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Dimer finite-difference bias competes with force-evaluation error

Example

On $R^{2}$ , let $V (x, y) = x^{4} / 24 + y^{2} / 2$ , take the point $(0, 0)$ and direction $v = e_{1}$ . The exact Hessian action is $\nabla^{2} V (0, 0) e_{1} = 0$ . With exact gradients, the central dimer estimate is $D_{ϵ} = (ϵ^{2} / 6) e_{1}$ . If the two gradient evaluations have errors $+ η e_{1}$ and $- η e_{1}$ , each of norm $η$ , the computed estimate becomes ${\hat{D}}_{ϵ} = (ϵ^{2} / 6 + η / ϵ) e_{1}$ . The truncation bias decreases with $ϵ$ , while this permitted noise contribution increases as $ϵ$ shrinks.

Facts & Assumptions

Given: The potential, point, direction, finite-difference distance $ϵ > 0$ , and opposite evaluation errors of magnitude $η \geq 0$ .

[F1]

The Hessian matrix and critical points of a scalar field: the Hessian is the Jacobian of the gradient; at the origin its action on $e_{1}$ is zero.

[F2]

Paired force evaluations give a second-order Hessian-vector estimate: for $C^{4}$ potentials the paired central gradient difference has truncation error $O (ϵ^{2})$ and force-evaluation contribution at most $η / ϵ$ .

Given: The data in the Example.

Proof

1.1F1givenalgebra

The gradient is $\nabla V (x, y) = (x^{3} / 6, y)$ . Thus $\nabla V (ϵ e_{1}) = (ϵ^{3} / 6, 0)$ and $\nabla V (- ϵ e_{1}) = (- ϵ^{3} / 6, 0)$ . Their central difference is $((ϵ^{3} / 3) / (2 ϵ), 0) = (ϵ^{2} / 6) e_{1}$ . The Hessian at the origin is $diag (0, 1)$ , so [F1] gives $H e_{1} = 0$ .

2.1F2step 1.1givenalgebra

Adding the stated force errors changes the numerator by $2 η e_{1}$ ; division by $2 ϵ$ adds $η e_{1} / ϵ$ . Hence the computed estimate is exactly ${\hat{D}}_{ϵ} = (ϵ^{2} / 6 + η / ϵ) e_{1}$ . This realizes both contributions in [F2] for this example.

3.1step 2.1algebra∎

For fixed $η > 0$ , the deterministic term $ϵ^{2} / 6$ tends to zero as $ϵ \to 0$ , but the permitted error term $η / ϵ$ grows without bound. Therefore reducing the dimer separation does not monotonically improve the Hessian-vector estimate when force evaluations are inexact.

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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)