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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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References

Quapp, W. , and Heidrich, D. , 1984, “ Analysis of the Concept of Minimum Energy Path on the Potential Energy Surface of Chemically Reacting Systems,” Theor. Chim. Acta, 66(3–4), pp. 245–260. [CrossRef]
Berne, B. J. , Ciccotti, G. , and Coker, D. F. , 1998, Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore.
Laidler, K. J. , and King, M. C. , 1983, “ Development of Transition-State Theory,” J. Phys. Chem., 87(15), pp. 2657–2664. [CrossRef]
Henkelman, G. , Jóhannesson, G. , and Jónsson, H. , 2002, “ Methods for Finding Saddle Points and Minimum Energy Paths,” Theoretical Methods in Condensed Phase Chemistry, Springer, Dordrecht, The Netherlands, pp. 269–302. [CrossRef]
Schlegel, H. B. , 2003, “ Exploring Potential Energy Surfaces for Chemical Reactions: An Overview of Some Practical Methods,” J. Comput. Chem., 24(12), pp. 1514–1527. [CrossRef] [PubMed]
Alhat, D. , Lasrado, V. , and Wang, Y. , 2008, “ A Review of Recent Phase Transition Simulation Methods: Saddle Point Search,” ASME Paper No. DETC2008-49411.
Lasrado, V. , Alhat, D. , and Wang, Y. , 2008, “ A Review of Recent Phase Transition Simulation Methods: Transition Path Search,” ASME Paper No. DETC2008-49410.
Jónsson, H. , Mills, G. , and Jacobsen, K. W. , 1998, “ Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions,” Classical and Quantum Dynamics in Condensed Phase Simulations, Vol. 385, World Scientific, Singapore.
Henkelman, G. , and Jónsson, H. , 2000, “ Improved Tangent Estimate in the Nudged Elastic Band Method for Finding Minimum Energy Paths and Saddle Points,” J. Chem. Phys., 113(22), pp. 9978–9985. [CrossRef]
Ionova, I. V. , and Carter, E. A. , 1993, “ Ridge Method for Finding Saddle Points on Potential Energy Surfaces,” J. Chem. Phys., 98(8), pp. 6377–6386. [CrossRef]
Dewar, M. J. , Healy, E. F. , and Stewart, J. J. , 1984, “ Location of Transition States in Reaction Mechanisms,” J. Chem. Soc., Faraday Trans. 2, 80(3), pp. 227–233. http://pubs.rsc.org/-/content/articlelanding/1984/f2/f29848000227/unauth#!divAbstract
Henkelman, G. , and Jónsson, H. , 1999, “ A Dimer Method for Finding Saddle Points on High Dimensional Potential Surfaces Using Only First Derivatives,” J. Chem. Phys., 111(15), pp. 7010–7022. [CrossRef]
Simons, J. , Joergensen, P. , Taylor, H. , and Ozment, J. , 1983, “ Walking on Potential Energy Surfaces,” J. Phys. Chem., 87(15), pp. 2745–2753. [CrossRef]
Perdew, J. P. , Ruzsinszky, A. , Constantin, L. A. , Sun, J. , and Csonka, G. I. , 2009, “ Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed,” J. Chem. Theory Comput., 5(4), pp. 902–908. [CrossRef] [PubMed]
Queipo, N. V. , Haftka, R. T. , Shyy, W. , Goel, T. , Vaidyanathan, R. , and Tucker, P. K. , 2005, “ Surrogate-Based Analysis and Optimization,” Prog. Aerosp. Sci., 41(1), pp. 1–28. [CrossRef]
Clarke, S. M. , Griebsch, J. H. , and Simpson, T. W. , 2005, “ Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses,” ASME J. Mech. Des., 127(6), pp. 1077–1087. [CrossRef]
Levin, D. , 1998, “ The Approximation Power of Moving Least-Squares,” Math. Comput. Am. Math. Soc., 67(224), pp. 1517–1531. [CrossRef]
Kim, C. , Wang, S. , and Choi, K. K. , 2005, “ Efficient Response Surface Modeling by Using Moving Least-Squares Method and Sensitivity,” AIAA J., 43(11), pp. 2404–2411. [CrossRef]
Keane, A. , 2004, “ Design Search and Optimisation Using Radial Basis Functions With Regression Capabilities,” Adaptive Computing in Design and Manufacture VI, Springer, London, pp. 39–49. [CrossRef]
Haykin, S. , 2004, Neural Networks: A Comprehensive Foundation, 2nd ed., Prentice Hall, Upper Saddle River, NJ.
Sacks, J. , Welch, W. J. , Mitchell, T. J. , and Wynn, H. P. , 1989, “ Design and Analysis of Computer Experiments,” Statist. Sci., 4(4), pp. 409–423. https://projecteuclid.org/euclid.ss/1177012413
Lu, G. Y. , and Wong, D. W. , 2008, “ An Adaptive Inverse-Distance Weighting Spatial Interpolation Technique,” Comput. Geosci., 34(9), pp. 1044–1055. [CrossRef]
Cressie, N. , and Johannesson, G. , 2008, “ Fixed Rank Kriging for Very Large Spatial Data Sets,” J. R. Stat. Soc. Ser. B, 70(1), pp. 209–226. [CrossRef]
Sakata, S. , Ashida, F. , and Zako, M. , 2004, “ An Efficient Algorithm for Kriging Approximation and Optimization With Large-Scale Sampling Data,” Comput. Methods Appl. Mech. Eng., 193(3), pp. 385–404. [CrossRef]
Myers, D. E. , 1982, “ Matrix Formulation of Co-Kriging,” J. Int. Assoc. Math. Geol., 14(3), pp. 249–257. [CrossRef]
Forrester, A. I. , Sóbester, A. , and Keane, A. J. , 2007, “ Multi-Fidelity Optimization Via Surrogate Modelling,” Proc. R. Soc. London A, 463(2088), pp. 3251–3269. [CrossRef]
van Stein, B. , Wang, H. , Kowalczyk, W. , Bäck, T. , and Emmerich, M. , 2015, “ Optimally Weighted Cluster Kriging for Big Data Regression,” International Symposium on Intelligent Data Analysis (IDA), Saint-Etienne, France, Oct. 22–24, pp. 310–321.
Nguyen-Tuong, D. , Seeger, M. , and Peters, J. , 2009, “ Model Learning With Local Gaussian Process Regression,” Adv. Rob., 23(15), pp. 2015–2034. [CrossRef]
Lee, I. , Choi, K. , Du, L. , and Gorsich, D. , 2008, “ Dimension Reduction Method for Reliability-Based Robust Design Optimization,” Comput. Struct., 86(13), pp. 1550–1562. [CrossRef]
Huang, B. , and Du, X. , 2008, “ Probabilistic Uncertainty Analysis by Mean-Value First Order Saddlepoint Approximation,” Reliab. Eng. Syst. Saf., 93(2), pp. 325–336. [CrossRef]
Wang, Z. , and Wang, P. , 2014, “ A Maximum Confidence Enhancement Based Sequential Sampling Scheme for Simulation-Based Design,” ASME J. Mech. Des., 136(2), p. 021006. [CrossRef]
He, L. , and Wang, Y. , 2013, “ A Concurrent Search Algorithm for Multiple Phase Transition Pathways,” ASME Paper No. DETC2013-12362.
He, L. , and Wang, Y. , 2015, “ A Curve Swarm Algorithm for Global Search of State Transition Paths,” Third World Congress on Integrated Computational Materials Engineering (ICME), Colorado Springs, CO, May 31–June 4, pp. 139–146.
He, L. , 2015, “ Multiple Phase Transition Path and Saddle Point Search in Computer Aided Nano Design,” Ph.D. dissertation, Georgia Institute of Technology, Atlanta, GA. https://smartech.gatech.edu/handle/1853/53967
Hardy, G. H. , Littlewood, J. E. , and Pólya, G. , 1952, Inequalities, Cambridge University Press, Cambridge, UK.
Fasshauer, G. E. , and Zhang, J. G. , 2007, “ On Choosing ‘Optimal’ Shape Parameters for RBF Approximation,” Numer. Algorithms, 45(1–4), pp. 345–368. [CrossRef]
Ankenman, B. , Nelson, B. L. , and Staum, J. , 2010, “ Stochastic Kriging for Simulation Metamodeling,” Oper. Res., 58(2), pp. 371–382. [CrossRef]
Ba, S. , and Joseph, V. R. , 2012, “ Composite Gaussian Process Models for Emulating Expensive Functions,” Ann. Appl. Stat., 6(4), pp. 1838–1860. [CrossRef]

Figures

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.

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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

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Fig. 5

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

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Fig. 7

Illustration for two possible transition paths between two states

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