endpoint minus next segment start, followed by $B (s_{0}, s_{N})$ .

Proof

1.1givenF1F4algebra

For an interior row $F_{i} (s_{0}, \dots, s_{N}) = Ψ_{i} (s_{i}) - s_{i + 1}$ , varying only block $s_{i}$ changes that row by $D Ψ_{i} (s_{i})$ , varying only block $s_{i + 1}$ changes it by $- I_{d}$ , and varying any other block leaves it unchanged. These are precisely the stated nonzero and zero blocks.

1.2givenF1F2F3algebra

For the boundary row $F_{N} = B (s_{0}, s_{N})$ , the chain rule gives derivatives $D_{1} B$ and $D_{2} B$ with respect to the first and last blocks; all intermediate blocks are absent from its formula, so their derivatives are zero. The differentiability assertions for exact and approximate segment maps follow from [F2] and [F3].

2.1givenstep 1.1step 1.2algebra∎

The residual has $(N + 1) d$ scalar rows and $(N + 1) d$ scalar variables, so the displayed block matrix is square.

Depends on

Used by

Finite-budget multiple shooting for a nonlinear two-point BVP Algorithm

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

R. Verfürth, Numerical Methods and Stochastics, Part I: Numerical Methods, Lecture Notes (standard reference, not scraped)