A Hessian-free Newton-Raphson method for the configuration of physics systems featured by numerically asymmetric force field
Y. Liang, Z. Shi and W. P. Chung
Mathematics and Computers in Simulation, 140, 1-23 (2017).

ABSTRACT
Numerically asymmetric force field is examined in particle-oriented problems
such as Quasicontinuum modeling and simulation. To configure the ground state of
a large-scale physical system featured by numerically asymmetric force field, we
propose a Hessian-free Newton-Raphson method where the Newton equation is solved
using central-difference based BiCGstab algorithm (denoted as HFNR-BiCGstab-diff
for simplicity). A detailed analytical and experimental investigation on the
convergence performance of the HFNR-BiCGstab-diff algorithm is given in this
paper. A pure HFNR-BiCGstab-diff algorithm may suffer from unreliable start-up,
particularly in the case that the initial guess is far from the minimizer. As a
remedy, a hybrid method that couples HFNR-BiCGstab-diff with preconditioned
nonlinear conjugate gradient algorithm (PNCG) is developed to achieve optimal
computational performance. The algorithms addressed in this paper have been
implemented using POSIX-Thread C++. Their performance has been evaluated using
three-dimensional Quasicontinuum simulation problems, which are featured by
asymmetric force field and large dimensional sizes up to 122,808
degree-of-freedom, as benchmarks. The numerical experiment on IBM SP2
demonstrates that, compared to alternative unconstrained optimization methods
such as preconditioned nonlinear conjugate gradient algorithm, the hybrid
algorithm saves 20%-60% running time.