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

Abstract

Abstract-The advent of general purpose graphics processing units (GPGPU's) brings about a whole new platform for running 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 because 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 consequence 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.

Accelerating Molecular Dynamics Simulation Using Graphics Processing Unit

“…Therefore we have concentrated on two subroutines responsible for those calculations. We used a general purpose MD simulation code mm_par 18 for GPU implementation using CUDA. The subroutines are v_real() and v_pme().…”

“…Recently several studies have reported GPU acceleration of MD simulation. [12][13][14][15][16][17][18][19] However, most studies considered only relatively simple systems like a system interacting only through Lennard-Jones interaction, which are lack of electrostatic interactions. The purpose of this paper is to implement a general purpose MD simulation code 20 for GPU using NVIDIA's CUDA 21 (Compute Unified Device Architecture).…”

First steps towards more numerical reproducibility

“…Questions whether such simulated results are reproducible or not have been reported more or less recently, e.g. in energy science [1], dynamic weather forecasting [2], atomic or molecular dynamic [3,4], fluid dynamic [5]. This paper focuses on numerical non-reproducibility due to the finite precision of computer arithmetic -see [6] for other issues about "reproducible research" in computational mathematics.…”

Estimation of numerical reproducibility on CPU and GPU

“…Such differences are particularly noticeable with new computing architectures such as multicore processors, GPUs (Graphics Processing Units) and APUs (Accelerated Processing Units). In high performance numerical simulations, reproducibility problems have been identified in various domains: energy science [1], climate science [2], atomic or molecular dynamics [3], [4], fluid dynamics [5]. Various studies have been carried out on numerical reproducibility on different architectures.…”

“…Various studies have been carried out on numerical reproducibility on different architectures. On the one hand, strategies have been proposed [2], [3], [4], [5] to improve numerical accuracy, using for instance accurate summations. Other works aim at forcing the reproducibility of results, either affected by the same rounding errors [6], [7] or correctly rounded [8], [9], [10], [11].…”