Data Compression for Optimization of a Molecular Dynamics System: Preserving Basins of Attraction

Understanding the evolution of atomistic systems is essential in various fields such as materials science, biology, and chemistry. The gold standard for these calculations is molecular dynamics, which simulates the dynamical interaction between pairs of molecules. The main challenge of such simulation is the numerical complexity, given a vast number of atoms over a long time scale. Furthermore, such systems often contain exponentially many optimal states, and the simulation tends to get trapped in local configurations. Recent developments leverage the existing temporal evolution of the system to improve the stability and scalability of the method; however, they suffer from large data storage requirements. To efficiently compress the data while retaining the basins of attraction, we have developed a framework to determine the acceptable level of compression for an optimization method by application of a Kantorovich-type theorem, using binary digit rounding as our compression technique. Choosing the Lennard-Jones potential function as a model problem, we present a method for determining the local Lipschitz constant of the Hessian with low computational cost, thus allowing the use of our technique in real-time computation.

[1]  Tetsuro Yamamoto,et al.  Historical developments in convergence analysis for Newton's and Newton-like methods , 2000 .

[2]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[3]  L. Wille,et al.  Computational complexity of the ground-state determination of atomic clusters , 1985 .

[4]  Janet E. Jones.  On the determination of molecular fields. —II. From the equation of state of a gas , 1924 .

[5]  Aneesur Rahman,et al.  Correlations in the Motion of Atoms in Liquid Argon , 1964 .

[6]  Xueguang Shao,et al.  Structural Distribution of Lennard-Jones Clusters Containing 562 to 1000 Atoms. , 2004, The journal of physical chemistry. A.

[7]  A. Voter.  Parallel replica method for dynamics of infrequent events , 1998 .

[8]  B. Alder,et al.  Studies in Molecular Dynamics. I. General Method , 1959 .

[9]  A. Voter.  Hyperdynamics: Accelerated Molecular Dynamics of Infrequent Events , 1997 .

[10]  Florian A. Potra,et al.  A Superquadratic Variant of Newton's Method , 2017, SIAM J. Numer. Anal..

[11]  C. Floudas,et al.  A global optimization approach for Lennard‐Jones microclusters , 1992 .

[12]  A. Voter,et al.  Temperature-accelerated dynamics for simulation of infrequent events , 2000 .

[13]  M. Hoare,et al.  Physical cluster mechanics: Statics and energy surfaces for monatomic systems , 1971 .

[14]  B. P. Zhang,et al.  Estimation of the Lipschitz constant of a function , 1996, J. Glob. Optim..

[15]  M. Hoare.  Structure and Dynamics of Simple Microclusters , 2007 .

[16]  J. Northby.  Structure and binding of Lennard‐Jones clusters: 13≤N≤147 , 1987 .

[17]  Xinghua Wang,et al.  Convergence of Newton's method and uniqueness of the solution of equations in Banach space , 2000 .

[18]  Danny Perez,et al.  Long-Time Dynamics through Parallel Trajectory Splicing. , 2016, Journal of chemical theory and computation.

[19]  L. Kantorovich,et al.  Functional analysis and applied mathematics , 1963 .

[20]  D. Wales,et al.  Packing schemes for Lennard-Jones clusters of 13 to 150 atoms: minima, transition states and rearrangement mechanisms , 1991 .

[21]  Daan Frenkel,et al.  Visualizing basins of attraction for different minimization algorithms. , 2013, The journal of physical chemistry. B.

[22]  Santhosh George,et al.  Ball convergence comparison between three iterative methods in Banach space under hypothese only on the first derivative , 2015, Appl. Math. Comput..

[23]  Henryk Wozniakowski,et al.  Convergence and Complexity of Newton Iteration for Operator Equations , 1979, JACM.