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Comparison of SQVV and L-BFGS Minimisers for the MB Surface

We tested the NEB/L-BFGS method by minimising a 17-image NEB for the two-dimensional Müller-Brown surface [25]. Our calculations were carried out using the OPTIM program [138]. The NEB method in its previous formulation [28] and a modified L-BFGS minimiser [9] were implemented in OPTIM in a previous discrete path sampling study [8]. We used the same number of images, initial guess and termination criteria as described in Section 2.5.1 to make the results directly comparable.

Figure 2.5 shows the performance of the L-BFGS minimiser as a function of $ k_{spr}$. We used the following additional L-BFGS specific settings. The number of corrections in the BFGS update was set to $ m=4$ (Nocedal's recommendation for the number of corrections is $ 3 \leqslant m \leqslant 7$, see Reference lbfgs), the maximum step size was $ 0.1$, and we limited the step size for each image separately, i.e.

$\displaystyle \lvert {\bf p}_j \rvert \leqslant 0.1,$ (4.38)

where $ {\bf p}_j$ is the step for image $ j$. The diagonal elements of the inverse Hessian were initially set to $ 0.1$.

Figure: (a) Number of iterations, $ \ell $, and (b) average deviation from average image separation, $ \varsigma $, as a function of the spring force constant, $ k_{spr}$, obtained using a 17-image NEB for the Müller-Brown surface [25]. Minimisation was performed using L-BFGS with number of corrections $ m=4$, maximum step size $ 0.1$ and RMS force termination criterion $ 0.01$.
\begin{psfrags} \psfrag{k} [Bc][Bc]{$k_{spr}/10^3$} \psfrag{l} [Bc][Bc]{$\el... ...terline{\includegraphics[width=.40\textheight]{dneb/lbfgsMB.eps}} \end{psfrags}

From Figure 2.5 it can be seen that the performance of L-BFGS minimisation is relatively independent of the choice of force constant. All the optimisations with $ 30 \leqslant k_{spr} \leqslant 10,000$ converged to the steepest-descent path, and, for most of this range, in less than 100 iterations. This method therefore gives roughly an order of magnitude improvement in speed over SQVV minimisation [see Figure 2.4 (a)].

We found it helpful to limit the step size while optimising the NEB with the L-BFGS minimiser. The magnitude and direction of the gradient on adjacent images can vary significantly. Taking bigger steps can cause the appearance of temporary discontinuities and kinks in the NEB. The NEB still converges to the correct path, but it takes a while for these features to disappear and the algorithm does not converge any faster.

next up previous contents
Next: Doubly Nudged Elastic Bands Up: Results Previous: Choice of the Force   Contents
Semen A Trygubenko 2006-04-10