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A powerful truncated Newton method for potential energy minimization
Author(s) -
Schlick Tamar,
Overton Michael
Publication year - 1987
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.540080711
Subject(s) - conjugate gradient method , newton's method , maxima and minima , newton's method in optimization , local convergence , quasi newton method , gradient descent , relaxation (psychology) , mathematics , convergence (economics) , nonlinear system , conjugate residual method , nonlinear conjugate gradient method , rate of convergence , minification , iterative method , computation , mathematical optimization , steffensen's method , computer science , algorithm , mathematical analysis , key (lock) , physics , artificial neural network , quantum mechanics , computer security , economic growth , psychology , social psychology , machine learning , economics
With advances in computer architecture and software, Newton methods are becoming not only feasible for large‐scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second‐derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.