4,963 research outputs found
Pipelining the Fast Multipole Method over a Runtime System
Fast Multipole Methods (FMM) are a fundamental operation for the simulation
of many physical problems. The high performance design of such methods usually
requires to carefully tune the algorithm for both the targeted physics and the
hardware. In this paper, we propose a new approach that achieves high
performance across architectures. Our method consists of expressing the FMM
algorithm as a task flow and employing a state-of-the-art runtime system,
StarPU, in order to process the tasks on the different processing units. We
carefully design the task flow, the mathematical operators, their Central
Processing Unit (CPU) and Graphics Processing Unit (GPU) implementations, as
well as scheduling schemes. We compute potentials and forces of 200 million
particles in 48.7 seconds on a homogeneous 160 cores SGI Altix UV 100 and of 38
million particles in 13.34 seconds on a heterogeneous 12 cores Intel Nehalem
processor enhanced with 3 Nvidia M2090 Fermi GPUs.Comment: No. RR-7981 (2012
2HOT: An Improved Parallel Hashed Oct-Tree N-Body Algorithm for Cosmological Simulation
We report on improvements made over the past two decades to our adaptive
treecode N-body method (HOT). A mathematical and computational approach to the
cosmological N-body problem is described, with performance and scalability
measured up to 256k () processors. We present error analysis and
scientific application results from a series of more than ten 69 billion
() particle cosmological simulations, accounting for
floating point operations. These results include the first simulations using
the new constraints on the standard model of cosmology from the Planck
satellite. Our simulations set a new standard for accuracy and scientific
throughput, while meeting or exceeding the computational efficiency of the
latest generation of hybrid TreePM N-body methods.Comment: 12 pages, 8 figures, 77 references; To appear in Proceedings of SC
'1
Geometry-Oblivious FMM for Compressing Dense SPD Matrices
We present GOFMM (geometry-oblivious FMM), a novel method that creates a
hierarchical low-rank approximation, "compression," of an arbitrary dense
symmetric positive definite (SPD) matrix. For many applications, GOFMM enables
an approximate matrix-vector multiplication in or even time,
where is the matrix size. Compression requires storage and work.
In general, our scheme belongs to the family of hierarchical matrix
approximation methods. In particular, it generalizes the fast multipole method
(FMM) to a purely algebraic setting by only requiring the ability to sample
matrix entries. Neither geometric information (i.e., point coordinates) nor
knowledge of how the matrix entries have been generated is required, thus the
term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme
for hierarchical matrix computations that reduces synchronization barriers. We
present results on the Intel Knights Landing and Haswell architectures, and on
the NVIDIA Pascal architecture for a variety of matrices.Comment: 13 pages, accepted by SC'1
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