LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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Second-order centered and unequal-step tangent estimates

Statement

Let $R : J \to R^{d}$ be a unit-speed $C^{3}$ curve on an open interval $J$ containing $[0, 1]$ . Let $J_{0} \subseteq J$ be an interval on which $\max_{j} \sup_{u \in J_{0}} ∣ R_{j}^{'''} (u) ∣ \leq M_{3} / \sqrt{d}$ . For the centered estimate, take $s$ and $h > 0$ with $[s - h, s + h] \subseteq J_{0}$ and define

$d_{h} : = \frac{R (s + h) - R (s - h)}{2 h}, {\hat{t}}_{h} : = \frac{d_{h}}{∥ d_{h} ∥_{2}} .$

If $δ_{h} : = M_{3} h^{2} / 6 \leq 1 / 2$ , then $d_{h} \neq 0$ and

$∥ d_{h} - R^{'} (s) ∥_{2} \leq δ_{h}, ∥ {\hat{t}}_{h} - R^{'} (s) ∥_{2} \leq 4 δ_{h} .$

For the unequal-step estimate, take positive $h_{-}, h_{+}$ with $[s - h_{-}, s + h_{+}] \subseteq J_{0}$ and define

$d_{h_{-}, h_{+}} : = \frac{h_{+}}{h_{-} (h_{-} + h_{+})} (R (s) - R (s - h_{-})) + \frac{h_{-}}{h_{+} (h_{-} + h_{+})} (R (s + h_{+}) - R (s)) .$

If $δ_{h_{-}, h_{+}} : = M_{3} h_{-} h_{+} / 6 \leq 1 / 2$ , then it is nonzero and its unit normalization differs from $R^{'} (s)$ by at most $4 δ_{h_{-}, h_{+}}$ . In particular, equal steps give an $O (h^{2})$ unit-tangent error; quasiuniform unequal steps give an $O (h_{-} h_{+})$ error. No second-order claim is made for a one-sided tangent branch.

Facts & Assumptions

Given: The unit-speed $C^{3}$ curve, stencil bounds and steps in the Statement.

[F1]

Regular minimum-energy paths on a declared collision-free tube defines the unit tangent of a regular path as $R^{'} (s) / ∥ R^{'} (s) ∥_{2}$ . For the present curve, the separate unit-speed hypothesis means $∥ R^{'} (s) ∥_{2} = 1$ on all of $J$ , so its unit tangent there is $R^{'} (s)$ .

[F2]

Multivariable Taylor formula with a Lagrange remainder along a line segment: applied coordinatewise to a $C^{3}$ curve, the degree-two Taylor remainder at displacement $u$ is bounded in Euclidean norm by $M_{3} ∣ u ∣^{3} / 6$ under the stated coordinate bound.

[A1]

The interval $J_{0}$ lies in the domain of the $C^{3}$ curve and every coordinate third derivative satisfies $∣ R_{j}^{'''} (u) ∣ \leq M_{3} / \sqrt{d}$ for $u \in J_{0}$ ; the $C^{3}$ convention is $C^{k}$ maps and multi-index derivative notation in Euclidean space.

Given: The hypotheses of the Statement.

Proof

1.1A1F2givenalgebra

Taylor's formula [F2] at $s + h$ and $s - h$ gives $R (s \pm h) = R (s) \pm h R^{'} (s) + \frac{h^{2}}{2} R^{''} (s) + r_{\pm}$ with $∥ r_{\pm} ∥_{2} \leq M_{3} h^{3} / 6$ by [A1]. Subtracting cancels the second-derivative terms, so $d_{h} = R^{'} (s) + (r_{+} - r_{-}) / (2 h)$ and $∥ d_{h} - R^{'} (s) ∥_{2} \leq M_{3} h^{2} / 6 = δ_{h}$ .

1.2A1F2givenalgebra

For unequal steps, write $R (s) - R (s - h_{-}) = h_{-} R^{'} - \frac{h_{-}^{2}}{2} R^{''} + r_{-}$ and $R (s + h_{+}) - R (s) = h_{+} R^{'} + \frac{h_{+}^{2}}{2} R^{''} + r_{+}$ with $∥ r_{\pm} ∥_{2} \leq M_{3} h_{\pm}^{3} / 6$ by [A1, F2]. Substitution into $d_{h_{-}, h_{+}}$ gives coefficient $1$ on $R^{'} (s)$ and $0$ on $R^{''} (s)$ ; the two remainder coefficients yield $∥ d_{h_{-}, h_{+}} - R^{'} (s) ∥_{2} \leq M_{3} h_{-} h_{+} / 6 = δ_{h_{-}, h_{+}}$ .

2.1F1step 1.1algebra

Since $∥ R^{'} (s) ∥_{2} = 1$ , step 1.1 gives $∥ d_{h} ∥_{2} \geq 1 - δ_{h} \geq 1 / 2$ , so $d_{h} \neq 0$ . Moreover, $∣ ∥ d_{h} ∥_{2} - 1 ∣ \leq δ_{h}$ , and therefore $∥ d_{h} / ∥ d_{h} ∥_{2} - R^{'} (s) ∥_{2} \leq ∥ d_{h} - R^{'} (s) ∥_{2} / ∥ d_{h} ∥_{2} + ∣ 1 - ∥ d_{h} ∥_{2} ∣ / ∥ d_{h} ∥_{2} \leq 4 δ_{h}$ .

3.1F1step 2.1step 1.2algebra∎

The same normalization argument as in step 2.1, using $δ_{h_{-}, h_{+}} \leq 1 / 2$ , gives nonzero $d_{h_{-}, h_{+}}$ and normalized error at most $4 δ_{h_{-}, h_{+}}$ . Setting $h_{-} = h_{+} = h$ recovers the centered formula, and if both steps are bounded by a fixed multiple of their maximum mesh size, their product is $O (h^{2})$ .

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Sources

Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, String method for the study of rare events (standard reference, not scraped)
Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, Simplified and improved string method for computing the minimum energy paths in barrier-crossing events (standard reference, not scraped)