LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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Paired force evaluations give a second-order Hessian-vector estimate

Statement

Let $V \in C^{4} (U, R)$ on an open set $U \subseteq R^{d}$ , let $x \in U$ , and let $v \in R^{d}$ satisfy $∥ v ∥_{2} = 1$ and $x \pm ϵ v \in U$ for $ϵ > 0$ . Assume the segments $x + [- ϵ, ϵ] v$ lie in $U$ . Define

$D_{ϵ} (x, v) : = \frac{\nabla V (x + ϵ v) - \nabla V (x - ϵ v)}{2 ϵ} .$

If approximate gradient evaluations ${\hat{g}}_{\pm}$ satisfy $∥ {\hat{g}}_{\pm} - \nabla V (x \pm ϵ v) ∥_{2} \leq η$ , and $M_{4}$ bounds the third derivatives of each gradient coordinate in the Taylor remainders on these segments, then

${∥ \frac{{\hat{g}}_{+} - {\hat{g}}_{-}}{2 ϵ} - \nabla^{2} V (x) v ∥}_{2} \leq \frac{\sqrt{d} M_{4}}{6} ϵ^{2} + \frac{η}{ϵ} .$

Thus the deterministic truncation error is $O (ϵ^{2})$ and the force-evaluation error contributes $O (η / ϵ)$ . This is an error bound for the Hessian action on the supplied direction; it is not an operator-norm bound for a full Hessian approximation.

Facts & Assumptions

Given: The smooth potential, admissible line segment, unit direction, finite-difference step, and force discrepancy bound in the Statement.

[F1]

$C^{k}$ maps and multi-index derivative notation in Euclidean space, The Hessian matrix and critical points of a scalar field, and The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: $V \in C^{4}$ gives continuous coordinate partials of each gradient coordinate through order three. Applying If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative to those iterated partials through order two gives their total derivatives, whose Jacobian entries are continuous. A total derivative computes every directional derivative, and its matrix is the Jacobian identifies the matrix of $D (\nabla V)$ with the Jacobian, and The Hessian of a $C^{2}$ scalar field is symmetric makes its Jacobian entries agree with the Hessian convention $H_{i j} = \partial_{i} \partial_{j} V$ .

[F2]

By [F1], $C^{k}$ maps and multi-index derivative notation in Euclidean space, and repeated application of The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , each restriction $t \mapsto \partial_{j} V (x + t v)$ is $C^{3}$ on its open interval domain. The one-dimensional case of Multivariable Taylor formula with a Lagrange remainder along a line segment then gives its coordinatewise degree-two Taylor expansion with Lagrange remainder.

[A1]

For each coordinate $j$ , the third derivative of $t \mapsto \partial_{j} V (x + t v)$ is bounded in absolute value by $M_{4}$ for $t \in [- ϵ, ϵ]$ ; this is the remainder bound required for [F2].

Given: The hypotheses of the Statement.

Proof

1.1A1F1F2givenalgebra

Let $I$ be the open interval component of ${t \in R : x + t v \in U}$ containing $[- ϵ, ϵ]$ ; it exists because that line segment lies in the open set $U$ . Each $g_{j} (t) : = \partial_{j} V (x + t v)$ is $C^{3}$ on the open, convex interval $I$ . Apply [F2] to $g_{j}$ at $0$ with displacements $\pm ϵ$ . The linear terms are $\pm ϵ (H v)_{j}$ by [F1], the quadratic terms agree, and the coordinate remainders satisfy $∣ (r_{\pm})_{j} ∣ \leq M_{4} ϵ^{3} / 6$ by [A1]. Thus $∥ r_{\pm} ∥_{2} \leq \sqrt{d} M_{4} ϵ^{3} / 6$ . Subtracting and dividing by $2 ϵ$ cancels the constant and quadratic terms and gives $∥ D_{ϵ} (x, v) - \nabla^{2} V (x) v ∥_{2} \leq \sqrt{d} M_{4} ϵ^{2} / 6$ .

2.1givenstep 1.1algebra∎

Write $e_{\pm} = {\hat{g}}_{\pm} - g (x \pm ϵ v)$ . Then $∥ e_{+} - e_{-} ∥_{2} \leq ∥ e_{+} ∥_{2} + ∥ e_{-} ∥_{2} \leq 2 η$ , so their contribution after division by $2 ϵ$ is at most $η / ϵ$ . Adding this to step 1.1 proves the inequality. A bound on this single direction alone supplies no bound on an operator acting on all directions.

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Sources

Graeme Henkelman and Hannes Jónsson, A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives (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)