Improving numerical reproducibility and stability in large-scale numerical simulations on GPUs

Abstract—The advent of general purpose graphics processing units (GPGPU’s) brings about a whole new platform for run-ning numerically intensive applications at high speeds. Their multi-core architectures enable large degrees of parallelism via a massively multi-threaded environment. Molecular dynamics (MD) simulations are particularly well-suited for GPU’s be-cause their computations are easily parallelizable. Significant performance improvements are observed when single precision floating-point arithmetic is used. However, this performance comes at the cost of accuracy: it is widely acknowledged that constant-energy (NVE) MD simulations accumulate errors as the simulation proceeds due to the inherent errors associated with integrators used for propagating the coordinates. A con-sequence of this numerical integration is the drift of potential energy as the simulation proceeds. Double precision arithmetic partially corrects this drifting, but is significantly slower than single precision, comparable to CPU performance. To address this problem, we extend the approaches of previous literature to improve numerical reproducibility and stability in MD simulations, while assuring efficiency and performance comparable to that when using the GPU hardware implementation of single precision arithmetic. We present development of a library of mathematical functions that use fast and efficient algorithms to fix the error produced by the equivalent operations performed by GPU. We successfully validate the library with a suite of synthetic codes emulating the MD behavior on GPUs. Keywords-Parallel programming; GPU programming; Molecular Dynamics; Floating-point arithmetic. I