Automatic Tuning of the Fast Multipole Method Based on Integrated Performance Prediction

The Fast Multipole Method (FMM) is an efficient, widely used method for the solution of N-body problems. One of the main data structures is a hierarchical tree data structure describing the separation into near-field and far-field particle interactions. This article presents a method for automatic tuning of the FMM by selecting the optimal FMM tree depth based on an integrated performance prediction of the FMM computations. The prediction method exploits benchmarking of significant parts of the FMM implementation to adapt the tuning to the specific hardware system being used. Furthermore, a separate analysis phase at runtime is used to predict the computational load caused by the specific particle system to be computed. The tuning method was integrated into an FMM implementation. Performance results show that a reliable determination of the tree depth is achieved, thus leading to minimal execution times of the FMM algorithm

Corrected article: An error-controlled fast multipole method

We present a two-stage error estimation scheme for the fast multipole method (FMM). This scheme can be applied to any particle system. It incorporates homogeneous as well as inhomogeneous distributions. The FMM error as a consequence of the finite representation of the multipole expansions and the operator error is correlated with an absolute or relative user-requested energy threshold. Such a reliable error control is the basis for making reliable simulations in computational physics. Our FMM program on the basis of the two-stage error estimation scheme is available on request

GROMEX: A Scalable and Versatile Fast Multipole Method for Biomolecular Simulation

Atomistic simulations of large biomolecular systems with chemical variability such as constant pH dynamic protonation offer multiple challenges in high performance computing. One of them is the correct treatment of the involved electrostatics in an efficient and highly scalable way. Here we review and assess two of the main building blocks that will permit such simulations: (1) An electrostatics library based on the Fast Multipole Method (FMM) that treats local alternative charge distributions with minimal overhead, and (2) A λ-dynamics module working in tandem with the FMM that enables various types of chemical transitions during the simulation. Our λ-dynamics and FMM implementations do not rely on third-party libraries but are exclusively using C++ language features and they are tailored to the specific requirements of molecular dynamics simulation suites such as GROMACS. The FMM library supports fractional tree depths and allows for rigorous error control and automatic performance optimization at runtime. Near-optimal performance is achieved on various SIMD architectures and on GPUs using CUDA. For exascale systems, we expect our approach to outperform current implementations based on Particle Mesh Ewald (PME) electrostatics, because FMM avoids the communication bottlenecks caused by the parallel fast Fourier transformations needed for PME

Comparison of scalable fast methods for long-range interactions

Based on a parallel scalable library for Coulomb interactions in particle systems, a comparison between the fast multipole method (FMM), multigrid-based methods, fast Fourier transform (FFT)-based methods, and a Maxwell solver is provided for the case of three-dimensional periodic boundary conditions. These methods are directly compared with respect to complexity, scalability, performance, and accuracy. To ensure comparable conditions for all methods and to cover typical applications, we tested all methods on the same set of computers using identical benchmark systems. Our findings suggest that, depending on system size and desired accuracy, the FMM- and FFT-based methods are most efficient in performance and stability