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Centered-tangent string residuals are second-order consistent on a regular MEP

Statement

Let $L > 0$ , let $M \geq 2$ be an integer, and let $R$ be a regular MEP as in Regular minimum-energy paths on a declared collision-free tube, now parameterized by arc length on $[0, L]$ , with $R \in C^{3}$ under the endpoint convention of Polyline interpolation and equal-length redistribution approximate a smooth arc-length path and the derivative bounds of Second-order centered and unequal-step tangent estimates and Polyline interpolation and equal-length redistribution approximate a smooth arc-length path. Assume the cited lemmas' mesh-size conditions hold, in particular $M_{3} h^{2} / 6 \leq 1 / 2$ , every interior centered chord is nonzero, and $0 < h \leq 1$ . Let $R_{i} = R (i h)$ for $i = 0, \dots, M$ , $M h = L$ , and suppose $∥ \nabla V (R (s)) ∥_{2} \leq G$ along the curve. Use the centered unit tangent ${\hat{t}}_{i} = (R_{i + 1} - R_{i - 1}) / ∥ R_{i + 1} - R_{i - 1} ∥_{2}$ , and define the adjacent chord lengths $ℓ_{i}^{+} = ∥ R_{i + 1} - R_{i} ∥_{2}$ and $ℓ_{i}^{-} = ∥ R_{i} - R_{i - 1} ∥_{2}$ . Then the projected string force at each interior sample satisfies

$∥ (I - {\hat{t}}_{i} {\hat{t}}_{i}^{T}) \nabla V (R_{i}) ∥_{2} \leq C_{F} h^{2} .$

If the equal-spacing spring residual $k (ℓ_{i}^{+} - ℓ_{i}^{-}) {\hat{t}}_{i}$ is also evaluated on these samples, then its norm is at most $C_{S} h^{3}$ for any fixed $k \in R$ , with $C_{S} = 2 ∣ k ∣ C_{1}$ . Equal-length polyline redistribution of the exact samples changes each image by at most $C_{R} h^{2}$ . The constants depend on the stated derivative bounds, $G$ , $∣ k ∣$ , and the path, not on $M$ . The projected-force conclusion is for the centered-tangent string residual; it does not assert second-order consistency for the improved energy-selected NEB tangent.

Facts & Assumptions

Given: The $C^{3}$ unit-speed regular MEP, mesh, derivative bounds, and gradient bound in the Statement.

[F1]

Regular minimum-energy paths on a declared collision-free tube: $P_{t (s)}^{⊥} \nabla V (R (s)) = 0$ and $t (s) = R^{'} (s)$ for the unit-speed parameterization.

[F2]

Second-order centered and unequal-step tangent estimates: the open-interval $C^{3}$ extension, bounded third derivatives, and $M_{3} h^{2} / 6 \leq 1 / 2$ give $∥ {\hat{t}}_{i} - t (i h) ∥_{2} \leq C_{t} h^{2}$ for the centered chord.

[F3]

Polyline interpolation and equal-length redistribution approximate a smooth arc-length path: under its endpoint $C^{3}$ convention, sampled chord lengths satisfy $∣ ℓ_{i} - h ∣ \leq C_{1} h^{3}$ , and equal-length interpolation has image error at most $C_{R} h^{2}$ .

[F4]

Finite-image projected-gradient string with arc-length redistribution: the string residual uses the normal projection formed from the centered chord, with endpoints fixed and linear equal-length redistribution.

[F5]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ u^{T} w ∣ \leq ∥ u ∥_{2} ∥ w ∥_{2}$ , and the Euclidean norm satisfies the triangle inequality.

Given: The data and conventions in the Statement.

Proof

1.1F1F2F5givenalgebra

For any vector $w$ , $(u u^{T} - v v^{T}) w = (u - v) (u^{T} w) + v ((u - v)^{T} w)$ . Since $u$ and $v$ are unit vectors, [F5] and the triangle inequality give $∥ (u u^{T} - v v^{T}) w ∥_{2} \leq 2 ∥ u - v ∥_{2} ∥ w ∥_{2}$ . Apply this with $u = {\hat{t}}_{i}$ , $v = t (i h)$ , and $w = \nabla V (R_{i})$ . Since $P_{v}^{⊥} \nabla V (R_{i}) = 0$ by [F1], we obtain $∥ P_{{\hat{t}}_{i}}^{⊥} \nabla V (R_{i}) ∥_{2} \leq 2 G ∥ {\hat{t}}_{i} - t (i h) ∥_{2} \leq 2 G C_{t} h^{2}$ .

1.2F3givenalgebra

By [F3], write each neighboring chord length as $ℓ_{i}^{\pm} = h + e_{i}^{\pm}$ with $∣ e_{i}^{\pm} ∣ \leq C_{1} h^{3}$ . Hence the spring residual has norm $∣ k ∣ ∣ ℓ_{i}^{+} - ℓ_{i}^{-} ∣ \leq 2 ∣ k ∣ C_{1} h^{3}$ , because the triangle inequality bounds the difference by $∣ e_{i}^{+} ∣ + ∣ e_{i}^{-} ∣$ .

2.1F3F4step 1.1step 1.2algebra∎

The spring term is tangent and has the bound in step 1.2; the normal component has the bound in step 1.1. Since $h \leq 1$ , $h^{3} \leq h^{2}$ , so their sum is bounded by $(2 G C_{t} + 2 ∣ k ∣ C_{1}) h^{2}$ . The string redistribution error is the last assertion of [F3]. These statements concern the centered-tangent string discretization specified by [F4]; the improved NEB tangent is a different branch and is treated separately.

Depends on

Used by

Dependency tree · two levels

24 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

Weinan E, Weiqing Ren, and Eric Vanden-Eijnden, String method for the study of rare events (standard reference, not scraped)
Xuanyu Liu, Huajie Chen, and Christoph Ortner, Convergence of the Discrete Minimum Energy Path (standard reference, not scraped)