Using pattern search methods for surface structure determination of nanomaterials

Abstract

Atomic scale surface structure plays an important role in describing many properties of materials, especially in the case of nanomaterials. One of the most effective techniques for surface structure determination is low-energy electron diffraction (LEED), which can be used in conjunction with optimization to fit simulated LEED intensities to experimental data. This optimization problem has a number of characteristics that make it challenging: it has many local minima, the optimization variables can be either continuous or categorical, the objective function can be discontinuous, there are no exact analytic derivatives (and no derivatives at all for categorical variables), and function evaluations are expensive. In this study, we show how to apply a particular class of optimization methods known as pattern search methods to address these challenges. These methods do not explicitly use derivatives, and are particularly appropriate when categorical variables are present, an important feature that has not been addressed in previous LEED studies. We have found that pattern search methods can produce excellent results, compared to previously used methods, both in terms of performance and locating optimal results.

Derivative‐free optimization methods for surface structure determination of nanosystems

“…Due to symmetry, there are only 14 atoms on this surface, resulting in 42 (14×3) optimization variables. We compared our results against the numerical experiments performed by Zhao et al [8]. Table 1 displays the results for using the additive surrogate versus no search and a simple LHS search, in which columns 2-4 contain the number of initial LHS points used, the best objective function value f (x * ) found, and the number of true function evaluations (fevals).…”

“…Several methods have been proposed for this problem including simulated simulated annealing [3], fast simulated annealing [4,5], a modified random sampling algorithm [6] and genetic algorithms (GAs) [7]. Recently, Zhao et al [8] applied a generalized pattern search method (GPS) to the problem of determining the structure of a Ni(001)-(5x5)-Li surface. Pattern search methods were found to have better performance than both simulated annealing and GA, generating better trial structures with significantly fewer evaluation functions required.…”

Global search algorithms in surface structure determination using photoelectron diffraction

“…Moreover, the experimental data show relatively strong (±40%) modulations in one (normal) emission direction, indicating a high-symmetry (atop) adsorption site. The original published analysis of the data did find the oxygen atom of the water (O w ) to be directly atop the five-fold coordinated surface Ti atoms (Fig 3), with a Ti-O w bond length of 2.21±0.02Å, but also found four different substrate surface relaxations to be significant, involving displacements perpendicular to the surface (∆z), and parallel to the surface in the direction 11 (∆x). Specifically, the z coordinate of the five-fold coordinated Ti atom, the x and z coordinates of the first layer planar O atoms, and the z coordinate of the bridging O atom below the five-fold coordinated Ti atom, were all found to differ from those of an ideal bulk-terminated solid.…”

A New Method for Solving Supervised Data Classification Problems

“…Since these functions are nonsmooth and estimate of subgradients is difficult, direct search methods of optimization seem to be the best option for solving them. The main attraction of direct search methods is their ability to find optimal solutions without the need for computing derivatives, in contrast to the more familiar gradient-based methods [24]. Direct search algorithms can be applied for problems that are difficult to be solved with traditional optimization techniques, including problems that are difficult to model mathematically or are not well defined.…”