Analytic Integration of the Newton Potential over Cuboids and an Application to Fast Multipole Methods

We present simplified formulæ for the analytic integration of the Newton potential of polynomials over boxes in two- and three-dimensional space. These are implemented in an easy-to-use C++ library that allows computations in arbitrary precision arithmetic which is also documented here. We describe how these results can be combined with fast multipole methods to evaluate the Newton potential of more general, non-polynomial densities.

Dual tree traversal on integrated GPUs for astrophysical N-body simulations

In astrophysical N-body simulations, O( N) fast multipole methods (FMMs) with dual tree traversal (DTT) on multi-core CPUs are faster than O( N log N) CPU tree-codes but can still be outperformed by GPU ones. In this article, we aim at combining the best algorithm, namely FMM with DTT, with the most powerful hardware currently available, namely GPUs. In the astrophysical context requiring low accuracies and non-uniform particle distributions, we show that such combination can be achieved thanks to a hybrid CPU-GPU algorithm on integrated GPUs: while the DTT is performed on the CPU cores, the far- and near-field computations are all performed on the GPU cores. We show how to efficiently expose the interactions resulting from the DTT to the GPU cores, how to deploy both the far- and near-field computations on GPU, and how to overlap the parallel DTT on CPU with GPU computations. Based on the falcON code and using OpenCL on AMD Accelerated Processing Units and on Intel integrated GPUs, this first heterogeneous deployment of DTT for FMM outperforms standard multi-core CPUs and matches GPU and high-end CPU performance, being hence more cost- and power-efficient.