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).


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.