Research Papers

An Efficient First-Principles Saddle Point Searching Method Based on Distributed Kriging Metamodels

[+] Author and Article Information
Anh Tran

George W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: anh.vt2@gatech.edu

Lijuan He

George W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: helijuan130@gmail.com

Yan Wang

George W. Woodruff School of
Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: yan.wang@me.gatech.edu

1Corresponding author.

Manuscript received January 14, 2017; final manuscript received June 25, 2017; published online September 7, 2017. Assoc. Editor: Sankaran Mahadevan.

ASME J. Risk Uncertainty Part B 4(1), 011006 (Sep 07, 2017) (8 pages) Paper No: RISK-17-1002; doi: 10.1115/1.4037459 History: Received January 14, 2017; Revised June 25, 2017

Searching for local minima, saddle points, and minimum energy paths (MEPs) on the potential energy surface (PES) is challenging in computational materials science because of the complexity of PES in high-dimensional space and the numerical approximation errors in calculating the potential energy. In this work, a local minimum and saddle point searching method is developed based on kriging metamodels of PES. The searching algorithm is performed on both kriging metamodels as the approximated PES and the calculated one from density functional theory (DFT). As the searching advances, the kriging metamodels are further refined to include new data points. To overcome the dimensionality problem in classical kriging, a distributed kriging approach is proposed, where clusters of data are formed and one metamodel is constructed within each cluster. When the approximated PES is used during the searching, each predicted potential energy value is an aggregation of the ones from those metamodels. The dimension of each metamodel is further reduced based on the observed symmetry in materials systems. The uncertainty associated with the ground-state potential energy is quantified using the statistical mean-squared error in kriging to improve the robustness of the searching method.

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Grahic Jump Location
Fig. 6

Transition pathway with local minima and saddle points from initial to final states and its corresponding configurations for Fe8H system. The saddle points are denoted as circles, whereas the local minima are denoted as squares.

Grahic Jump Location
Fig. 2

The invariant property of PES with respect to the atomistic component and the construction of symmetry-enhanced cluster kriging metamodel: (a) The potential energy of an arbitrary material systems is invariant to the permutation of its atomistic component and (b) The construction of symmetry-enhanced cluster kriging metamodel by sorting each of its component by x-, y-, and z-coordinates separately and construct the PES metamodel based on the permutated configurations

Grahic Jump Location
Fig. 1

Flowchart of the searching method with metamodel involves three stages. The first and second stages are combined into a single transition pathway search, and third stage is the climbing process. A PES metamodel is utilized in all three stages to accelerate the searching process.

Grahic Jump Location
Fig. 3

Comparison between classical and distributed kriging on the comparison domains shows very similar response surfaces (a) Classical kriging on the comparison domain with 1800 sampling points and (b) Distributed kriging on the comparison domain, where the sampling points on odd clusters are plotted as diamonds, and the sampling points on even clusters are plotted as upside down triangle. There are totally nine clusters, where each cluster contains 200 sampling points.

Grahic Jump Location
Fig. 4

Comparison between the distributed kriging and the true value of Schwefel function shows an excellent agreement with a small number of cluster size Ndata of 150

Grahic Jump Location
Fig. 5

Comparison of computational cost shows that the distributed kriging scheme is scalable

Grahic Jump Location
Fig. 7

Illustration for two possible transition paths between two states




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