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Global optimization-based dimer method for finding saddle points

  • * Corresponding author: Lei Zhang

    * Corresponding author: Lei Zhang
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  • Searching saddle points on the potential energy surface is a challenging problem in the rare event. When there exist multiple saddle points, sampling different initial guesses are needed in local search methods in order to find distinct saddle points. In this paper, we present a novel global optimization-based dimer method (GOD) to efficiently search saddle points by coupling ant colony optimization (ACO) algorithm with optimization-based shrinking dimer (OSD) method. In particular, we apply OSD method as a local search algorithm for saddle points and construct a pheromone function in ACO to update the global population. By applying a two-dimensional example and a benchmark problem of seven-atom island on the (111) surface of an FCC crystal, we demonstrate that GOD shows a significant improvement in computational efficiency compared with OSD method. Our algorithm is the first try to apply the global optimization technique in searching saddle points and it offers a new framework to open up possibilities of adopting other global optimization methods.

    Mathematics Subject Classification: 37M05, 49K35, 37N30, 34K28, 65P99.

    Citation:

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  • Figure 1.  An example of $ B_2 $ function that has one minimum M0, four maxima M1-M4, and four index-1 saddle points S1-S4. (A): modified $ \kappa $ in Eq. (15). (B): pheromone function in Eq. (16). The black lines are the contours of the $ B_2 $ function. The parameters used in Eq. (16) are $ \alpha = 0.75,a = 0.05,b = 100 $

    Figure 2.  2D example with three different initial samplings: (A) circle sampling; (B) uniform sampling; (C) random sampling. White dots represent the initial points, green dots are the points deleted in the halfway, and red dots are the saddle points. The yellow lines represent the dynamic paths by GOD

    Figure 3.  Two initial samplings for the seven-atom island: (A) sample 1: shift the parallel $ (2,7) $ and $ (4,5) $ atoms simultaneously to get $ 100 $ initial points; (B) sample 2: shift parallel atoms along six directions to get $ 300 $ initial points

    Figure 4.  Seven-atom island example with two different initial samplings: (A) sample 1 with 100 initial points; (B) sample 2 with 300 initial points. Blue line represents the number of moving points, yellow line represents the number of deleted points, and red line corresponds to the number of saddle points. Some parameters are $ \delta_1 = 1.0, a = b = 10, \alpha = 0.5 $. $ step_{ls} = 20 $ for sample 1 and $ step_{ls} = 30 $ for sample 2

    Table 1.  Comparison of OSD method and GOD method in the 2D example

    sample method initial points force evaluations saddles
    circle OSD 50 7270 4
    circle GOD 50 1249 4
    uniform OSD 400 44212 22
    uniform GOD 400 8716 22
    random OSD 400 45581 22
    random GOD 400 8352 22
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    Table 2.  Comparison of OSD method and GOD method in the seven-atom island problem

    sample method cputime(s) force evaluations saddles
    sample 1 OSD 325.781 80946 29
    sample 1 GOD 188.375 45072 29
    sample 2 OSD 1022.720 250941 92
    sample 2 GOD 618.906 140624 92
     | Show Table
    DownLoad: CSV

    Table 3.  Parameter sensitivity of GOD method for Sample 1 in the seven-atom island problem

    parameters cputime(s) force evaluations saddles
    $ \delta_1 $ 1.0 188.375 45072 29
    0.75 226.781 49225 29
    0.5 234.141 51353 29
    $ \alpha $ 0.25 226.547 49972 29
    0.5 226.781 49225 29
    0.75 226.875 49929 29
    $ a $ 1 223.688 49036 29
    10 226.781 49225 29
    100 224.531 48894 29
    1000 234.359 51131 29
    $ b $ 1 223.688 49036 29
    10 226.781 49225 29
    100 204.297 48282 29
    1000 191.656 45857 29
     | Show Table
    DownLoad: CSV
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