A Hessian-free Newton–Raphson method for the configuration of physics systems featured by numerically asymmetric force field

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.